JP3632093B2 - Projector using central projection method - Google Patents

Description

【００４７】
で与えられる。
【００４８】
したがって、ｚ軸とのなす角度ωは、
【００４９】
【数２】

【００５０】
で与えられる。
【００５１】
また、直線ＯＳの、ｘ軸からの方位角αは、
α＝ｔａｎ^−１（ｙ_０／ｘ_０） ………（３）
で与えられる。
【００５２】
一方、回転体２１に関し、図７において、直線ＯＱと直線ＯＮを含む断面上で考えると、図８のようになる。いま、図８において、回転体２１の元となる曲線を ｆ（ｚ）とする。原点Ｏを通り、ｚ軸からの角度φの直線は、
ｘ＝ｚｔａｎφ ………（４）
で与えられる。
【００５３】
両者の交点Ｑのｚ軸上の座標は、
ｆ（ｚＱ）＝ｚＱｔａｎφ ………（５）
で与えられる。
【００５４】
次に、曲線ｆ（ｚ）の頂点（回転体２１の頂点Ｎ）から点Ｑまでの曲線の長さをＬとすると、Ｌは、
【００５５】
【数３】

【００５６】
の線積分で求まる。（６）式でｆ´（ｚ）は曲線ｆの一次微分である。
【００５７】
一例として、半径Ｒの球の場合は、
【００５８】
【数４】

【００５９】
のようになる。
【００６０】
したがって、図６において、一次投影曲面２１ａが球の場合の曲線ＮＱの長さＬは 、断層像２３上の座標（ｘ_０，ｙ_０，ｚ_０）を用いて、
【００６１】
【数５】

【００６２】
で与えられる。
【００６３】
断層像２３上の各点について（７）式で計算することにより、球の曲面（球面）への一次投影像の位置が得られる。すなわち、一次投影像の球面での座標は（α，Ｌ）により全て与えられる。
【００６４】
また、原点Ｏから断層像２３上の点Ｓ（ｘ_０，ｙ_０，ｚ_０）までの距離ｒｓは、
【００６５】
【数６】

【００６６】
で与えられる。この距離は、陰影付け時に利用される。
【００６７】
このようにして、断層像２３上の対応する点を全て回転体曲面２１ａへ投影し、これらの点を二次投影平面４への再投影（二次投影）に用いる。
なお、図７の上側の図において、曲線４０は回転体曲面２１ａの回転軸を通る断面であり、その他は図６と同様である。
【００６８】
図９は、二次投影平面４への回転体曲面／平面座標変換の一例を示す。図９においては、原点Ｎを中心として半径方向の距離をＬ、原点Ｎより定めた基準直線ｗからの角度（方位角）をαで表し、極座標を使用している。これにより、回転体曲面２１ａへの一次投影像は原点Ｎからの距離Ｌと角度αで表される。これをＸ，Ｙ座標に再投影し、モニタに表示すればよい。
【００６９】
図９において、矩形領域５０はモニタ表示の領域の例であり、この領域５０はモニタ（図示せず）の表示メモリの数、例えば５１２×５１２画素あるいは１０２４×１０２４画素に対応させる。図示複数の同心円のうちどれだけの円を含む矩形領域５０とし、表示するかは任意に決めることができる。
【００７０】
モニタ上のＸ，Ｙ座標はＬを用いて、
Ｘ＝ξ_１＝Ｌ´ｃｏｓα ………（９）
Ｙ＝η_１ ＝Ｌ´ｓｉｎα ………（１０）
により与えられる。
【００７１】
この、η_１，ξ_１が表示メモリ上の画素アドレスとなる。すなわち、一次投影曲面２１ａの点Ｑはモニタ画面上では点（η_１，ξ_１）に表示することになるもので、これにより回転体曲面／平面座標変換による表示が行われたことになる。
【００７２】
図１０は、二次投影平面４への回転体曲面／平面座標変換の他の例を示す。ここでは、二次投影平面４上に再投影するとき、回転体曲面２１ａ上の像をそのまま二次投影平面４に平行移動するように、ｚ軸に平行に投影する方法を示すもので、これによると、回転体２１として球を用いた場合、図１１に示すように、モニタ上では画像の中心部分が拡大される効果をもつ。なお、図１１において図９と同一符号は同一又は相当部分を示す。
【００７３】
図１０の回転体曲面２１ａ上の一次投影像の各点は、回転軸（ｚ軸）から回転体曲面２１ａまでの距離ＯＱを用いて二次投影平面４へ変換する。二次投影平面４への変換式は、
Ｌ´＝ＯＱ・ｓｉｎω ………（１２）
で与える。
ここでωは、図６に示すように、視点ｅと投影対象を結ぶ直線とｚ軸とのなす角である。また、方位角はそのまま用いる。
【００７４】
陰影付けは、回転体曲面（一次投影曲面）２１ａ上で行う方法と、二次投影平面４上で行う方法とがある。計算の基礎となる視点ｅから断層像２３上の各点までの距離は既知のため、どちらの方法においても可能である。
【００７５】
この陰影付けには、原点Ｏから投影対象の各点への距離に応じて濃淡を付ける方法や、各投影直線（光線）と投影対象面とのなす角度に応じて輝度変換する方法や、投影中心線と投影対象面とのなす角度に応じて輝度変換する方法、更には原点Ｏからの距離に応じて等高線のように輝度変換を行う方法等が用いられる。
また、視点ｅからの光線に対して、投影対象の組織の透過率を用いるボリュームレンダリング処理を行う方法を用いてもよい。
【００７６】
図１２〜図２１は、複数の断層像２３（２３Ａ〜２３Ｈ）から上述投影方法を適用して三次元画像を構成する方法の一例を説明するための図である。図１３〜図１４，図１５〜図２１のいずれの場合においても、最初に回転体曲面２１ａ上に断層像２３を一次投影し、次に回転体曲面／平面座標変換により二次投影平面４へ再投影（二次投影）することになる。
【００７７】
まず、三次元画像の構成方法における回転体曲面／平面座標変換について述べる。
図１２に示す例では、説明を簡易化するため回転体曲面２１ａの大円２０と断層像２３面、更には二次投影平面４が各々平行であるように座標系をとっている。また、回転体曲面２１ａの中心線をｘ軸と平行にとる。
【００７８】
この図１２において、視点ｅの位置は、三次元座標系（ｘ，ｙ，ｚ）の原点Ｏである。
また、
２１は回転体
ｘ，ｙ，ｚは三次元座標系（ｘ，ｙ，ｚ）の各軸、
Ｑは、断層像２３上のＳ点に対応する回転体曲面２１ａ上の点、
Ｓ´は視点ｅとＱ点を結ぶ直線２２と断層像２３Ａの交わる点、
Ｎは回転体２１の頂点、
２０はｘ−ｙ平面（ｚ＝_０）、
４は二次投影平面、
ｗはｘ−ｙ平面上での回転角の基準直線（基準軸）、
ｄ_０は断層像２３Ａのｚ軸上の距離、
である。
【００７９】
いま、視点ｅを通り、二次投影平面４に直交する平面（直交平面）を考える。図１２における断層像２３Ａ〜２３Ｃの他にも断層像２３Ｄ〜２３Ｈを用意しての、上記直交平面上の図を図１３に示す。
【００８０】
この図１３において、断層像２３Ａ〜２３Ｈは同一投影対象（ないし対象物）について同一方向に等間隔で得られた断層像であり、断層像２３には臓器領域Ｂ１，Ｂ２，Ｂ３，Ｂ４が強調して書いてある。なお、断層像２３Ａ〜２３Ｈは必ずしも等間隔である必要はない。
【００８１】
上記臓器領域Ｂ１，Ｂ２，Ｂ３，Ｂ４を、まず一次投影曲面である回転体曲面２１ａに投影し、更にそれを回転体曲面／平面座標変換して二次投影平面４に投影すると、二次投影データＢ１´，Ｂ２´，Ｂ３´，Ｂ４´となる。
【００８２】
この二次投影データＢ１´，Ｂ２´，Ｂ３´，Ｂ４´を表示メモリ（図示せず）に書き込むときは、三次元的効果を出すために、視点ｅからみてより遠くに存在する投影データを先に書き込み、それより近くの投影データは後から上書きする。その後、陰影付け処理を行う。
【００８３】
図１４は図１３よりも一般化して示したもので、回転体２１の頂点Ｎの方向が断層像２３面と直交せず、ｙ軸を中心として任意の角度だけ回転した場合の例である。この場合は、補間演算により求めた回転体曲面２１ａへの一次投影像を、二次投影平面４へ再投影（二次投影）することになる。
【００８４】
図１５は、視点ｅ、断層像２３及び一次投影曲面２１ａがより複雑な位置関係をもった場合の回転体曲面／平面座標変換を説明するための図で、断層像２３上のＳｉ（ｘｉ，ｙｉ，ｄｉ）の一次投影像が最終的に二次投影平面４上の点Ｐｉ（Ｘ，Ｙ）になることを示す。
【００８５】
図１５において、本発明投影方法による断層像２３の回転体曲面２１ａへの投影に当たって、断層像２３の各画素座標の回転体曲面２１ａ上の座標への変換は次のように行われる。
【００８６】
ここで、説明に一般性をもたせるため次のように位置関係を設定する。
図１５において、投影対象である断層像２３はｚ軸に直交する面内にあり、原点Ｏからｄ_０の距離にあるものとする。複数の断層像２３…に対しては原点Ｏからの距離ｄ_０が変わる。ただし、複数の断層像２３…が等間隔である必要はない。
【００８７】
また、原点Ｏ上の視点ｅから、ｚ軸とは異なる任意の投影方向の中心Ｎを決め、その方向をＯＮとし、直線ＯＮをｈとする。そして、原点Ｏから距離Ｒの点を起点とする任意の線ｔを引く。この線ｔは、曲線であっても、あるいは直線であってもよく、直線ｈと平行な直線であってもよい。
【００８８】
次にこの線ｔを、直線ｈを回転軸として回転させた回転体２１を想定する。そして、この回転体２１の曲面２１ａに断層像（投影対象）２３を投影する。一方、回転体曲面２１ａとは別に二次投影平面４を与え、その平面４上に直交座標系Ｘ，Ｙを決める。この平面４上の各点（Ｘ，Ｙ）がモニタ上の各表示座標となる。
【００８９】
図１６は上記回転体曲面２１ａ上の座標関係を示す。いま、図１５における座標系において、原点Ｏを中心とする単位球Ｃを考える。そして、ｚ軸の方向をＯＮ_１とし、このＯＮ_１を基準の方向とする。また、図１５における座標系に対して、ＯＮ_１の向きを任意に変えた新しい方向をＯＮ_２とする。なお、このＯＮ_２の方向を任意に変えることにより、その方向を中心とした三次元画像を表示可能である。
【００９０】
次に、原点Ｏと単位球Ｃ上の点ｑを結ぶ直線を延長し、その直線と回転体曲面２１ａとの交点をＱとする。回転体２１の性質から、その頂点Ｎから点Ｑまでの表面上の距離Ｌは、点Ｎ，Ｏ，Ｑを含む平面ｖ内で回転体２１の元となる線ｔを用いて（６）式で示す積分により求めることができる。
【００９１】
また図１６において、直線ＯＮ_２に対し、原点Ｏを通りＯＮ_２に直交する単位球Ｃの中心を通る平面（大円）を２０´、ＯＮ_２のｘ軸からの回転角をα_０、ｚ軸となす角をωとする。言い換えると、直線ＯＮ_１をｚ軸からω、ｘ軸からα_０だけ回転させた直線がＯＮ_２である。ここで、ｘ軸をｘ−ｙ平面内で角度α_０だけ回転させた直線を新しい方位角の基準軸Ｏｘ´とする。
【００９２】
点ｑ_１，ｑ_２は単位球Ｃ上の点であり、直線ＯＮ_２と平面２０´に対して基準線Ｏｘ´から測った方位角λ、平面２０´からの角度βで表される。同様に、これらの点は元の座標系でｘ軸からの方位角α、平面２０からの角度δで表わされる。直線ＯＮ_２を回転軸とした一次投影曲面２１ａへの投影像は、λ，βを等間隔に変えて計算すればよい。
【００９３】
いま、点ｑ_１の元の座標系での角度を各々α，δとし、平面２０´の平面２０からの傾きをεとする。そして、直線ＯＮ_２とｚ軸とのなす角をω、直線Ｏｘ´のｘ−ｙ平面内での回転角をα_０、また、Ｑ点のｘ−ｙ平面内での基準軸Ｏｘ´からの角度をα´とする。更に、平面２０´上での基準軸Ｏｘ´と直線ＯＮを元とした新しい座標系において、点Ｑ_１の角度を各々λ，βとする。
【００９４】
このようにすると、ε＝ωであり、球面三角形の性質からλ，βとα，δ，ωの間には、次の
ｃｏｓβｃｏｓλ＝ｃｏｓδｃｏｓα´
ｃｏｓβｓｉｎλ＝ｓｉｎδｓｉｎω＋ｃｏｓδｓｉｎα´ｃｏｓω ………（１３）
ｓｉｎβ ＝ｓｉｎδｃｏｓω−ｃｏｓδｓｉｎα´ｓｉｎω
の関係式が成り立つ。
【００９５】
変換後のλ，βを指定して逆に元のｘ，ｙ，ｚ座標系のα，δを求める場合は、λ，βとα，δを逆にした
ｃｏｓδｃｏｓα´＝ｃｏｓβｃｏｓλ
ｃｏｓδｓｉｎα´＝−ｓｉｎβｓｉｎω＋ｃｏｓβｓｉｎλｃｏｓω ………（１４）
ｓｉｎδ ＝ｓｉｎβｃｏｓω＋ｃｏｓβｓｉｎλｓｉｎω
を用いればよい。
互いに該当する位置に座標が一致しなければ周辺の点から内挿計算により求めればよい。
【００９６】
また、図１６から、元の座標系と新しい座標系での回転角αの間には、
α＝α´＋α_０ ………（１５）
の関係式が成り立つ。
【００９７】
すなわち、球面上の任意の点の回転前の座標と回転後の座標は（１３）式〜（１５）式により関係付けられる。これらの式により回転後の座標は回転前の座標系に直して容易に計算可能である。また、その逆も可能である。
【００９８】
断層像上の各点Ｓに対して２つの角度α，δが与えられ、更に（１３）式を用いてβ，λが求まる。原点Ｏから前記の点Ｓを通る直線と回転体曲面２１ａとの交点Ｑは、直線ｈと点Ｓを含む断面（平面ｖ）内で考えればよいため、直線ＯＮを新しくｘ軸として考えた
ｆ（ｘ）＝ｘｔａｎφ ………（１６）
で求まる。
ここで、φは直線Ｏｚと、直線ＯＮ_２とのなす角である。
【００９９】
また、回転体２１の性質を用いて、回転体２１の頂点Ｎから点Ｑまでの表面距離Ｌ（＝ＮＱ）は、前記平面ｖ上で線積分により計算すればよい。この計算は、（６）式で行えばよい。
一方、二次投影平面４への変換における角度λは、（１３）式から求まる。
【０１００】
以上により求められたλ，Ｌを用いて図１７の二次投影平面４上の像が求まる。図１７は、極座標形式であるが、直交座標系への変換は、
ｘ＝η＝Ｌｃｏｓλ ………（１７）
ｙ＝ξ＝Ｌｓｉｎλ ………（１８）
で求まる。
そして、（１７），（１８）式のη，ξを用いてモニタへ投影像を表示すればよい。
【０１０１】
このようにして、頂点Ｎと原点Ｏを結ぶ直線を中心軸として角度λ，βを変えて行くことで断層像２３を走査して交点を求めることにより、一次投影像が回転体曲面２１ａ上に求まる。また、逆に断層像２３の各点を走査して回転体曲面２１ａへの一次投影像を求めてもよい。このようにして、一次投影曲面２１ａに得られた一次投影像を回転体曲面／平面座標変換により二次投影平面４に再投影（二次投影）し、表示メモリ（図示せず）に書き込むことによりモニタに二次投影像が表示される。
【０１０２】
図１８は、回転体曲面２１ａへの投影像を角度λ，βにより表示したものである。この２つの角度をＸ，Ｙ座標に割当ててモニタに表示する場合は１：１の変換表示となる。すなわち、これらのデータを用い、二次投影平面４へ変換することにより、視覚による像とは一部異なり周辺部が拡大された三次元画像が得られる。
【０１０３】
図１９〜図２１は、図１７における距離Ｌを一定の関数により変換してＬ´を求め、画像表示する例を示す。図１９では頂点Ｎからの距離のある範囲が拡大されて表示される。図１９〜図２１においてＬ´軸のｎはモニタ上での表示画素数に対応するものであり、最大画素数の１／２にすればよい。図２０では、係数ｋを変えることにより頂点Ｎからの距離の拡大率が変わり、中心部分が任意に拡大できるという特徴をもつ。また、図２１では注目する領域を拡大できるが、その前後が滑らかに変わり、見やすい画像となる。
【０１０４】
図１〜図２１を参照して以上に説明した試案においては、一次投影曲面として用いる回転体の種類を限定しなかったが、
以下に、回転体が球である場合、円錐体である場合、楕円回転体である場合、放物線回転体である場合、および円柱である場合のそれぞれを区分して詳しく述べる。
なお、回転体２１が球である場合を除き、一次投影曲面への投影方法のみを述べるが、二次投影平面４への再投影（二次投影）処理は、試案（図１〜図２１）におけると各々同様である。
【０１０５】
例１：回転体２１が球である場合
視点ｅ（原点Ｏ）を中心とした半径Ｒの円を回転させると図２２に示すような球となる。半円ｔの方程式は、
【０１０６】
【数７】

【０１０７】
で与えられる。
【０１０８】
半円ｔのｘ軸を回転軸とする回転体は球Ｃとなる。この場合は球の曲面、すなわち球面に一次投影することになる。原点Ｏからｘ軸上で距離ａの点を中心とする円を回転させてなる球の曲面（球面）への一次投影も可能である。
球面への一次投影は、視点ｅからの距離に関して等方性の強い臓器（対象物）に対し、広い範囲の三次元画像を一度に見たいときに適している。
【０１０９】
図２２における直線２０の方程式はｙ＝ｘｔａｎφであるから、ｘ軸上の座標ｘ_０はＲｃｏｓφとなる。
球Ｃ上の頂点Ｎから点Ｑまでの円周上の距離Ｌは、（６）式と（１９）式の一次微分とを用いて、
【０１１０】
【数８】

【０１１１】
から計算で与える。
【０１１２】
回転体が球の場合の一次投影と二次投影について、以下に詳しく説明する。図２３〜図２５は回転体が球の場合の実施例を示している。このうち図２３は全体の概要を示している。
【０１１３】
図２３において、まず投影の対象物である臓器２３ａに対して、任意の位置に視点ｅを設定する。
そして、この視点ｅを中心とした球２１を考える。球２１の半径は任意であるが、半径を決めることにより、どれだけの範囲の臓器２３ａを投影するかが決まる。半径を大きくすれば、広い範囲の臓器２３ａが投影可能であるが、モニタの表示領域は有限であり、臓器２３ａの微小部分は分からなくなるので、適宜の大きさに選定される。
【０１１４】
次に、視点ｅから放射状の直線（光線）を引き、その放射状直線の臓器２３ａの各点との交点を、その放射状直線の前記球２１の曲面（球面）２１ａとの交点に投影することで、球面２１ａを一次投影曲面とした一次投影（中心投影）を行う。
【０１１５】
また、視点ｅを通る任意の平面として大円２０を決め、その中央から垂線ｈを立て、その垂線ｈに対して直交する平面をＴとする。この平面Ｔは、前記球面２１ａへの一次投影像が、回転体曲面（ここでは球面）／平面座標変換により投影される二次投影平面４である。
この中心を通る平面Ｔ，大円２０の対の方向を変えることにより、種々の方向からの投影像が作成可能である。
【０１１６】
図２４は、垂線ｈを含む平面内における断層像２３と一次投影曲面２１ａ、二次投影平面４、臓器２３ａの投影対象５、視点ｅの間の関係を示している。一次投影曲面２１ａは球２１の曲面である。視点ｅは球２１の中心であり、また三次元座標の原点Ｏでもあり、ここでは投影の対象物である臓器２３ａの投影対象５を上記球面２１ａに投影する。
【０１１７】
この場合、原点Ｏを囲む球面２１ａは立体角４π（球）になっているが、この角度は任意である。球面２１ａに投影された一次投影像は、回転体曲面／平面座標変換により球面２１ａを二次投影平面４上へ展開することにより平面４に再投影（二次投影）する。
【０１１８】
図２５は、二次投影平面４の例であり、中心を頂点Ｎとして球面２１ａに投影された一次投影像は、頂点Ｎからの円弧を用いた距離ｒと任意に定めた基準直線ｗからの角度αとの２つのパラメータにより平面４に展開する。球２１は、完全な球の場合は原点Ｏを中心とした立体角４πの球となるが、半球（立体角２π）、あるいは任意の立体角の球でもよい。
【０１１９】
一方、球面２１ａから平面４への展開において、球面２１ａの立体角のうち、どこまでを平面４へ展開するかは任意である。図２５の例では立体角２πである。直感的な視覚による像と一致するのは立体角２π付近までである。２πを越えると二次投影平面４上で反対側の頂点が円周全体に展開され、視覚による像と一致しなくなるため、４πより小さい方が分かりやすい。平面４への展開の中心である頂点Ｎは、任意に与える。
【０１２０】
以下、図２６〜図３０を参照して一次投影像の球面２１ａから平面４への再投影（球面／平面座標変換）の実施例について説明する。
まず、図２６を参照して説明する。図２６において、２１ａは、三次元座標系の各軸ｘ，ｙ，ｚの原点Ｏ上にある視点ｅに中心をもち、半径Ｒの球２１の曲面である。更に、球２１の大円２０はｘ−ｙ平面（ｚ＝_０）に一致しているものとする。
【０１２１】
４は、ｚ軸に直交する平面からなる投影面で、球面２１ａへの一次投影後のデータ（一次投影像）を更に投影（再投影）する二次投影平面である。この二次投影平面４は、モニタの表示画面に相当する。
【０１２２】
また、
２３は断層像、
ｈは視点ｅから二次投影平面に引いた垂線
Ｑは視点ｅから発した光線による球面２１ａ上の点（α，ω）、
Ｎは球２１の頂点、
Ｓは断層像２３上の点であり、直線ＯＱとの交点、
Ｒは球面２１ａの半径、
ＰはＱ点に対応する二次投影平面４上の直交座標系上の点（η，ξ）、
αはｘ−ｙ平面上でｘ軸から測った回転角、
ωはＱ点と原点Ｏを結んだ線とｚ軸のなす角、
Ｌは球面２１ａ上の円弧ＮＱ、
Ｌ´はＬと同じ長さをもった線分で、二次投影平面４上の直線Ｎ´Ｐ、
ηはＰ点に対応するモニタ上のＸ座標、
ξはＰ点に対応するモニタ上のＹ座標、
である。
【０１２３】
いま、図２６において、原点Ｏを通り、ｘ−ｙ平面に垂直な平面に着目すると図２７の下側の図となる。
図２７の下側の図において、ｚ軸は図示面に垂直で、図示面から上方に向かう方向をＮにとっている。半円４０は球面２１ａの一断面である。
【０１２４】
図２７の下側の図で、断層像２３上の各点と原点Ｏを結んだ直線を球面２１ａまで延ばし、球面２１ａとの交点をＱとする。この方法では断層像２３上での点は等間隔になるが、角度座標は等間隔にはならない。角度座標を等間隔に求めるには内挿計算を用いればよい。
【０１２５】
一方、半円４０を等角度で分割し、直線ＯＱと断層像２３との交点を求める方法では、逆に断層像２３上の点は等間隔にならないが、これも断層像２３上での近傍点から内挿計算により計算可能である。
【０１２６】
いずれの方法でも、球面（一次投影曲面）２１ａ上での角度座標は全て求めることが可能であり、得られた点は二次投影平面４への計算に使用する。
【０１２７】
図２７において、断層像２３上の点Ｓ（ｘ_０，ｙ_０，ｚ_０）に着目すると、原点Ｏと点Ｓを結ぶ直線ＯＳとｘ−ｙ平面となす角度θは、
【０１２８】
【数９】

【０１２９】
で与えられる。
【０１３０】
したがって、ｚ軸とのなす角度ωは、
【０１３１】
【数１０】

【０１３２】
で与えられる。
【０１３３】
一方、直線ＯＳの、ｘ軸からの方位角αは、
α＝ｔａｎ^−１（ｙ_０／ｘ_０） ………（２３）
で与えられる。
【０１３４】
したがって、図２６の円弧ＮＱの長さＬは、
【０１３５】
【数１１】

【０１３６】
で与えられる。
【０１３７】
断層像２３上の各点について、（２４）式で計算することにより、球面への一次投影像の位置が得られる。すなわち、一次投影像の球面での座標は（α，Ｌ）により全て与えられる。
【０１３８】
また、原点Ｏから断層像２３上の点Ｓまでの距離ｒｓは、
【０１３９】
【数１２】

【０１４０】
で与えられる。この距離は、陰影付け時に利用される。
このようにして、断層像２３上の対応する点を全て球面２１ａへ投影し、これらの点を二次投影平面４への再投影に用いる。
【０１４１】
図２８は、二次投影平面４への球面／平面座標変換の一例を示す。図２８においては、頂点Ｎを中心として半径方向の距離をＬ、頂点Ｎより定めた基準直線ｗからの角度（方位角）をαで表し、等距離（円弧の長さを等しくとってある）、等角度の座標を使用している。これにより、球面２１ａへの一次投影像は頂点Ｎからの距離Ｌ´と角度αで表される。これをＸ，Ｙ座標に再投影（二次投影）し、モニタに表示すればよい。
【０１４２】
図２８において、矩形領域５０はモニタ表示領域の例であり、この領域５０はモニタの表示メモリの数、例えば５１２×５１２画素あるいは１０２４×１０２４画素に対応させる。矩形領域５０を、図示複数の同心円のうちどれだけの円を含む領域とし、モニタ表示するかは任意に決められる。
【０１４３】
一例として、次のように立体角２π（半球）全体をモニタ表示する場合は、半球の円周π・Ｒの長さの範囲をモニタ画面の縦横の表示の長さに対応させる。
【０１４４】
モニタ上のＸ，Ｙ座標はＬを用いて、
ＮＱ＝Ｌ´＝Ｒω ………（２６）
Ｘ＝ξ_１＝Ｌ´ｃｏｓα ………（２７）
Ｙ＝η_１ ＝Ｌ´ｓｉｎα ………（２８）
により与えられる。
【０１４５】
この、η_１，ξ_１が表示メモリ上の画素アドレスとなる。すなわち、二次投影平面４の点Ｐはモニタ画面上では点（η_１，ξ_１）に表示することになるもので、これにより球面／平面座標変換による表示が行われたことになる。
【０１４６】
図２８において、半径ＯＰの最大値は、 立体角＝２πの場合はπＲ、立体角＝４πの場合は２πＲであるが、任意の立体角の場合はその立体角に応じてＯＰの長さを変える。
【０１４７】
図２９は、二次投影平面４への球面／平面座標変換の他の例を示す。ここでは、二次投影平面４上に再投影するとき、球面２１ａ上の一次投影像をそのまま二次投影平面４に平行移動するよう、ｚ軸に平行に投影する方法を示すもので、これによると、図３０に示すように、モニタ上では画像の中心部分が拡大される効果をもつ。
【０１４８】
図２９の球面（一次投影曲面）２１ａ上の一次投影像の各点は、頂点Ｎからの円弧の長さＬ´と方位角αにより表わされるが、二次投影平面４への変換式は、Ｌ´´＝Ｌ´・ｓｉｎω ………（２９）
で与える。ここでωは、図２６に示すように、視点ｅと投影物を結ぶ直線とｚ軸とのなす角である。また、方位角はそのまま用いる。
【０１４９】
陰影付けは、一次投影曲面２１ａ上で行う方法と、二次投影平面４上で行う方法とがあり、どちらの方法を用いてもよい。
陰影付けには、原点Ｏから投影対象物の各点への距離に応じて濃淡を付ける方法や、各投影直線と対象物面とのなす角度に応じて輝度変換する方法や、投影中心線と対象物面とのなす角度に応じて輝度変換する方法、更には原点Ｏからの距離に応じて等高線のように輝度変換を行う方法等が用いられる。
また、視点ｅからの光線に対して、投影物の組織の透過率を用いるボリュームレンダリング処理を行う方法を用いてもよい。
【０１５０】
図３１〜図３７は、複数の断層像２３…から上述投影方法により三次元画像を構成する方法の一例を説明するための図である。図３２〜３３，図３４〜３７のいずれの場合においても、最初に球面２１ａ上に断層像２３を投影し、次に球面／平面座標変換により二次投影平面４へ再投影することになる。
【０１５１】
まず、上記三次元画像の構成方法における球面／平面座標変換について述べる。
図３１に示す例では、説明を簡易化するため球面２１ａの大円２０と断層像２３面、更には二次投影平面４が各々平行であるように座標系をとっている。また、球面２１ａの中心線をｘ軸と平行にとっている。
【０１５２】
この図３１において、視点ｅの位置は、三次元座標系（ｘ，ｙ，ｚ）の原点Ｏ（０，０，０）である。
また、
ｘ，ｙ，ｚは三次元座標系（ｘ，ｙ，ｚ）の各軸、
Ｑは、断層像２３上のＳ点に対応する球面２１ａ上の点、
Ｓ´は視点ｅとＱ点を結ぶ直線２２と断層像２３Ａの交わる点、
Ｎは球２１の頂点、
２０はｘ−ｙ平面（ｚ＝_０）、
４は二次投影平面、
ｗはｘ−ｙ平面上での回転角の基準直線、
ｄ_０は断層像２３Ａのｚ軸上の距離、
である。
【０１５３】
いま、視点ｅを通り、二次投影平面４に直交する平面（直交平面）を考える。図３１における断層像２３Ａ〜２３Ｃの他にも断層像２３Ｄ〜２３Ｈを用意しての、上記直交平面上の図を図３２に示す。
【０１５４】
この図３２において、断層像２３Ａ〜２３Ｈは同一投影対象（ないし対象物）について同一方向に等間隔で得られた断層像であり、断層像２３には臓器領域Ｂ１，Ｂ２，Ｂ３，Ｂ４が強調して書いてある。なお、断層像２３Ａ〜２３Ｈは必ずしも等間隔である必要はない。
【０１５５】
上記臓器領域Ｂ１，Ｂ２，Ｂ３，Ｂ４を、まず球面２１ａに投影し、更にそれを球面／平面座標変換して二次投影平面４に投影すると、Ｂ１´，Ｂ２´，Ｂ３´，Ｂ４´となる。
【０１５６】
ここで、投影データ（ここではＢ１´，Ｂ２´，Ｂ３´，Ｂ４´）を表示メモリ（図示せず）に書き込むときは、三次元的効果を出すために、視点ｅからみてより遠くに存在する投影データを先に書き込み、それより近くの投影データは後から上書きする。その後、陰影付け処理を行う。
【０１５７】
図３３は図３２よりも一般化して示したもので、球２１の頂点Ｎの方向が断層像２３面と直交せず、ｙ軸を中心として任意の角度だけ回転した場合の例である。この場合は、補間演算により求めた球面２１ａへの一次投影像を、二次投影平面４へ再投影することになる。
【０１５８】
図３４は、視点ｅ、断層像２３及び一次投影曲面２１ａがより複雑な位置関係をもった場合の上述球面／平面座標変換を説明するための図で、断層像２３上のＳｉ（ｘｉ，ｙｉ，ｄｉ）の一次投影像が最終的に二次投影平面４上の点Ｐｉ（Ｘ，Ｙ）になることを示す。
【０１５９】
図３４において、本発明投影方法による断層像２３の球面２１ａへの投影に当たって、断層像２３の各画素座標の球面２１ａ上の座標への変換は次のように行われる。
【０１６０】
ここで、説明に一般性をもたせるため次のように位置関係を設定する。
図３４において、投影物である断層像２３はｚ軸に直交する面内にあり、原点Ｏからｄ_０の距離にあるものとする。複数の断層像２３…に対しては原点Ｏからの距離ｄ_０が変わる。ただし、複数の断層像２３…が等間隔である必要はない。
【０１６１】
また、原点Ｏ上の視点ｅから、ｚ軸とは異なる任意の投影方向の中心Ｎを決め、その方向をＯＮとし、直線ＯＮをｈとする。そして、原点Ｏから球２１の半径Ｒの点で直線ｈに直交する二次投影平面４を与える。この平面４と直線ｈとの交点の座標Ｎが二次投影平面４での直交座標系Ｘ，Ｙの中心座標となる。この平面４上の点（Ｘ，Ｙ）がモニタ上の表示座標となる。
【０１６２】
図３５は上記球面２１ａ上の座標関係を示す。いま、図３４における座標系に対して、ＯＮ_１の向きを任意に変えた新しい位置をＯＮ_２とする。なお、このＯＮ２の方向を任意に変えることにより、その方向を中心とした三次元画像を表示可能である。
【０１６３】
また図３５において、直線ＯＮ_２に対し、原点Ｏを通りＯＮ_２に直交する平面（大円）を２０´、ＯＮ_２のｘ軸からの回転角をα_０、ｚ軸となす角をωとする。言い換えると、直線ＯＮ_１をｚ軸からω、ｘ軸からα_０だけ回転させた直線がＯＮ２である。ここで、ｘ軸をｘ−ｙ平面内で角度α_０だけ回転させた直線を新しい方位角の基準軸Ｏｘ´とする。
【０１６４】
点Ｑ_１，Ｑ_２は球面２１ａ上の点であり、直線ＯＮ_２に対して基準線Ｏｘ´から測った方位角λ、平面２０´からの角度βで表される。同様に、これらの点は元の座標系でｘ軸からの方位角α、平面２０からの角度δで表わされる。直線ＯＮ２を回転軸とした一次投影曲面２１ａへの投影像は、λ，βを等間隔に変えて計算すればよい。
【０１６５】
いま、点Ｑ_１の元の座標系での角度を各々α，δとし、平面２０´の平面２０からの傾きをεとする。そして、直線ＯＮ_２とｚ軸とのなす角をω、ｘ−ｙ平面内での回転角をα_０、また、Ｑ点のｘ´軸からの角度をα´とする。更に、新しい座標系でのＳ点の角度を各々β，λとする。
【０１６６】
このようにすると、ε＝ωであり、β，λとα，δ，ωの間には、次の
ｃｏｓβｃｏｓλ＝ｃｏｓδｃｏｓα´
ｃｏｓβｓｉｎλ＝ｓｉｎδｓｉｎω＋ｃｏｓδｓｉｎα´ｃｏｓω ………（３０）
ｓｉｎβ ＝ｓｉｎδｃｏｓω−ｃｏｓδｓｉｎα´ｓｉｎω
の関係式が成り立つ。
【０１６７】
ここで変換後のλ，βを指定して逆に元のｘ，ｙ，ｚ座標系のα，δを求める場合は、β，λとα，δを逆にした
ｃｏｓδｃｏｓα´＝ｃｏｓβｃｏｓλ
ｃｏｓδｓｉｎα´＝−ｓｉｎβｓｉｎω＋ｃｏｓβｓｉｎλｃｏｓω ………（３１）
ｓｉｎδ ＝ｓｉｎβｃｏｓω＋ｃｏｓβｓｉｎλｓｉｎω
を用いればよい。
該当する位置に座標が一致しなければ周辺の点から内挿計算により求める。
【０１６８】
また、元の座標系と新しい座標系での回転角αの間には、α´＝α−α_０であるから、
α＝α´＋α_０ ………（３２）
の関係式が成り立つ。
【０１６９】
一方、β，λから、二次投影平面４への変換は、次のようにして求まる。
すなわち、βから弧の長さｌを求めると、
ｌ＝Ｒ（π ／２−β） ………（３３）
のようになる。
このｌとβを用いて二次投影像を得、モニタ表示すればよい。
【０１７０】
図３６は球面２１ａへの投影像をα，δにより表示した場合の像である。このままモニタ表示してもよいが、δがπ／２に近づくにつれて歪みが大きくなり直感的な視覚による像と一致しなくなる。そこで、上記α，δによる像データを用い、更に二次投影平面４へ再投影することにより視覚による像と一致する三次元画像が得られる。
【０１７１】
任意方向を中心とする二次投影平面４へ再投影した三次元画像は、既に述べたように新しい方位角λ，βを用いて（３１）式から求まる。また、二次投影平面４において、一次投影曲面２１ａのどれだけの領域を表示するかによって再投影の際に用いる一次投影像データの範囲が異なる。
【０１７２】
図３７は、上記一次投影曲面２１ａ全体への投影像データを用いて、任意の方向を中心とした三次元画像を計算するときに利用する投影像データの範囲を示す。この図３７において、点Ｐ_１〜Ｐ_３は任意の方向ベクトルＯＮの中心Ｎを表わす。例えば、Ｐ_１は元の座標系において、方位角α＝π／２、平面２０からの角度ω＝０を中心Ｎの方向とする二次投影平面４への三次元画像構築で用いるデータの範囲を示している。
【０１７３】
前記（２１）〜（３３）式から各断層像２３…上の各点の一次投影曲面２１ａでの座標が求まる。断層像２３は複数あるため輪郭点は複数存在することもあるが、視点ｅから遠いところから表示メモリに書き込むことにより、視点ｅに一番近い投影像が表示される。
【０１７４】
このようにして、頂点を中心Ｎとして角度β，λを変えて行くことにより、一次投影像が球面２１ａ上に求まる。この一次投影像を前記球面／平面座標変換により二次投影平面４に再投影し、対応する画像表示メモリに書き込むことにより二次投影像がモニタに表示される。
【０１７５】
例２：回転体２１が円錐体である場合
図３８に示すような、視点ｅ（原点Ｏ）からｘ軸上の距離Ｒのところからｘ軸となす角Ωの直線ｔ３８を引き、その直線ｔ３８を、ｘ軸を回転軸として回転させると円錐体２１となる。この場合は、円錐体２１の曲面２１ａに一次投影することになる。このような円錐体曲面２１ａへの投影は、管状の臓器（対象物）の投影に適する。
【０１７６】
この場合、直線ｔ３８は、
ｙ＝−ｘｔａｎΩ＋ＲｔａｎΩ ………（３４）
で与えられる。
【０１７７】
直線の式ｙ＝ｘｔａｎφを用いることにより、座標ｘ_０は、
ｘ_０＝ＲｔａｎΩ／（ｔａｎΩ＋ｔａｎφ） ………（３５）
で与えられる。
円錐体曲面２１ａ上の距離Ｌ（＝ＮＱ）は、（６）式におけるｆ´として、（３４）式の一次微分を用いることにより、
Ｌ＝（Ｒ−ｘ_０）ｓｅｃΩ ………（３６）
で求まる。
【０１７８】
例３：回転体２１が楕円回転体である場合
図３９に示すように、視点ｅ（原点Ｏ）を中心とした楕円ｔ３９を、ｘ軸を回転軸として回転させると、図４０に示すような楕円回転体２１となる。図４０において、２１ａは楕円回転体曲面、４０は対象物中の投影対象、４１はその一次投影像、Ｎはｘ軸上にとった頂点である。この場合は、楕円回転体２１の曲面に一次投影することになる。原点Ｏに中心をもたず、ｘ軸上で原点Ｏから距離ｋの点を中心とする楕円回転体２１も可能である。
【０１７９】
楕円回転体２１への投影は、平面の部分が多く広がりをもった臓器（対象物）に適する。図３９において、ａ，ｂは各々ｘ軸及びｙ軸との交点であり、このａ，ｂを変えることにより広い視野を得たり、奥行きのある立体感のある三次元画像が得られる。
【０１８０】
次の
【０１８１】
【数１３】

【０１８２】
で与えられる曲線を、ｘ軸を回転軸として回転させることで楕円回転体２１が与えられる。
【０１８３】
直線の式ｙ＝ｘｔａｎφを用いることにより、座標ｘ_０は計算で求められる。また、楕円回転体曲面２１ａ上の距離Ｌは、（６）式と（３７）式の一次微分から計算で求まる。
【０１８４】
例４：回転体２１が放物線回転体である場合
図４１に示すように、視点ｅ（原点Ｏ）からｘ軸上の距離Ｒのところから（３８）式で示す放物線ｔ４１を引いてそれを、ｘ軸を回転軸として回転させると、図４２に示すような放物線回転体２１となる。この場合は、放物線回転体２１の曲面に一次投影することになる。図４２において、２１ａは放物線回転体曲面、４０は対象物中の投影対象、４１はその投影像、Ｎはｘ軸上にとった頂点である。
【０１８５】
これは比較的太い管状の臓器（対象物）や、内部の充填した臓器（対象物）に仮想的な管状の穴を明けながら三次元画像を構成するのに適している。
【０１８６】
放物線回転体曲面２１ａ上で、頂点Ｎから点Ｑまでの距離Ｌは、（６）式と、
【０１８７】
【数１４】

【０１８８】
の一次微分から計算で求まる。
【０１８９】
例５：回転体２１が円柱である場合
図４３に示すように、視点ｅ（原点Ｏ）からｘ軸上の距離Ｒのところに、
ｙ＝Ｒ ………（３９）
で示す直線ｔ４３を引き、その直線ｔ４３を、ｘ軸を回転軸として回転させると円柱２１となる。この場合は、図４４に示すような円柱２１の曲面に一次投影することになる。
【０１９０】
図４４において、２１ａは円柱曲面、４０は対象物中の投影対象、４１はその投影像である。この場合は、頂点Ｎはｘ軸上には定めず、円柱２１上の任意の１点を決めればよい。
【０１９１】
円柱曲面２１ａを用いた投影方法では、管状の対象物を投影して三次元画像を表示する。円柱曲面２１ａ上の距離Ｌ（＝ＮＱ）は、この場合は頂点Ｎの位置から測る。
Ｌ＝ ( ａ＋ｘ _０ ) で求められる。
以上に説明した「円柱曲面２１ａを用いた投影方法」では、管状の対象物を投影して三次元画像を表示するのに適しているが、この方式は地図学におけるメルカトル図法に対応し、極付近の歪みが大きいので臓器観察には適しない。
このため、本発明においては、試案の投影技術の中から「円柱面を一次投影曲面とすること」を除外する。
以上に、試案における例１として回転体が球である場合を、例２として回転体が円錐体である場合を、例３として回転体が楕円回転体である場合を、例４として回転体が放物線回転体である場合を、例５として回転体が円柱である場合を、それぞれ考察してきたが、前述のごとく円柱を除外するので、一次投影面として適正な曲面は「球，円錐体，楕円回転体，または放物線回転体」形状の曲面ということになる。
【０１９３】
図４５は本発明投影方法が適用可能なハードウェア構成例を示すブロック図である。この図４５において、５０はＣＰＵ、５１は主メモリ、５２は磁気ディスク、５３は表示メモリ、５５はマウスコントローラで、これらは共通バス５７に接続されている。磁気ディスク５２には、複数の断層像及び本発明投影方法の実行演算のためのプログラム等が格納されている。
【０１９４】
ＣＰＵ５０は、これら複数の断層像及び本発明投影方法の実行演算のためのプログラムを読み出し、主メモリ５１を用いて回転体曲面／平面座標変換等の演算を行い、その結果を表示メモリ５３に送り、ＣＲＴモニタ５４に表示させる。マウスコントローラ５５に接続されたマウス５６は、回転体曲面／平面座標変換等の演算の際の、視点や直線・曲線の始端，終端の位置等を指定する。回転体曲面／平面座標変換された画像は、必要に応じて磁気ディスク５２に格納される。
【０１９５】
マウスコントローラ５５，マウス５６に代えて、他のポインティングデバイスコントローラ，ポインティングデバイス、例えばトラックボール等を用いてもよい。
【０１９６】
【発明の効果】
以上説明したように本発明によれば、パノラマ状の広い範囲の対象物内部が見えるような三次元画像が得られ、しかも対象物の形状や観察したい部分、あるいは観察目的に応じて投影面の形状を任意に変え、それら対象物の形状、観察したい部分、あるいは観察目的等に適した投影面により見やすい三次元画像を得ることができる。特に、血管壁、食道壁等の管状の臓器の投影に適用したとき、あたかもそれら臓器の内部に入って見ているような三次元画像を得ることができるという効果がある。
【図面の簡単な説明】
【図１】試案の投影方法の例を示すフローチャートである。
【図２】試案の投影方法の例を示す説明図である。
【図３】試案の投影方法の詳細を説明するための図である。
【図４】試案の投影方法の詳細を説明するための図である。
【図５】試案の投影方法の詳細を説明するための図である。
【図６】試案の投影方法における回転体曲面／平面座標変換の原理説明図である。
【図７】試案の投影方法における回転体曲面／平面座標変換の原理説明図である。
【図８】試案の投影方法における回転体曲面／平面座標変換の原理説明図である。
【図９】試案の投影方法における、回転体曲面／平面座標変換の一例の説明図である。
【図１０】試案の投影方法における回転体曲面／平面座標変換の他の例の説明図である。
【図１１】試案の投影方法における回転体曲面／平面座標変換の他の例の説明図である。
【図１２】複数の断層像から試案の投影方法を適用して三次元画像を構成する方法の一例を説明するための図である。
【図１３】複数の断層像から試案の投影方法を適用して三次元画像を構成する方法の一例を説明するための図である。
【図１４】複数の断層像から試案の投影方法を適用して三次元画像を構成する方法の一例を説明するための図である。
【図１５】複数の断層像から試案の投影方法を適用して三次元画像を構成する方法の一例において視点、断層像、及び一次投影曲面がより複雑な位置関係をもった場合の回転体曲面／平面座標変換を説明するための図である。
【図１６】図１５における回転体曲面上の座標関係を示す図である。
【図１７】図１６におけるλ，Ｌを用いて二次投影平面上の像が求まることの説明図である。
【図１８】回転体曲面への投影像を図１６における角度λ，βにより表示した図である。
【図１９】図１７における距離Ｌを一定の関数により変換してＬ´を求め、画像表示する例を示す図である。
【図２０】図１７における距離Ｌを一定の関数により変換してＬ´を求め、画像表示する例を示す図である。
【図２１】図１７における距離Ｌを一定の関数により変換してＬ´を求め、画像表示する例を示す図である。
【図２２】回転体が球である場合の本発明投影方法の説明図である。
【図２３】回転体が球である場合の本発明投影方法の説明図である。
【図２４】回転体が球である場合の本発明投影方法の説明図である。
【図２５】回転体が球である場合の本発明投影方法の説明図である。
【図２６】回転体が球である場合の本発明投影方法における一次投影像の球面から平面への再投影の実施例の説明図である。
【図２７】回転体が球である場合の本発明投影方法における一次投影像の球面から平面への再投影の実施例の説明図である。
【図２８】回転体が球である場合の本発明投影方法における一次投影像の球面から平面への再投影の実施例の説明図である。
【図２９】回転体が球である場合の本発明投影方法における一次投影像の球面から平面への再投影の実施例の説明図である。
【図３０】回転体が球である場合の本発明投影方法における一次投影像の球面から平面への再投影の実施例の説明図である。
【図３１】回転体が球である場合の本発明投影方法を適用して複数の断層像から三次元画像を構成する方法の一例を説明するための図である。
【図３２】回転体が球である場合の本発明投影方法を適用して複数の断層像から三次元画像を構成する方法の一例を説明するための図である。
【図３３】回転体が球である場合の本発明投影方法を適用して複数の断層像から三次元画像を構成する方法の一例を説明するための図である。
【図３４】回転体が球である場合の本発明投影方法を適用して複数の断層像から三次元画像を構成する方法の一例において、視点、断層像及び一次投影曲面がより複雑な位置関係をもった場合の回転体曲面／平面座標変換を説明するための図である。
【図３５】図３４における球面上の座標関係を示す図である。
【図３６】図３４における球面への投影像を図３５における角度λ，βにより表示した図である。
【図３７】図３６における一次投影曲面全体への投影像データを用いて、任意の方向を中心とした三次元画像を計算するときに利用する投影像データの範囲を示す図である。
【図３８】回転体が円錐体である場合の本発明投影方法の説明図である。
【図３９】回転体が楕円回転体である場合の本発明投影方法の説明図である。
【図４０】回転体が楕円回転体である場合の本発明投影方法の説明図である。
【図４１】回転体が放物線回転体の場合の本発明投影方法の説明図である。
【図４２】回転体が放物線回転体の場合の本発明投影方法の説明図である。
【図４３】試案において回転体が円柱である場合の投影方法の説明図である。
【図４４】試案において回転体が円柱である場合の投影方法の説明図である。
【図４５】本発明投影方法が適用可能なハードウェア構成例を示すブロック図である。
【図４６】中心投影法の説明図である。
【図４７】平行投影法の説明図である。
【図４８】中心投影法と投影像の移動との関係の説明図である。
【符号の説明】
ｅ 視点
ｘ，ｙ，ｚ 三次元座標系の各軸
Ｏ 三次元座標系の原点
２１ 回転体
２１ａ 回転体曲面（一次投影曲面）
２２ 直線
２３ 断層像
４ 二次投影平面
５ 投影対象
６ 一次投影像
Ｃ 三次元座標系の原点を中心として単位球（半径１）
２０ 単位球の中心を通りｘ−ｙ平面に一致する平面（大円）
２０´ 原点を通りｚ軸からの角度ωの回転体の回転軸に直交する平面
Ｑ 視点ｅから発した光線による回転体曲面２１ａ上の点
Ｎ 回転体の頂点（中心）
Ｓ 断層像上の点であり直線ＯＱとの交点
Ｒ 回転体曲面の半径
Ｐ Ｑ点に対応する二次投影平面上の直交座標系上の点
α ｘ−ｙ平面上でｘ軸から測った回転角
ω Ｑ点と原点を結んだ線とｚ軸のなす角
Ｌ 回転体曲面上の円弧ＮＱ
Ｌ´ Ｌと同じ長さをもった線分で二次投影平面上の直線Ｎ´Ｐ
η Ｐ点に対応するモニタ上のＸ座標
ξ Ｐ点に対応するモニタ上のＹ座標[0001]
[Industrial application fields]
The present invention relates to computer graphics, and in particular, a projection method using a central projection method suitable for tomographic image projection in a three-dimensional image configuration.And projection apparatusIt is about.
[0002]
[Prior art]
As shown in FIG. 46, the central projection method is a method of projecting projection objects 40a and 40b onto the projection plane 4 by emitting light rays radially from one point (viewpoint e) set at an arbitrary position. This method shades a two-dimensional image obtained by arranging an array of three-dimensional images 23 in a predetermined direction into a two-dimensional image viewed from an arbitrary direction (a three-dimensional image is a pixel). Is a two-dimensional array, and is used for projection of each tomographic image 23 when it is configured as an image shaded on it and displayed as a three-dimensional image, that is, a pseudo three-dimensional image (for example, a special feature). Application No. 6-3492).
[0003]
In addition, as shown in FIG. 47, the parallel projection method in which the projection objects 40a and 40b are projected onto the projection plane 4 by parallel rays is also used to project each tomographic image 23 in the configuration of the three-dimensional image from a plurality of tomographic images 23. Has been used.
[0004]
46 and 47, 40A and 40B are projection images of the projection objects 40a and 40b projected on the projection plane 4, and x, y, and z are axes of the three-dimensional coordinate system.
[0005]
In both the above-described projection methods, when observing bones, skin, body surfaces, and the like from the outside, a parallel projection method in which the external shape can be intuitively understood by looking at them from the outside is suitable. On the other hand, when observing a three-dimensional display of blood vessels, esophageal lining, intestine, bronchi, and minute parts in the body, place the viewpoint e inside the projection object such as the organs, and project the surroundings from that one point. A central projection method in which the objects 40a and 40b can be seen is suitable.
[0006]
By the way, conventionally, the projection plane in the above-described central projection method (general central projection method) and parallel projection method is “Computer Graphics” (JD Foley & A. Van Dam), which is a typical document relating to computer graphics. Written by Tomomi Imamiya, translated by the Japan Computer Association on July 15, 1984. It was flat.
[0007]
For this reason, when projecting an object using the center projection method, the projected image is enlarged from the center of the viewpoint toward the periphery, and is rapidly moved away from the center on the projection surface. This is because the amount of movement increases as the distance between the viewpoint and the object is shorter.
[0008]
For example, in FIG. 48, the distance from the viewpoint e to the projection plane 4 is R, the distance to the tomographic image (object) 23 is r, and the projection center 50 of a point (pixel) on the tomographic image 23 is in the circumferential direction. The distance H from the projection center 50 on the projection plane 4 of a certain point on the tomographic image 23 is expressed as follows.
H = Rl / r
As r decreases, H increases rapidly and moves away from the center 50 on the projection plane 4. In addition, even if l increases, H increases in the same way at a ratio of R / r. Since the display area of an image display monitor (hereinafter abbreviated as a monitor) such as a CRT monitor is finite, when H is determined, the area projected by R is determined, but the width of 180 degrees from the viewpoint e is projected. To achieve this, R = 0, and only a narrow portion can be projected.
[0009]
[Problems to be solved by the invention]
Three-dimensional display is often used for three-dimensional objects such as the stomach, lungs, abdomen, and heart. For such a projection object, a three-dimensional display that gives a panoramic field of view so that the viewpoint enters inside and the inside of the object in a wide range of 180 degrees in the vertical and horizontal directions can be seen is desired.
[0010]
In addition, a three-dimensional display is desired so that the interior of a wide range of objects can be seen by projecting onto a curved surface having a smooth shape approximating the shape of a specific object and developing it.
Furthermore, the shape of the projection surface is changed according to the shape of the object and the portion to be observed (projection target), and an easy-to-see 3D image is obtained using the shape of the object, the portion to be observed, or a projection surface suitable for the purpose of observation. It is desirable.
[0011]
However, in the prior art, none of the above-mentioned requests has been realized, and there has been a demand for improvement in this regard.
[0012]
An object of the present invention is to obtain a panoramic three-dimensional image in which the inside of an object can be seen, and arbitrarily change the shape of the object, the portion to be observed, or the shape of the projection surface according to the observation purpose. To obtain a three-dimensional image that is easy to see with the projection surface suitable for the shape of the object, the portion to be observed, or the purpose of observationAnd suitable for observation of organsUsing central projectionThrowIt is to provide a shadow device.
[0013]
[Means for Solving the Problems]
In order to achieve the above object, the present inventionAccording to claim 1projectionapparatusSets a viewpoint at an arbitrary position inside the projection object and a projection plane outside the projection object, respectively,
Sphere, cone, ellipsoidal rotator, or parabolic rotator (hereinafter abbreviated as specific rotator) shapeIs placed between the viewpoint and the projection plane.meansWhen,
Projecting the projection object or at least one tomographic image of the projection object onto the projection curved surface as viewed from the viewpoint by a central projection methodmeansWhen,
Reprojecting the projection image obtained on the projection curved surface onto the projection plane by rotating body curved surface / plane coordinate transformationmeansAnd consist of
Further, in the projection device according to claim 2, in addition to the configuration requirements of claim 1, the projection curved surface is a spherical surface,
In addition, the radius of the spherical surface is a structure that can be selected as “a dimension capable of discriminating a minute part of an organ that is a projection target”.
[0014]
[Action]
The present inventionAccording to the projection device of claim 1In the process of projecting onto a plane corresponding to the display screen of the monitor, a straight line in an arbitrary projection direction is drawn from the viewpoint.Can beAnd draw a line from any pointAnd, The line is formed by rotating the straight line as a rotation axisSpecific rotating body (hereinafter simply referred to as rotating body if not confusing)As a result, the projection target is projected onto the curved surface (primary projection curved surface) of the rotating body by the central projection method.Furthermore, this primary projection image can be reprojected onto the projection plane (secondary projection plane).
When the invention according to claim 2 is applied to the projection apparatus according to claim 1, since the primary projection curved surface is a spherical surface, the radius dimension thereof is constant, the calculation can be performed quickly and easily. It is possible to make a trade-off between being able to discriminate a minute part of "and observing a wide range of organs".
[0015]
Therefore, according to this first process, the position of the line forming the rotator and the shape of the panoramic shape allows the viewpoint to enter the inside of the projection object and to see the interior of the object in a wide range of 180 degrees above, below, right and left 3D display that gives a field of view is possible, and it is projected onto a curved surface with a smooth shape approximating the shape of a specific object. It becomes. Furthermore, the shape of the target object, the part to be observed, or the shape of the projection surface according to the observation purpose can be changed to a three-dimensional image that is easy to see using the shape of the target object, the part to be observed, or the projection surface suitable for the observation purpose. Can be obtained.
[0016]
In addition, a process (second process) of re-projecting (secondary projection) a primary projection image onto a curved surface (primary projection curved surface) of a rotator onto a predetermined plane (secondary projection plane) by rotating body curved surface / plane coordinate conversion. Prepare. Here, the rotator curved surface / planar coordinate conversion means that each coordinate of the primary projection image projected on the rotator curved surface (primary projection curved surface) is converted into a secondary projection plane by a fixed relational expression. This means that the primary projection image is developed and re-projected (secondary projection) on the secondary projection plane, and parameters for this rotating body curved surface / planar coordinate conversion can be arbitrarily set.
[0017]
Therefore, according to the second process, combined with the function in the first process, the shape of the object and the shape of the projection surface corresponding to the portion to be observed (projection target) can be changed. It becomes possible to obtain an easy-to-see three-dimensional image by using a projection plane suitable for a desired portion or the like.
[0018]
The projected image according to the projection method of the present invention is shaded during or after the rotating body curved surface / planar coordinate conversion, and is displayed on the monitor screen as a pseudo three-dimensional image.
[0019]
【Example】
Embodiments of the present invention will be described below with reference to the drawings.For convenience of explanation,
First, the central projection technique of the tentative plan will be described with reference to FIGS. 1 to 21, and then the embodiment of the present invention will be described with reference to FIGS.
The central projection technique of the tentative plan uses “arbitrary rotating curved surface” as the primary projected curved surface. On the other hand, the central projection technique of the present invention uses a “specific rotation curved surface” as a primary projection curved surface (a specific rotation curved surface is a rotation curved surface in the shape of a sphere, a cone, an elliptical rotation body, or a parabolic rotation body. Details are explained below).
FIG.In the draftA flowchart showing an example of a projection method using the central projection method, and FIG. 2 are also explanatory diagrams. Here, the case where the center projection method of the tentative plan is applied to a three-dimensional image will be described.
[0020]
First, a viewpoint e is set at an arbitrary position with respect to a projection target such as an organ (here, a three-dimensional image formed by arranging a plurality of tomographic images 23 in a certain direction) (step 101 in FIG. 1). reference).
[0021]
Next, a projection center N is set in an arbitrary direction from the viewpoint e, and a straight line h is drawn from the viewpoint e toward the projection center N. Further, a line having an arbitrary shape different from the straight line h, here, a curve f, is drawn in an arbitrary direction from an arbitrary point different from the viewpoint e (see Step 102).
Then, the rotating body 21 is obtained by rotating the curve f about the straight line h as a rotation axis (see step 103).
[0022]
Next, by the central projection method, the projection target 5 on the object is primarily projected onto a curved surface (also referred to as a rotator curved surface or a primary projection curved surface) 21a of the rotator 21 (see step 104).
[0023]
Further, the projection image (primary projection image) onto the rotating body curved surface 21a is reprojected (also referred to as secondary projection) onto a predetermined plane (secondary projection plane) 4 using the rotating body curved surface / plane coordinate conversion. .
[0024]
Here, the re-projected plane 4 corresponds to the monitor screen, and the rotating body curved surface / plane coordinate conversion corresponds to the coordinate conversion of the projected image (primary projection image) onto the rotating body curved surface 21a to the monitor screen. As a result of this conversion, the projection image of the projection target is displayed on the monitor screen (see step 105).
[0025]
That is, the rotating body curved surface / planar coordinate conversion is performed by converting each coordinate of the primary projection image projected onto the rotating body curved surface (primary projection curved surface) 21a onto the secondary projection plane 4 by a fixed relational expression. This means that the primary projection image is developed on the secondary projection plane 4 and reprojected (secondary projection).
The projected image is shaded during or after the rotating body curved surface / planar coordinate conversion, and is displayed on the monitor screen as a pseudo three-dimensional image.
[0026]
As described above, in the projection method of the present invention, the projection target is first projected by the central projection method using the rotating curved surface as the primary projected curved surface, and then the primary projection image is converted into a predetermined secondary projection plane by the rotating curved surface / plane coordinate conversion. Are reprojected (secondary projection). In other words, each point (pixel position) on the secondary projection plane is expanded in correspondence with the display pixel (address) on the monitor screen, and shading is performed during or after the rotary body curved surface / plane coordinate conversion. As a result, the image is displayed as a pseudo three-dimensional image on the monitor screen, and the effect of stereoscopic image display can be obtained on a plane (monitor screen).
Each point (positional coordinate) of the primary projection image is represented by, for example, a distance from the vertex of the rotator and an angle from a reference line in the angular direction.
[0027]
Note that FIG. 2 shows a case where all the objects to be projected (or projection objects) are located inside the rotating body, but this is not an essential requirement. On the contrary, the rotating body may be between the viewpoint e and the object. In this case, the light beam from the viewpoint e passes through the rotating body curved surface and reaches the object. no problem.
[0028]
Hereinafter, the projection method of the present invention will be described in more detail with reference to FIGS.
FIG. 3 is an overall explanatory diagram. In FIG. 3, first, an organ image that is a projection target (a three-dimensional image of an organ formed by arranging a plurality of organ tomographic images 23 in a fixed direction, here a stacked three-dimensional image) And simply referred to as an organ.) A viewpoint e is set at an arbitrary position with respect to 23a.
[0029]
Next, the center N of the projection direction is determined from the viewpoint e, a straight line h is drawn in that direction, and a line of an arbitrary shape apart from the straight line h,(Here the curve f)pull. A rotating body 21 obtained by rotating the curve f around the straight line h is assumed.
Although the curve f is described as “a line having an arbitrary shape”, as is clear from the procedure described with reference to FIG. 3, the curve f can be formed as “a rotating body can be formed when the straight line h is rotated about the rotation axis. It is obvious that if it is not a “like line”, it will be unwell.
[0030]
The distance from the straight line h to the curve f serving as the rotation axis is the radius of the rotating body 21. This radius varies depending on the position on the straight line h. If this radius is increased, a wide range of organs 23a can be projected. However, the image display area of the monitor screen is limited, and if the radius is increased, the minute portion of the organ 23a is not known, and therefore the size is selected to be appropriate.
[0031]
Next, a radial line is drawn from the viewpoint e, and the intersection of the radial line with each point of the organ 23a is projected onto the intersection of the radial line with the rotating body curved surface 21a. Perform primary projection (center projection) as a projection curved surface.
[0032]
Further, a plane T is assumed separately from the rotating body curved surface 21a. This plane T is the secondary projection plane 4 on which the primary projection image on the rotator curved surface 21a is projected by the rotator curved surface / plane coordinate conversion, and an image (secondary projection image) on the secondary projection plane 4 Is written to the display memory and displayed on the monitor. Shading is performed when the secondary projection image is written in a display memory (not shown) or displayed on a monitor (not shown).
[0033]
FIG. 4 shows the relationship between the tomographic image 23, the primary projection curved surface 21a, the secondary projection plane 4, the projection target 5, and the viewpoint e in the plane including the straight line h. As shown in the figure, the viewpoint e is one point on the straight line h that is the rotation axis of the rotator 21 and also the origin O of the three-dimensional coordinates (x, y, z). The primary projection curved surface 21a is a curved surface of the rotator 21, and the projection target 5 on the tomographic image 23 is projected onto the rotator curved surface 21a. The primary projection image 6 projected onto the rotator curved surface 21a is converted into the secondary projection plane 4 by developing the rotator curved surface (primary projection curved surface) 21 onto the secondary projection plane 4 by rotator curved surface / plane coordinate conversion. Reprojected (secondary projection).
[0034]
FIG. 5 is an explanatory diagram of parameter examples in the rotating body curved surface / planar coordinate conversion when the primary projection image 6 is developed on the secondary projection plane 4 and reprojected. Here, the image (primary image) projected onto the rotator curved surface 21a when the intersection N between the straight line h forming the rotation axis and the secondary projection plane 4 when forming the rotator 21 in FIGS. An example is shown in which the projection image 6) is developed on the secondary projection plane 4 by two parameters of the surface distance L from the vertex N and an angle α from an arbitrarily defined axis 81 orthogonal to the straight line h. ing.
[0035]
In the development from the rotating body curved surface 21a to the plane 4, the range of the rotating body curved surface 21a region to be expanded is arbitrarily determined.
Further, the vertex N that is the center of the development onto the secondary projection plane 4 can be arbitrarily given, but is usually given to the center of the plane 4. In other words, the center of the display image on the monitor is taken in the direction of the straight line h.
[0036]
In FIG. 3, x, y, and z indicate the respective axes of the three-dimensional coordinate system.
The projection object or object may be data input by an input device such as a mouse, or a tomographic image as described above (including a tomographic image obtained by decomposing a volume image by three-dimensional measurement). Data obtained as a result of calculation by an arithmetic unit may be used. Further, it may be one, or two or more as described above.
[0037]
Hereinafter, the principle of the rotating body curved surface / planar coordinate conversion will be described with reference to FIGS.
First, a description will be given with reference to FIG. In FIG. 6, reference numeral 21a denotes a curved surface (primary projection curved surface) of the rotating body 21 having the z axis as a rotation axis among the respective axes x, y, and z of the three-dimensional coordinate system. The viewpoint e is located at the origin O of the three-dimensional coordinate system. Furthermore, it is assumed that the circle 20 whose center is located at the origin O (z = 0) of the rotation axis (z axis) of the rotator 21 is parallel to the xy plane.
[0038]
Reference numeral 4 denotes a secondary projection plane that re-projects the data (primary projection image) after projection onto the rotator curved surface 21a by rotator curved surface / plane coordinate conversion. This secondary projection plane 4 corresponds to the display screen of the monitor.
[0039]
Also,
23 is a tomographic image,
Q is a point on the rotating body curved surface 21a by a light ray emitted from the viewpoint e.
N is the apex of the rotating body 21,
S is a point on the tomographic image 23, and the intersection with the straight line OQ,
R is the radius of the rotating body curved surface 21a,
P is a point on the Cartesian coordinate system on the secondary projection plane 4 corresponding to the Q point,
α is a rotation angle measured from the x-axis on the xy plane,
ω is the angle between the line connecting the Q point and the origin O and the z axis,
L is an arc NQ on the rotating body curved surface 21a,
L ′ is a line segment having the same length as L, and a straight line N′P on the secondary projection plane 4,
η is the X coordinate on the monitor corresponding to point P,
ξ is the Y coordinate on the monitor corresponding to point P,
It is.
[0040]
Now, in FIG. 6, focusing on a plane that includes the x-axis and is perpendicular to the xy plane, the lower diagram in FIG. 7 is obtained. In the lower diagram of FIG. 7, the z-axis is perpendicular to the drawing surface, and N is the direction upward from the drawing surface.
[0041]
In the lower diagram of FIG. 7, a straight line 22 connecting each point S on the tomographic image 23 and the origin O is extended to the rotating body curved surface 21a, and an intersection point with the rotating body curved surface 21a is defined as Q. In this method, the points on the tomographic image 23 are equally spaced, but the angular coordinates are not equally spaced. Interpolation calculation may be used to obtain angular coordinates at equal intervals.
[0042]
On the other hand, assuming a unit sphere centered at the origin O and dividing the center angles (α, ω) at equal angles to find the intersection of the straight line OQ and the tomographic image 23, conversely, Although the points are not evenly spaced, this can also be calculated by interpolation from neighboring points on the tomographic image 23.
[0043]
In any method, all of the angle coordinates on the rotating body 21 and the intersection point Q on the rotating body curved surface 21a can be obtained, and the obtained points are used for calculation to the secondary projection plane 4.
[0044]
Further, the distance from the origin O to each point on the tomographic image 23 is used at the time of shading. As a shading method using the distance, for example, there is a shading method in which a point closer to the viewpoint e is brighter and a farther point is darker.
[0045]
In FIG. 7, a point S (x₀, Y₀, Z₀), The angle θ between the straight line OS (straight line 22) connecting the origin O and the point S and the xy plane is
[0046]
[Expression 1]

[0047]
Given in.
[0048]
Therefore, the angle ω made with the z-axis is
[0049]
[Expression 2]

[0050]
Given in.
[0051]
The azimuth angle α of the straight line OS from the x axis is
α = tan^-1(Y₀/ X₀) ……… (3)
Given in.
[0052]
On the other hand, regarding the rotator 21 in FIG. 7, a cross section including the straight line OQ and the straight line ON is as shown in FIG. Now, in FIG. 8, let the curve which becomes the origin of the rotary body 21 be f (z). A straight line passing through the origin O and having an angle φ from the z axis is
x = ztanφ (4)
Given in.
[0053]
The coordinate on the z-axis of the intersection Q of both is
f (zQ) = zQtanφ (5)
Given in.
[0054]
Next, assuming that the length of the curve from the vertex of the curve f (z) (the vertex N of the rotating body 21) to the point Q is L, L is
[0055]
[Equation 3]

[0056]
Is obtained by the line integral of. In equation (6), f ′ (z) is the first derivative of the curve f.
[0057]
As an example, for a sphere of radius R,
[0058]
[Expression 4]

[0059]
become that way.
[0060]
Accordingly, in FIG. 6, the length L of the curve NQ when the primary projection curved surface 21 a is a sphere is expressed by coordinates (x₀, Y₀, Z₀)Using,
[0061]
[Equation 5]

[0062]
Given in.
[0063]
By calculating each point on the tomographic image 23 by the equation (7), the position of the primary projection image on the curved surface (spherical surface) of the sphere can be obtained. That is, the coordinates on the spherical surface of the primary projection image are all given by (α, L).
[0064]
Further, the point S (x on the tomographic image 23 from the origin O₀, Y₀, Z₀) To)
[0065]
[Formula 6]

[0066]
Given in. This distance is used when shading.
[0067]
In this way, all corresponding points on the tomographic image 23 are projected onto the rotator curved surface 21a, and these points are used for reprojection (secondary projection) onto the secondary projection plane 4.
In the upper diagram of FIG. 7, a curved line 40 is a cross section passing through the rotation axis of the rotating body curved surface 21a, and the others are the same as in FIG.
[0068]
FIG. 9 shows an example of rotating body curved surface / planar coordinate conversion to the secondary projection plane 4. In FIG. 9, the distance in the radial direction from the origin N is represented by L, the angle (azimuth angle) from the reference straight line w determined from the origin N is represented by α, and polar coordinates are used. Thus, the primary projection image on the rotating body curved surface 21a is represented by the distance L from the origin N and the angle α. This may be re-projected to the X and Y coordinates and displayed on the monitor.
[0069]
In FIG. 9, a rectangular area 50 is an example of a monitor display area, and this area 50 corresponds to the number of display memories of a monitor (not shown), for example, 512 × 512 pixels or 1024 × 1024 pixels. The rectangular region 50 including how many concentric circles shown in the drawing can be arbitrarily determined.
[0070]
Use X for the X and Y coordinates on the monitor.
X = ξ₁= L'cosα ......... (9)
Y = η₁  = L'sinα ......... (10)
Given by.
[0071]
This η₁, Ξ₁Becomes the pixel address on the display memory. That is, the point Q of the primary projection curved surface 21a is a point (η on the monitor screen).₁, Ξ₁), And thus, display by rotating body curved surface / planar coordinate conversion is performed.
[0072]
FIG. 10 shows another example of rotating body curved surface / planar coordinate conversion to the secondary projection plane 4. Here, a method of projecting parallel to the z-axis so that the image on the rotating body curved surface 21a is translated as it is to the secondary projection plane 4 when reprojecting onto the secondary projection plane 4 is shown. According to the above, when a sphere is used as the rotating body 21, as shown in FIG. 11, the central portion of the image is enlarged on the monitor. In FIG. 11, the same reference numerals as those in FIG. 9 denote the same or corresponding parts.
[0073]
Each point of the primary projection image on the rotator curved surface 21a in FIG. 10 is converted into the secondary projection plane 4 using the distance OQ from the rotation axis (z axis) to the rotator curved surface 21a. The conversion formula to the secondary projection plane 4 is
L ′ = OQ · sin ω (12)
Give in.
Here, as shown in FIG. 6, ω is an angle formed by a straight line connecting the viewpoint e and the projection target and the z axis. The azimuth angle is used as it is.
[0074]
There are a method of performing shading on the rotating body curved surface (primary projection curved surface) 21 a and a method of performing on the secondary projection plane 4. Since the distance from the viewpoint e which is the basis of the calculation to each point on the tomographic image 23 is known, either method is possible.
[0075]
For this shading, a method of adding shading according to the distance from the origin O to each point of the projection target, a method of changing the luminance according to the angle between each projection line (ray) and the projection target surface, A method of performing luminance conversion according to an angle formed by the center line and the projection target surface, a method of performing luminance conversion like a contour line according to a distance from the origin O, and the like are used.
Also, a method of performing volume rendering processing using the transmittance of the tissue to be projected with respect to the light beam from the viewpoint e may be used.
[0076]
12 to 21 are diagrams for explaining an example of a method for constructing a three-dimensional image by applying the above-described projection method from a plurality of tomographic images 23 (23A to 23H). 13 to 14 and 15 to 21, first, the tomographic image 23 is first projected onto the rotator curved surface 21 a, and then to the secondary projection plane 4 by the rotator curved surface / planar coordinate conversion. It will be reprojected (secondary projection).
[0077]
First, the rotating body curved surface / planar coordinate conversion in the three-dimensional image construction method will be described.
In the example shown in FIG. 12, a coordinate system is adopted so that the great circle 20 of the rotating body curved surface 21a, the tomographic image 23 plane, and the secondary projection plane 4 are parallel to each other for the sake of simplicity. The center line of the rotating body curved surface 21a is set parallel to the x axis.
[0078]
In FIG. 12, the position of the viewpoint e is the origin O of the three-dimensional coordinate system (x, y, z).
Also,
21 is a rotating body
x, y, z are axes of the three-dimensional coordinate system (x, y, z),
Q is a point on the rotating body curved surface 21a corresponding to the S point on the tomographic image 23;
S ′ is a point where the straight line 22 connecting the viewpoint e and the point Q intersects with the tomographic image 23A,
N is the apex of the rotating body 21,
20 is an xy plane (z =₀),
4 is the secondary projection plane,
w is the reference straight line (reference axis) of the rotation angle on the xy plane,
d₀Is the distance on the z-axis of the tomographic image 23A,
It is.
[0079]
Now, a plane (orthogonal plane) passing through the viewpoint e and orthogonal to the secondary projection plane 4 is considered. FIG. 13 is a diagram on the orthogonal plane in which tomographic images 23D to 23H are prepared in addition to the tomographic images 23A to 23C in FIG.
[0080]
In FIG. 13, tomographic images 23A to 23H are tomographic images obtained at equal intervals in the same direction with respect to the same projection target (or object). It is written. Note that the tomographic images 23A to 23H do not necessarily have equal intervals.
[0081]
When the organ regions B1, B2, B3, and B4 are first projected onto the rotator curved surface 21a, which is a primary projection curved surface, and further converted into a rotator curved surface / plane coordinate and projected onto the secondary projection plane 4, a secondary projection is performed. Data B1 ′, B2 ′, B3 ′, and B4 ′ are obtained.
[0082]
When writing the secondary projection data B1 ′, B2 ′, B3 ′, and B4 ′ to a display memory (not shown), in order to produce a three-dimensional effect, the projection data existing farther from the viewpoint e is used. Write first, and overwrite the projection data near it later. Thereafter, a shading process is performed.
[0083]
FIG. 14 shows a more general form than FIG. 13, and shows an example in which the direction of the vertex N of the rotator 21 is not orthogonal to the plane of the tomographic image 23 and is rotated by an arbitrary angle around the y axis. In this case, the primary projection image on the rotating body curved surface 21a obtained by the interpolation calculation is reprojected (secondary projection) onto the secondary projection plane 4.
[0084]
FIG. 15 is a diagram for explaining the rotator curved surface / planar coordinate conversion when the viewpoint e, the tomographic image 23, and the primary projection curved surface 21a have a more complicated positional relationship, and Si (xi, It shows that the primary projection image of yi, di) finally becomes a point Pi (X, Y) on the secondary projection plane 4.
[0085]
In FIG. 15, in projecting the tomographic image 23 onto the rotator curved surface 21a by the projection method of the present invention, conversion of each pixel coordinate of the tomographic image 23 into coordinates on the rotator curved surface 21a is performed as follows.
[0086]
Here, in order to give generality to the description, the positional relationship is set as follows.
In FIG. 15, the tomographic image 23 to be projected is in a plane orthogonal to the z-axis, and is d from the origin O.₀It is assumed that the distance is. The distance d from the origin O for a plurality of tomographic images 23.₀Changes. However, the plurality of tomographic images 23 do not have to be equally spaced.
[0087]
Further, a center N of an arbitrary projection direction different from the z axis is determined from the viewpoint e on the origin O, the direction is set to ON, and the straight line ON is set to h. Then, an arbitrary line t starting from the point of the distance R from the origin O is drawn. The line t may be a curve, a straight line, or a straight line parallel to the straight line h.
[0088]
Next, a rotating body 21 in which the line t is rotated about the straight line h is assumed. Then, a tomographic image (projection target) 23 is projected onto the curved surface 21 a of the rotating body 21. On the other hand, a secondary projection plane 4 is provided separately from the rotating body curved surface 21 a, and orthogonal coordinate systems X and Y are determined on the plane 4. Each point (X, Y) on the plane 4 becomes each display coordinate on the monitor.
[0089]
FIG. 16 shows the coordinate relationship on the rotating body curved surface 21a. Now, consider a unit sphere C centered on the origin O in the coordinate system in FIG. And turn on z-axis direction₁And this ON₁Is the reference direction. Moreover, it is ON with respect to the coordinate system in FIG.₁ON with a new direction with any direction changed₂And This ON₂By arbitrarily changing the direction, it is possible to display a three-dimensional image centered on that direction.
[0090]
Next, a straight line connecting the origin O and the point q on the unit sphere C is extended, and an intersection of the straight line and the rotating body curved surface 21a is set as Q. Due to the nature of the rotating body 21, the distance L on the surface from the vertex N to the point Q is expressed by the following equation (6) using a line t that is the origin of the rotating body 21 in the plane v including the points N, O, Q. It can obtain | require by the integration shown by.
[0091]
In FIG. 16, the straight line is ON.₂On the other hand, it goes through the origin O₂20 ′, a plane (large circle) passing through the center of the unit sphere C orthogonal to₂The rotation angle from the x-axis of₀, The angle made with the z axis is ω. In other words, straight line ON₁From the z axis to ω, from the x axis to α₀Only a straight line rotated is ON₂It is. Where the x axis is the angle α in the xy plane₀The straight line rotated only by the angle is defined as a new azimuth angle reference axis Ox ′.
[0092]
Point q₁, Q₂Is a point on the unit sphere C, and the straight line is ON₂And an azimuth angle λ measured from the reference line Ox ′ with respect to the plane 20 ′ and an angle β from the plane 20 ′. Similarly, these points are represented by an azimuth angle α from the x-axis and an angle δ from the plane 20 in the original coordinate system. Straight line ON₂The projection image onto the primary projection curved surface 21a with the rotation axis as the rotation axis may be calculated by changing λ and β at equal intervals.
[0093]
Now point q₁The angles in the original coordinate system are α and δ, respectively, and the inclination of the plane 20 ′ from the plane 20 is ε. And straight line ON₂And the rotation angle of the straight line Ox ′ in the xy plane is α.₀In addition, an angle from the reference axis Ox ′ in the xy plane of the point Q is α ′. Further, in the new coordinate system based on the reference axis Ox ′ and the straight line ON on the plane 20 ′, the point Q₁Are λ and β, respectively.
[0094]
In this way, ε = ω, and due to the nature of the spherical triangle, the following between λ, β and α, δ, ω
cos β cos λ = cos δ cos α ′
cosβsinλ = sinδsinω + cosδsinα′cosω (13)
sin β = sin δ cos ω−cos δ sin α′sin ω
The following relational expression holds.
[0095]
To specify α and δ in the original x, y, and z coordinate systems by specifying λ and β after conversion, λ, β and α, δ are reversed.
cos δ cos α ′ = cos β cos λ
cos δ sin α ′ = − sin β sin ω + cos β sin λ cos ω (14)
sinδ = sinβcosω + cosβsinλsinω
May be used.
If the coordinates do not coincide with each other, they can be obtained by interpolation from surrounding points.
[0096]
Also, from FIG. 16, between the rotation angle α in the original coordinate system and the new coordinate system,
α = α ′ + α₀                                            ……… (15)
The following relational expression holds.
[0097]
That is, the coordinates before rotation and the coordinates after rotation of an arbitrary point on the spherical surface are related by Expressions (13) to (15). With these equations, the coordinates after rotation can be easily calculated by correcting the coordinate system before rotation. The reverse is also possible.
[0098]
Two angles α and δ are given to each point S on the tomographic image, and β and λ are obtained using equation (13). Since the intersection Q between the straight line passing through the point S from the origin O and the rotating body curved surface 21a may be considered in the cross section (plane v) including the straight line h and the point S, the straight line ON is newly considered as the x axis.
f (x) = xtanφ (16)
It is obtained by
Where φ is straight line Oz and straight line ON₂Is the angle between
[0099]
Further, the surface distance L (= NQ) from the vertex N to the point Q of the rotator 21 may be calculated by line integration on the plane v using the properties of the rotator 21. This calculation may be performed using equation (6).
On the other hand, the angle λ in the conversion to the secondary projection plane 4 is obtained from the equation (13).
[0100]
An image on the secondary projection plane 4 in FIG. 17 is obtained using λ and L obtained as described above. FIG. 17 shows the polar coordinate format, but the conversion to the Cartesian coordinate system is
x = η = L cos λ (17)
y = ξ = Lsin λ (18)
It is obtained by
Then, the projection image may be displayed on the monitor using η and ξ in the equations (17) and (18).
[0101]
In this way, by scanning the tomographic image 23 by changing the angles λ and β with the straight line connecting the vertex N and the origin O as the central axis, the intersection image is obtained by scanning the tomographic image 23 on the rotating body curved surface 21a. I want. Conversely, each point of the tomographic image 23 may be scanned to obtain a primary projection image on the rotating body curved surface 21a. In this way, the primary projection image obtained on the primary projection curved surface 21a is reprojected (secondary projection) onto the secondary projection plane 4 by rotating body curved surface / plane coordinate conversion, and is written in a display memory (not shown). Thus, the secondary projection image is displayed on the monitor.
[0102]
FIG. 18 shows a projection image onto the rotating body curved surface 21a displayed at angles λ and β. When these two angles are assigned to the X and Y coordinates and displayed on the monitor, the conversion display is 1: 1. That is, by using these data and converting to the secondary projection plane 4, a three-dimensional image in which the peripheral portion is enlarged is obtained, which is partially different from the visual image.
[0103]
19 to 21 show examples in which the distance L in FIG. 17 is converted by a fixed function to obtain L ′ and displayed as an image. In FIG. 19, a range having a distance from the vertex N is enlarged and displayed. 19 to 21, n on the L ′ axis corresponds to the number of display pixels on the monitor and may be ½ of the maximum number of pixels. In FIG. 20, by changing the coefficient k, the enlargement ratio of the distance from the vertex N changes, and the center portion can be arbitrarily enlarged. In FIG. 21, the region of interest can be enlarged, but the front and back thereof change smoothly, resulting in an easy-to-view image.
[0104]
In the tentative plan described above with reference to FIGS. 1 to 21, the type of rotating body used as the primary projection curved surface is not limited.
Hereinafter, the case where the rotator is a sphere, the case of a cone, the case of an ellipse rotator, the case of a parabola rotator, and the case of a cylinder will be described in detail.
In addition,Except for the case where the rotator 21 is a sphere, only the projection method onto the primary projection curved surface will be described, but the reprojection (secondary projection) processing onto the secondary projection plane 4In the tentative plan (Figures 1 to 21)Each is the same.
[0105]
Example 1: When the rotating body 21 is a sphere
When a circle with a radius R about the viewpoint e (origin O) is rotated, a sphere as shown in FIG. 22 is obtained. The equation for the semicircle t is
[0106]
[Expression 7]

[0107]
Given in.
[0108]
A rotating body having the x-axis of the semicircle t as a rotation axis is a sphere C. In this case, primary projection is performed on the curved surface of a sphere, that is, a spherical surface. Primary projection onto a curved surface (spherical surface) of a sphere formed by rotating a circle centered on a point at a distance a on the x axis from the origin O is also possible.
The primary projection onto the spherical surface is suitable when it is desired to view a wide range of three-dimensional images at once for an organ (object) that is highly isotropic with respect to the distance from the viewpoint e.
[0109]
Since the equation of the straight line 20 in FIG. 22 is y = x tan φ, the coordinate x on the x axis₀Becomes R cos φ.
The distance L on the circumference from the vertex N to the point Q on the sphere C is obtained by using the first derivative of the equation (6) and the equation (19).
[0110]
[Equation 8]

[0111]
From the calculation.
[0112]
The primary projection and secondary projection when the rotating body is a sphere will be described in detail below. 23 to 25 show an embodiment in which the rotating body is a sphere. Of these, FIG. 23 shows an overview of the whole.
[0113]
In FIG. 23, first, a viewpoint e is set at an arbitrary position with respect to an organ 23a that is an object to be projected.
Then, consider a sphere 21 centered on this viewpoint e. The radius of the sphere 21 is arbitrary, but the range of the organ 23a to be projected is determined by determining the radius. If the radius is increased, a wide range of organs 23a can be projected, but the display area of the monitor is finite, and the minute portions of the organs 23a are not known, so an appropriate size is selected.
[0114]
Next, a radial straight line (light ray) is drawn from the viewpoint e, and the intersection of each radial straight line with each point of the organ 23a is projected onto the intersection of the radial straight line with the curved surface (spherical surface) 21a. Then, primary projection (center projection) with the spherical surface 21a as the primary projection curved surface is performed.
[0115]
Further, the great circle 20 is determined as an arbitrary plane passing through the viewpoint e, a perpendicular line h is set from the center thereof, and a plane perpendicular to the perpendicular line h is defined as T. The plane T is a secondary projection plane 4 on which a primary projection image on the spherical surface 21a is projected by a rotating body curved surface (here, a spherical surface) / plane coordinate conversion.
By changing the direction of the pair of plane T and great circle 20 passing through the center, projection images from various directions can be created.
[0116]
FIG. 24 shows the relationship between the tomographic image 23, the primary projection curved surface 21a, the secondary projection plane 4, the projection target 5 of the organ 23a, and the viewpoint e in the plane including the perpendicular h. The primary projection curved surface 21 a is a curved surface of the sphere 21. The viewpoint e is the center of the sphere 21 and also the origin O of the three-dimensional coordinates. Here, the projection target 5 of the organ 23a, which is the projection target, is projected onto the spherical surface 21a.
[0117]
In this case, the spherical surface 21a surrounding the origin O has a solid angle 4π (sphere), but this angle is arbitrary. The primary projection image projected onto the spherical surface 21a is reprojected (secondary projection) onto the plane 4 by developing the spherical surface 21a onto the secondary projection plane 4 by rotating body curved surface / plane coordinate conversion.
[0118]
FIG. 25 is an example of the secondary projection plane 4, and the primary projection image projected on the spherical surface 21 a with the center as the vertex N is from the distance r using the arc from the vertex N and the arbitrarily determined reference straight line w. The plane 4 is developed by two parameters, the angle α. In the case of a perfect sphere, the sphere 21 is a sphere having a solid angle 4π centered on the origin O, but may be a hemisphere (solid angle 2π) or a sphere having an arbitrary solid angle.
[0119]
On the other hand, in the development from the spherical surface 21 a to the plane 4, it is arbitrary how far the solid angle of the spherical surface 21 a is developed to the plane 4. In the example of FIG. 25, the solid angle is 2π. It is up to the solid angle of 2π that coincides with the intuitive visual image. If it exceeds 2π, the vertex on the opposite side on the secondary projection plane 4 is developed over the entire circumference and does not coincide with the visual image. The vertex N that is the center of the development on the plane 4 is arbitrarily given.
[0120]
Hereinafter, an embodiment of reprojection (spherical / planar coordinate conversion) of the primary projection image from the spherical surface 21a to the plane 4 will be described with reference to FIGS.
First, a description will be given with reference to FIG. In FIG. 26, 21a is a curved surface of a sphere 21 having a radius R and having a center at a viewpoint e on the origin O of each axis x, y, z of the three-dimensional coordinate system. Further, the great circle 20 of the sphere 21 has an xy plane (z =₀).
[0121]
Reference numeral 4 denotes a projection plane composed of a plane orthogonal to the z-axis, which is a secondary projection plane on which data (primary projection image) after the primary projection onto the spherical surface 21a is further projected (reprojected). This secondary projection plane 4 corresponds to the display screen of the monitor.
[0122]
Also,
23 is a tomographic image,
h is a perpendicular drawn from the viewpoint e to the secondary projection plane
Q is a point (α, ω) on the spherical surface 21a due to a light ray emitted from the viewpoint e,
N is the apex of the sphere 21,
S is a point on the tomographic image 23, and the intersection with the straight line OQ,
R is the radius of the spherical surface 21a,
P is a point (η, ξ) on the orthogonal coordinate system on the secondary projection plane 4 corresponding to the Q point,
α is a rotation angle measured from the x-axis on the xy plane,
ω is the angle between the line connecting the Q point and the origin O and the z axis,
L is an arc NQ on the spherical surface 21a,
L ′ is a line segment having the same length as L, and a straight line N′P on the secondary projection plane 4,
η is the X coordinate on the monitor corresponding to point P,
ξ is the Y coordinate on the monitor corresponding to point P,
It is.
[0123]
Now, in FIG. 26, focusing on a plane passing through the origin O and perpendicular to the xy plane, the lower diagram in FIG. 27 is obtained.
In the lower diagram of FIG. 27, the z-axis is perpendicular to the drawing surface, and N is the direction upward from the drawing surface. The semicircle 40 is a cross section of the spherical surface 21a.
[0124]
27, a straight line connecting each point on the tomographic image 23 and the origin O is extended to the spherical surface 21a, and an intersection point with the spherical surface 21a is defined as Q. In this method, the points on the tomographic image 23 are equally spaced, but the angular coordinates are not equally spaced. Interpolation calculation may be used to obtain angular coordinates at equal intervals.
[0125]
On the other hand, in the method of dividing the semicircle 40 at an equal angle and obtaining the intersection of the straight line OQ and the tomographic image 23, the points on the tomographic image 23 are not equally spaced, but this is also the vicinity on the tomographic image 23. It can be calculated by interpolation from the points.
[0126]
In any method, all the angle coordinates on the spherical surface (primary projection curved surface) 21 a can be obtained, and the obtained points are used for the calculation to the secondary projection plane 4.
[0127]
In FIG. 27, a point S (x₀, Y₀, Z₀), The angle θ between the straight line OS connecting the origin O and the point S and the xy plane is
[0128]
[Equation 9]

[0129]
Given in.
[0130]
Therefore, the angle ω made with the z-axis is
[0131]
[Expression 10]

[0132]
Given in.
[0133]
On the other hand, the azimuth angle α of the straight line OS from the x-axis is
α = tan^-1(Y₀/ X₀) ……… (23)
Given in.
[0134]
Therefore, the length L of the arc NQ in FIG.
[0135]
## EQU11 ##

[0136]
Given in.
[0137]
For each point on the tomographic image 23, the position of the primary projection image on the spherical surface is obtained by calculating with the equation (24). That is, the coordinates on the spherical surface of the primary projection image are all given by (α, L).
[0138]
The distance rs from the origin O to the point S on the tomographic image 23 is
[0139]
[Expression 12]

[0140]
Given in. This distance is used when shading.
In this way, all corresponding points on the tomographic image 23 are projected onto the spherical surface 21a, and these points are used for reprojection onto the secondary projection plane 4.
[0141]
FIG. 28 shows an example of spherical / planar coordinate transformation to the secondary projection plane 4. In FIG. 28, the distance in the radial direction centered on the vertex N is represented by L, the angle (azimuth angle) from the reference straight line w determined from the vertex N is represented by α, and the equidistant (the length of the arc is made equal). , Using equiangular coordinates. Thus, the primary projection image on the spherical surface 21a is represented by the distance L ′ from the vertex N and the angle α. This may be re-projected (secondary projection) to the X and Y coordinates and displayed on the monitor.
[0142]
In FIG. 28, a rectangular area 50 is an example of a monitor display area, and this area 50 corresponds to the number of monitor display memories, for example, 512 × 512 pixels or 1024 × 1024 pixels. The rectangular area 50 is an area including how many concentric circles shown in the figure, and it is arbitrarily determined whether to display on the monitor.
[0143]
As an example, when the entire solid angle 2π (hemisphere) is displayed on the monitor as follows, the range of the length of the circumference π · R of the hemisphere is made to correspond to the display length of the monitor screen.
[0144]
Use X for the X and Y coordinates on the monitor.
NQ = L ′ = Rω (26)
X = ξ₁= L'cosα ......... (27)
Y = η₁  = L'sinα ......... (28)
Given by.
[0145]
This η₁, Ξ₁Becomes the pixel address on the display memory. That is, the point P on the secondary projection plane 4 is a point (η₁, Ξ₁In other words, display by spherical / planar coordinate conversion is performed.
[0146]
In FIG. 28, the maximum value of the radius OP is πR when the solid angle = 2π, and 2πR when the solid angle = 4π, but in the case of an arbitrary solid angle, the length of the OP is set according to the solid angle. Change.
[0147]
FIG. 29 shows another example of spherical / planar coordinate conversion to the secondary projection plane 4. Here, when re-projecting onto the secondary projection plane 4, a method of projecting the primary projection image on the spherical surface 21a in parallel with the z-axis so as to be translated into the secondary projection plane 4 as it is is shown. As shown in FIG. 30, the central portion of the image is enlarged on the monitor.
[0148]
Each point of the primary projection image on the spherical surface (primary projection curved surface) 21a in FIG. 29 is represented by the arc length L ′ from the vertex N and the azimuth α, but the conversion formula to the secondary projection plane 4 is L ″ = L ′ · sin ω (29)
Give in. Here, as shown in FIG. 26, ω is an angle formed by a straight line connecting the viewpoint e and the projection and the z axis. The azimuth angle is used as it is.
[0149]
There are a method of performing shading on the primary projection curved surface 21a and a method of performing on the secondary projection plane 4, and either method may be used.
For shading, a method of adding shading according to the distance from the origin O to each point of the projection object, a method of converting luminance according to the angle between each projection line and the object surface, A method of performing luminance conversion according to an angle formed with the object surface, a method of performing luminance conversion like a contour line according to a distance from the origin O, and the like are used.
Further, a method of performing volume rendering processing using the transmittance of the tissue of the projection with respect to the light beam from the viewpoint e may be used.
[0150]
FIGS. 31 to 37 are diagrams for explaining an example of a method for constructing a three-dimensional image from a plurality of tomographic images 23 by the above-described projection method. 32 to 33 and 34 to 37, the tomographic image 23 is first projected on the spherical surface 21a, and then re-projected onto the secondary projection plane 4 by spherical / planar coordinate conversion.
[0151]
First, spherical / planar coordinate conversion in the above three-dimensional image construction method will be described.
In the example shown in FIG. 31, the coordinate system is taken so that the great circle 20 of the spherical surface 21a, the tomographic image 23 plane, and the secondary projection plane 4 are parallel to each other for the sake of simplicity. The center line of the spherical surface 21a is parallel to the x axis.
[0152]
In FIG. 31, the position of the viewpoint e is the origin O (0, 0, 0) of the three-dimensional coordinate system (x, y, z).
Also,
x, y, z are axes of the three-dimensional coordinate system (x, y, z),
Q is a point on the spherical surface 21a corresponding to the S point on the tomographic image 23;
S ′ is a point where the straight line 22 connecting the viewpoint e and the point Q intersects with the tomographic image 23A,
N is the apex of the sphere 21,
20 is an xy plane (z =₀),
4 is the secondary projection plane,
w is the reference straight line of the rotation angle on the xy plane,
d₀Is the distance on the z-axis of the tomographic image 23A,
It is.
[0153]
Now, a plane (orthogonal plane) passing through the viewpoint e and orthogonal to the secondary projection plane 4 is considered. FIG. 32 shows a diagram on the orthogonal plane in which tomographic images 23D to 23H are prepared in addition to the tomographic images 23A to 23C in FIG.
[0154]
In FIG. 32, tomographic images 23A to 23H are tomographic images obtained at equal intervals in the same direction with respect to the same projection target (or object), and organ regions B1, B2, B3, and B4 are emphasized in the tomographic image 23. It is written. Note that the tomographic images 23A to 23H do not necessarily have equal intervals.
[0155]
When the organ regions B1, B2, B3, B4 are first projected onto the spherical surface 21a, and further converted into spherical / planar coordinates and projected onto the secondary projection plane 4, B1 ′, B2 ′, B3 ′, B4 ′ and Become.
[0156]
Here, when writing projection data (here B1 ′, B2 ′, B3 ′, B4 ′) to a display memory (not shown), it exists farther from the viewpoint e in order to produce a three-dimensional effect. Projection data to be written is written first, and projection data near it is overwritten later. Thereafter, a shading process is performed.
[0157]
FIG. 33 is shown in a more general manner than FIG. 32, and is an example in which the direction of the vertex N of the sphere 21 is not orthogonal to the plane of the tomographic image 23 but rotated by an arbitrary angle around the y axis. In this case, the primary projection image on the spherical surface 21 a obtained by the interpolation calculation is reprojected on the secondary projection plane 4.
[0158]
FIG. 34 is a diagram for explaining the above-described spherical / planar coordinate conversion when the viewpoint e, the tomographic image 23, and the primary projection curved surface 21a have a more complicated positional relationship, and Si (xi, yi) on the tomographic image 23. , Di) indicates that the primary projection image finally becomes a point Pi (X, Y) on the secondary projection plane 4.
[0159]
In FIG. 34, when the tomographic image 23 is projected onto the spherical surface 21a by the projection method of the present invention, the pixel coordinates of the tomographic image 23 are converted into coordinates on the spherical surface 21a as follows.
[0160]
Here, in order to give generality to the description, the positional relationship is set as follows.
In FIG. 34, a tomographic image 23 as a projection is in a plane orthogonal to the z-axis, and d from the origin O.₀It is assumed that the distance is. The distance d from the origin O for a plurality of tomographic images 23.₀Changes. However, the plurality of tomographic images 23 do not have to be equally spaced.
[0161]
Further, a center N of an arbitrary projection direction different from the z axis is determined from the viewpoint e on the origin O, the direction is set to ON, and the straight line ON is set to h. Then, a secondary projection plane 4 orthogonal to the straight line h is given from the origin O at a point of radius R of the sphere 21. The coordinates N of the intersection of the plane 4 and the straight line h are the central coordinates of the orthogonal coordinate systems X and Y on the secondary projection plane 4. The point (X, Y) on the plane 4 becomes display coordinates on the monitor.
[0162]
FIG. 35 shows the coordinate relationship on the spherical surface 21a. Now, for the coordinate system in FIG.₁ON a new position with any direction changed₂And Note that by arbitrarily changing the direction of ON2, a three-dimensional image centered on that direction can be displayed.
[0163]
In FIG. 35, the straight line is ON.₂On the other hand, it goes through the origin O₂20 ', ON the plane (large circle) orthogonal to₂The rotation angle from the x-axis of₀, The angle made with the z axis is ω. In other words, straight line ON₁From the z axis to ω, from the x axis to α₀The straight line rotated only by ON2 is ON2. Where the x axis is the angle α in the xy plane₀The straight line rotated only by the angle is defined as a new azimuth angle reference axis Ox ′.
[0164]
Point Q₁, Q₂Is a point on the spherical surface 21a and is a straight line ON₂Is expressed by an azimuth angle λ measured from the reference line Ox ′ and an angle β from the plane 20 ′. Similarly, these points are represented by an azimuth angle α from the x-axis and an angle δ from the plane 20 in the original coordinate system. The projected image on the primary projection curved surface 21a with the straight line ON2 as the rotation axis may be calculated by changing λ and β at equal intervals.
[0165]
Point Q now₁The angles in the original coordinate system are α and δ, respectively, and the inclination of the plane 20 ′ from the plane 20 is ε. And straight line ON₂And the angle between the z axis and ω, the rotation angle in the xy plane is α₀In addition, the angle of the Q point from the x ′ axis is α ′. Furthermore, the angles of the S point in the new coordinate system are β and λ, respectively.
[0166]
In this way, ε = ω, and between β, λ and α, δ, ω
cos β cos λ = cos δ cos α ′
cosβsinλ = sinδsinω + cosδsinα′cosω (30)
sin β = sin δ cos ω−cos δ sin α′sin ω
The following relational expression holds.
[0167]
In this case, when α and δ of the original x, y, and z coordinate systems are obtained by specifying λ and β after conversion, β, λ and α, δ are reversed.
cos δ cos α ′ = cos β cos λ
cos δ sin α ′ = − sin β sin ω + cos β sin λ cos ω (31)
sinδ = sinβcosω + cosβsinλsinω
May be used.
If the coordinates do not match the corresponding position, it is obtained by interpolation from surrounding points.
[0168]
In addition, between the rotation angle α in the original coordinate system and the new coordinate system, α ′ = α−α₀Because
α = α ′ + α₀                                          ......... (32)
The following relational expression holds.
[0169]
On the other hand, conversion from β, λ to the secondary projection plane 4 is obtained as follows.
That is, when obtaining the arc length l from β,
l = R (π / 2−β) (33)
become that way.
A secondary projection image may be obtained using these l and β and displayed on a monitor.
[0170]
FIG. 36 shows an image when the projection image onto the spherical surface 21a is displayed by α and δ. Although it may be displayed on the monitor as it is, the distortion increases as δ approaches π / 2 and does not match the intuitive visual image. Therefore, a three-dimensional image that matches the visual image is obtained by reprojecting to the secondary projection plane 4 using the image data of α and δ.
[0171]
A three-dimensional image re-projected on the secondary projection plane 4 centered on an arbitrary direction can be obtained from the equation (31) using the new azimuth angles λ and β as described above. In addition, the range of the primary projection image data used for reprojection differs depending on how many areas of the primary projection curved surface 21a are displayed on the secondary projection plane 4.
[0172]
FIG. 37 shows the range of projection image data used when calculating a three-dimensional image centered in an arbitrary direction using the projection image data on the entire primary projection curved surface 21a. In FIG. 37, the point P₁~ P₃Represents the center N of an arbitrary direction vector ON. For example, P₁Shows the range of data used in the construction of the three-dimensional image on the secondary projection plane 4 with the azimuth angle α = π / 2 and the angle ω = 0 from the plane 20 in the direction of the center N in the original coordinate system. .
[0173]
From the equations (21) to (33), the coordinates on the primary projection curved surface 21a of each point on each tomographic image 23 are obtained. Since there are a plurality of tomographic images 23, there may be a plurality of contour points, but by writing to the display memory from a position far from the viewpoint e, the projection image closest to the viewpoint e is displayed.
[0174]
In this way, by changing the angles β and λ with the apex as the center N, a primary projection image is obtained on the spherical surface 21a. The primary projection image is reprojected onto the secondary projection plane 4 by the spherical / planar coordinate conversion and written in the corresponding image display memory, whereby the secondary projection image is displayed on the monitor.
[0175]
Example 2: When the rotating body 21 is a cone
As shown in FIG. 38, when a straight line t38 having an angle Ω with the x-axis is drawn from a point R on the x-axis from the viewpoint e (origin O), and the straight line t38 is rotated about the x-axis as a rotation axis, a cone is obtained. It becomes the body 21. In this case, primary projection is performed on the curved surface 21 a of the cone 21. Such projection onto the conical curved surface 21a is suitable for projection of a tubular organ (object).
[0176]
In this case, the straight line t38 is
y = −xtanΩ + RtanΩ (34)
Given in.
[0177]
By using the linear equation y = x tan φ, the coordinate x₀Is
x₀= RtanΩ / (tanΩ + tanφ) (35)
Given in.
The distance L (= NQ) on the cone curved surface 21a is obtained by using the first derivative of the equation (34) as f ′ in the equation (6).
L = (R−x₀) SecΩ ………… (36)
It is obtained by
[0178]
Example 3: When the rotating body 21 is an elliptic rotating body
As shown in FIG. 39, when an ellipse t39 centered on the viewpoint e (origin O) is rotated about the x axis as a rotation axis, an ellipse rotating body 21 as shown in FIG. 40 is obtained. In FIG. 40, 21a is an elliptic rotating body curved surface, 40 is a projection target in an object, 41 is a primary projection image thereof, and N is a vertex taken on the x axis. In this case, primary projection is performed on the curved surface of the elliptical rotator 21. An ellipsoidal rotator 21 that does not have a center at the origin O and is centered on a point at a distance k from the origin O on the x axis is also possible.
[0179]
The projection onto the ellipsoidal rotator 21 is suitable for an organ (object) having a large number of planar portions. In FIG. 39, a and b are intersections with the x-axis and the y-axis, respectively. By changing a and b, a wide field of view can be obtained, or a three-dimensional image with a three-dimensional effect can be obtained.
[0180]
next
[0181]
[Formula 13]

[0182]
The elliptically rotating body 21 is given by rotating the curve given in (1) with the x axis as the rotation axis.
[0183]
By using the linear equation y = x tan φ, the coordinate x₀Is calculated. The distance L on the ellipsoidal rotator curved surface 21a can be obtained by calculation from the first derivative of the equations (6) and (37).
[0184]
Example 4: When the rotating body 21 is a parabolic rotating body
As shown in FIG. 41, when a parabola t41 shown by the equation (38) is subtracted from the viewpoint e (origin O) at a distance R on the x axis and rotated around the x axis as a rotation axis, FIG. A parabolic rotator 21 as shown is obtained. In this case, the primary projection is performed on the curved surface of the parabolic rotator 21. In FIG. 42, 21a is a parabolic rotator curved surface, 40 is a projection object in the object, 41 is a projection image thereof, and N is a vertex taken on the x-axis.
[0185]
This is suitable for constructing a three-dimensional image while making a virtual tubular hole in a relatively thick tubular organ (object) or an internal filled organ (object).
[0186]
On the parabolic rotator curved surface 21a, the distance L from the vertex N to the point Q is expressed by the following equation (6):
[0187]
[Expression 14]

[0188]
Calculated from the first derivative of.
[0189]
Example 5: When the rotating body 21 is a cylinder
As shown in FIG. 43, at a distance R on the x-axis from the viewpoint e (origin O),
y = R ……… (39)
When a straight line t43 shown in FIG. 6 is drawn and the straight line t43 is rotated about the x axis as a rotation axis, a cylinder 21 is obtained. In this case, primary projection is performed on the curved surface of the cylinder 21 as shown in FIG.
[0190]
In FIG. 44, 21a is a cylindrical curved surface, 40 is a projection target in an object, and 41 is a projection image thereof. In this case, the vertex N is not determined on the x axis, and an arbitrary point on the cylinder 21 may be determined.
[0191]
In the projection method using the cylindrical curved surface 21a, a tubular object is projected to display a three-dimensional image.TheThe distance L (= NQ) on the cylindrical curved surface 21a is measured from the position of the vertex N in this case.The
L = ( a + x ₀ ) Is required.
The “projection method using the cylindrical curved surface 21a” described above is suitable for displaying a three-dimensional image by projecting a tubular object. This method corresponds to the Mercator projection in cartography, Since the distortion in the vicinity is large, it is not suitable for organ observation.
For this reason, the present invention excludes “making the cylindrical surface a primary projection curved surface” from the proposed projection technique.
As described above, when the rotating body is a sphere as Example 1 in the tentative plan, when the rotating body is a cone as Example 2, when the rotating body is an elliptical rotating body as Example 3, The case where the rotating body is a parabolic rotator and the case where the rotating body is a cylinder as Example 5 have been considered. However, as described above, since a cylinder is excluded, an appropriate curved surface as a primary projection surface is “sphere, cone, ellipse”. This is a curved surface of a “rotating body, or parabolic rotating body”.
[0193]
FIG. 45 is a block diagram showing a hardware configuration example to which the projection method of the present invention can be applied. In FIG. 45, 50 is a CPU, 51 is a main memory, 52 is a magnetic disk, 53 is a display memory, 55 is a mouse controller, and these are connected to a common bus 57. The magnetic disk 52 stores a plurality of tomographic images, a program for executing calculation of the projection method of the present invention, and the like.
[0194]
The CPU 50 reads the plurality of tomographic images and the program for executing the calculation of the projection method of the present invention, performs calculation such as rotating body curved surface / planar coordinate conversion using the main memory 51, and sends the result to the display memory 53. And displayed on the CRT monitor 54. A mouse 56 connected to the mouse controller 55 designates the viewpoint, the position of the start and end of a straight line / curve, etc. at the time of computation such as rotating body curved surface / planar coordinate conversion. The image obtained by converting the rotating body curved surface / planar coordinates is stored in the magnetic disk 52 as necessary.
[0195]
Instead of the mouse controller 55 and the mouse 56, another pointing device controller or a pointing device such as a trackball may be used.
[0196]
【The invention's effect】
As described above, according to the present invention, a panoramic three-dimensional image can be obtained so that the inside of a wide range of an object can be seen, and the projection surface of the object can be determined depending on the shape of the object, the portion to be observed, or the observation purpose. By changing the shape arbitrarily, it is possible to obtain an easy-to-see three-dimensional image by the projection surface suitable for the shape of the object, the portion to be observed, or the observation purpose. In particular, when applied to the projection of tubular organs such as blood vessel walls and esophageal walls, there is an effect that it is possible to obtain a three-dimensional image as if it is seen inside these organs.
[Brief description of the drawings]
[Figure 1]TentativeIt is a flowchart which shows the example of the projection method.
[Figure 2]TentativeIt is explanatory drawing which shows the example of the projection method.
[Fig. 3]TentativeIt is a figure for demonstrating the detail of the projection method.
[Fig. 4]TentativeIt is a figure for demonstrating the detail of the projection method.
[Figure 5]TentativeIt is a figure for demonstrating the detail of the projection method.
[Fig. 6]TentativeIt is principle explanatory drawing of the rotary body curved surface / plane coordinate conversion in a projection method.
[Fig. 7]TentativeIt is principle explanatory drawing of the rotary body curved surface / plane coordinate conversion in a projection method.
[Fig. 8]TentativeIt is principle explanatory drawing of the rotary body curved surface / plane coordinate conversion in a projection method.
FIG. 9TentativeIt is explanatory drawing of an example of a rotary body curved surface / plane coordinate conversion in the projection method.
FIG. 10TentativeIt is explanatory drawing of the other example of the rotary body curved surface / plane coordinate conversion in a projection method.
FIG. 11TentativeIt is explanatory drawing of the other example of the rotary body curved surface / plane coordinate conversion in a projection method.
FIG. 12 From multiple tomographic imagesTentativeIt is a figure for demonstrating an example of the method of constructing a three-dimensional image by applying a projection method.
FIG. 13 From a plurality of tomogramsTentativeIt is a figure for demonstrating an example of the method of constructing a three-dimensional image by applying a projection method.
FIG. 14 From a plurality of tomographic imagesTentativeIt is a figure for demonstrating an example of the method of constructing a three-dimensional image by applying a projection method.
FIG. 15: From multiple tomographic imagesTentativeFIG. 6 is a diagram for explaining a rotating body curved surface / planar coordinate conversion in a case where a viewpoint, a tomographic image, and a primary projection curved surface have a more complicated positional relationship in an example of a method of constructing a three-dimensional image by applying a projection method. is there.
16 is a diagram showing a coordinate relationship on the rotating body curved surface in FIG. 15;
FIG. 17 is an explanatory diagram showing that an image on the secondary projection plane is obtained using λ and L in FIG. 16;
18 is a diagram showing a projected image on a rotating body curved surface by the angles λ and β in FIG.
FIG. 19 is a diagram illustrating an example in which L ′ is obtained by converting the distance L in FIG. 17 by a fixed function and an image is displayed.
20 is a diagram illustrating an example in which L ′ is obtained by converting the distance L in FIG. 17 by a certain function, and an image is displayed.
FIG. 21 is a diagram illustrating an example in which L ′ is obtained by converting the distance L in FIG. 17 by a fixed function and an image is displayed.
FIG. 22 is an explanatory diagram of the projection method of the present invention when the rotating body is a sphere.
FIG. 23 is an explanatory diagram of the projection method of the present invention when the rotating body is a sphere.
FIG. 24 is an explanatory diagram of the projection method of the present invention when the rotating body is a sphere.
FIG. 25 is an explanatory diagram of the projection method of the present invention when the rotating body is a sphere.
FIG. 26 is an explanatory diagram of an example of reprojection of a primary projection image from a spherical surface to a plane in the projection method of the present invention when the rotating body is a sphere.
FIG. 27 is an explanatory diagram of an example of reprojection of a primary projection image from a spherical surface to a plane in the projection method of the present invention when the rotator is a sphere.
FIG. 28 is an explanatory diagram of an example of reprojection of a primary projection image from a spherical surface to a plane in the projection method of the present invention when the rotator is a sphere.
FIG. 29 is an explanatory diagram of an example of reprojection from the spherical surface to the plane of the primary projection image in the projection method of the present invention when the rotating body is a sphere.
FIG. 30 is an explanatory diagram of an example of reprojection from the spherical surface to the plane of the primary projection image in the projection method of the present invention when the rotating body is a sphere.
FIG. 31 is a diagram for explaining an example of a method of constructing a three-dimensional image from a plurality of tomographic images by applying the projection method of the present invention when the rotating body is a sphere.
FIG. 32 is a diagram for explaining an example of a method for constructing a three-dimensional image from a plurality of tomographic images by applying the projection method of the present invention when the rotating body is a sphere.
FIG. 33 is a diagram for explaining an example of a method of constructing a three-dimensional image from a plurality of tomographic images by applying the projection method of the present invention when the rotating body is a sphere.
FIG. 34 shows an example of a method for constructing a three-dimensional image from a plurality of tomographic images by applying the projection method of the present invention when the rotator is a sphere, and the viewpoint, tomographic image, and primary projection curved surface have a more complicated positional relationship. It is a figure for demonstrating rotation body curved surface / plane coordinate conversion at the time of having.
35 is a diagram showing a coordinate relationship on the spherical surface in FIG. 34. FIG.
36 is a diagram in which the projection image onto the spherical surface in FIG. 34 is displayed with the angles λ and β in FIG. 35.
FIG. 37 is a diagram showing a range of projection image data used when calculating a three-dimensional image centered on an arbitrary direction using the projection image data on the entire primary projection curved surface in FIG.
FIG. 38 is an explanatory diagram of the projection method of the present invention when the rotating body is a cone.
FIG. 39 is an explanatory diagram of the projection method of the present invention when the rotator is an elliptical rotator.
FIG. 40 is an explanatory diagram of the projection method of the present invention when the rotator is an elliptic rotator.
FIG. 41 is an explanatory diagram of the projection method of the present invention when the rotator is a parabolic rotator.
FIG. 42 is an explanatory diagram of the projection method of the present invention when the rotator is a parabolic rotator.
FIG. 43In the tentative planIt is explanatory drawing of the projection method in case a rotary body is a cylinder.
FIG. 44In the tentative planIt is explanatory drawing of the projection method in case a rotary body is a cylinder.
FIG. 45 is a block diagram illustrating a hardware configuration example to which the projection method of the present invention can be applied.
FIG. 46 is an explanatory diagram of the center projection method.
FIG. 47 is an explanatory diagram of a parallel projection method.
FIG. 48 is an explanatory diagram of the relationship between the central projection method and the movement of the projected image.
[Explanation of symbols]
e Viewpoint
x, y, z Each axis of 3D coordinate system
O Origin of 3D coordinate system
21 Rotating body
21a Rotating body curved surface (primary projection curved surface)
22 straight lines
23 Tomographic image
4 Secondary projection plane
5 Projection target
6 Primary projection image
C Unit sphere centered on the origin of the 3D coordinate system (radius 1)
20 Plane (great circle) that passes through the center of the unit sphere and matches the xy plane
20 ′ A plane that passes through the origin and is orthogonal to the rotational axis of the rotating body at an angle ω from the z-axis
Q Points on the rotating body curved surface 21a due to light rays emitted from the viewpoint e
N Vertex (center) of rotating body
S A point on the tomographic image that intersects the straight line OQ
R radius of rotating body curved surface
Point on Cartesian coordinate system on quadratic projection plane corresponding to P Q point
Rotation angle measured from the x axis on the α xy plane
ω Angle between line connecting Q point and origin and z axis
L Circular arc NQ on the rotating body curved surface
A straight line N′P on the secondary projection plane with a line segment having the same length as L′ L
η X coordinate on the monitor corresponding to point P
ξ Y coordinate on the monitor corresponding to point P

Claims (2)

A viewpoint is set at an arbitrary position inside the projection object, a projection plane is set outside the projection object, and a projection curved surface having any one shape of a sphere, a cone, an ellipse rotation body, and a parabola rotation body is set. Means for setting between a viewpoint and the projection plane;
Means for projecting the projection object or at least one tomographic image of the projection object onto the projection curved surface as viewed from the viewpoint by a central projection method;
A projection apparatus using a central projection method, comprising: means for reprojecting a projection image obtained on the projection curved surface onto the projection plane by rotating body curved surface / planar coordinate conversion. The projected curved surface is a spherical surface;
2. The projection apparatus using the central projection method according to claim 1, wherein the radius of the spherical surface can be selected as a dimension capable of discriminating a minute part of an organ as a projection target .

Images