JP6441833B2 - Quantization method, quantization apparatus, and quantization program - Google Patents

Description

  The present invention relates to a quantization method, a quantization apparatus, and a quantization program.

  In recent years, with the improvement of image quality, expectations for wide dynamic range images are increasing. Expression using a high bit depth for an image signal having a wide dynamic range has been studied. Devices that can acquire a high bit depth signal expanded from the conventional 8 bits to 10 bits or more have appeared.

  The code amount of the image increases as the bit depth of the image signal is increased. For this reason, an efficient encoding method is required. FIG. 5 is a diagram illustrating a first conventional encoding method of a high bit depth signal. In the first conventional high bit depth signal encoding method, the encoding apparatus performs bit depth conversion processing on the image signal of the high bit depth signal, thereby converting the image signal into a low bit depth signal and encoding the image signal. And decryption processing. Further, the encoding device performs inverse bit depth conversion processing on the decoded image to generate a high bit depth image. Finally, the encoding device encodes a differential signal between the high bit depth image and the input signal. The output of the encoding device is an encoded stream of a differential signal and an encoded stream of a low bit depth signal. As described above, the first conventional encoding method of a high bit depth signal is a method corresponding to the bit depth scalable encoding.

  FIG. 6 is a diagram illustrating a second conventional encoding method of a high bit depth signal. In FIG. 6, the encoding device does not output the encoded stream of the difference signal. The encoding device performs a bit depth conversion process on the acquired image signal, thereby converting the image signal into a decoded signal and performing encoding and decoding processes. Thereafter, the encoding apparatus performs inverse bit depth conversion processing on the decoded signal of the low bit depth image, and outputs the decoded signal of the high bit depth image.

  The coding efficiency in the first coding method of the conventional high bit depth signal shown in FIG. 5 largely depends on the bit depth conversion process. In the first conventional high bit depth signal encoding method, the sum of squares of the difference value between the input signal and the decoded signal after the inverse bit depth conversion process (hereinafter referred to as “bit depth conversion error”) is suppressed. Thus, the code amount of the output signal of the encoder can be suppressed. Non-Patent Document 1 discloses a method using several tone mappings in bit depth conversion processing. In order to improve the coding efficiency, the bit depth conversion needs to be designed so as to minimize the bit depth conversion error.

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, "Photographic Tone Reproduction for Digital Images", In SIGGRAPH 2002 Conference Proceeding, ACM SIGGRAPH, Addison Wesley, pp. 267-277, August 2002.

  The conventional bit depth change process can be regarded as a kind of quantization process for a histogram of an image signal. In order to improve the coding efficiency, it is necessary to design the bit depth conversion so as to minimize the bit depth conversion error. However, if the bit depth conversion is designed so as to minimize the bit depth conversion error, there is a problem that the amount of calculation increases. That is, the conventional quantization method has a problem in that the amount of calculation required for quantization cannot be reduced when the conversion error generated when performing gradation conversion of a signal is minimized.

  In view of the above circumstances, the present invention provides a quantization method and a quantization apparatus that can reduce the amount of calculation required for quantization even when the conversion error that occurs when performing gradation conversion of a signal is minimized. And to provide a quantization program.

  According to one aspect of the present invention, a histogram of an input signal representing a value that can take a range expressed by the first number of levels is taken as a range expressed by a second number of levels smaller than the first level. A quantization method in a quantization device that approximates with a histogram of an output signal that represents a value to be obtained, comprising: forming a composite element that is a set of elements in the histogram of the input signal; and included in the plurality of composite elements An approximation error when approximating one or more values with a representative value is defined as an element in the composite element at the bottom end of the class that is a set of the composite elements, and an element in the composite element at the top end of the class. By defining the lower limit value of the approximation error for the histogram section of the input signal configured with an element as the upper end, class boundary candidates that are boundaries of the class are limited to the composite element, and the class Storing a quantized value for minimizing the accumulated value of the approximation error up to a boundary and a minimum value of the accumulated value of the approximate error in a memory, and the stored minimum value when selecting the class boundary And the minimum value called from the memory is used for the calculation of the approximation error in the selection of the class boundary, thereby minimizing the sum of the approximation errors for all the complex elements of the class boundary. Performing class boundary selection processing for selecting the class boundary to be converted, and when the composite element selected as the class boundary includes a plurality of elements, the selected composite element is defined as an upper subset and a lower subset. The upper subset and the lower subset obtained by the division are newly added to the subset composite element, the class boundary selection process is performed, and the class boundary is And repeating until the number of elements selected the composite element is one as a quantization method comprising a.

  One aspect of the present invention is the quantization method described above, in which, as the approximation error when a plurality of the composite elements are approximated by representative values, the element of the maximum level value of the composite element at the lowermost end is set as the lower end. The quantization method further includes the step of defining the approximation error with respect to the histogram section of the input signal configured with the element of the minimum level value of the composite element at the uppermost end as an upper end.

  One aspect of the present invention is the quantization method described above, wherein when the composite element selected as the class boundary is divided into the upper subset and the lower subset, the upper subset and the upper subset Calculating the approximation error for the set of composite elements determined by the upper subset obtained by dividing the composite element of the class boundary of the class one level above Based on the result of performing the process of selecting the class boundary in the subsequent stage by reusing the approximation error for the composite element and the set of other composite elements, the upper subset and the other composite elements are used. Calculating the approximation error for the set of composite elements.

  One aspect of the present invention is the above-described quantization method, wherein when the composite element selected as the class boundary is divided into the upper subset and the lower subset, the lower subset and the other composite The quantization method further includes a step of calculating the approximation error for the set of complex elements determined by an element and selecting the class boundary in the subsequent stage.

  One aspect of the present invention is the quantization method described above, wherein the set of composite elements includes an element of the composite element at the uppermost end, and a minimum level value of values of the elements of the composite element at the lowermost end. It is a quantization method further including the step comprised by the said composite element smaller than the maximum level value of a certain value.

  According to one aspect of the present invention, a histogram of an input signal representing a value that can take a range expressed by the first number of levels is taken as a range expressed by a second number of levels smaller than the first level. A quantization apparatus for approximating with a histogram of an output signal representing a value to be obtained, comprising: a histogram generation unit constituting a composite element that is a set of elements in the histogram of the input signal; and one included in the plurality of composite elements The approximation error when approximating the above values with representative values is defined as the element in the composite element at the bottom end of the class that is a set of the composite elements, and the element in the composite element at the top end of the class. Defining a lower limit value of an approximation error with respect to a section of the histogram of the input signal configured as an upper end, limiting class boundary candidates that are boundaries of the class to the composite element, and the class boundary A minimum value determining unit that stores in a memory a quantized value that minimizes the accumulated value of the approximate error in memory, a minimum value of the accumulated value of the approximate error, and the stored minimum when the class boundary is selected. By calling a value from the memory and using the minimum value called from the memory for calculating the approximation error in the selection of the class boundary, the sum of the approximation errors is calculated for all the complex elements of the class boundary. When class boundary selection processing for selecting the class boundary to be minimized is performed, and the composite element selected as the class boundary includes a plurality of elements, the selected composite element is divided into an upper subset and a lower subset. The upper subset and the lower subset obtained by the division are newly added to the subset composite element, the class boundary selection process is performed, and the class boundary is selected. It said number of elements of the composite element is a quantizer; and a upper track section repeated until one.

  According to one aspect of the present invention, a histogram of an input signal representing a value that can take a range expressed by the first number of levels is taken as a range expressed by a second number of levels smaller than the first level. A method of configuring a composite element that is a set of elements in the histogram of the input signal in a computer of a quantization device that approximates with a histogram of an output signal representing a value to be obtained, and one or more included in the plurality of composite elements The approximation error when the value is approximated with a representative value is defined as an element in the composite element at the bottom end of the class that is a set of the composite elements, and an element in the composite element at the top end of the class as the top. The lower limit value of the approximation error for the section of the histogram of the input signal to be configured is defined, the class boundary candidates that are the boundary of the class are limited to the composite element, and up to the class boundary The quantization value for minimizing the approximate value of the approximate error and the minimum value of the approximate value of the approximate error are stored in the memory, and when the class boundary is selected, the stored minimum value is stored from the memory. The class that minimizes the sum of the approximation errors for all the complex elements of the class boundary by using the minimum value called from the memory to calculate the approximation error in the selection of the class boundary A class boundary selection process for selecting a boundary; and when the composite element selected as the class boundary includes a plurality of elements, the selected composite element is divided into an upper subset and a lower subset, The upper subset and the lower subset obtained by the division are newly added to the subset composite element, the class boundary selection process is performed, and the class boundary selected The number of elements of application elements are quantized program for executing a procedure repeated until one.

  According to the present invention, it is possible to reduce the amount of calculation required for quantization even when the conversion error generated when performing gradation conversion of a signal is minimized.

It is a figure which shows the example of a structure of the quantization apparatus in embodiment. It is a figure which shows the procedure of the process which determines the minimum and upper limit of the element in the composite element in the m-th stage in embodiment. It is a flowchart which shows the adaptive quantization process in embodiment. It is a flowchart which shows the detail of step S205 in embodiment. It is a figure which shows the 1st encoding method of the conventional high bit depth signal. It is a figure which shows the 2nd encoding method of the conventional high bit depth signal.

Embodiments of the present invention will be described in detail with reference to the drawings.
In the following, symbols (^: hat, ~: tilde) attached to variables etc. in the figure or formula are described before the variables. For example, “c” with “^” on top is expressed as “^ c”. Hereinafter, the subscript symbols and the like are described first in the subscript symbols and the superscript symbols and the like of the variables or symbols in the drawings or formulas. For example, a variable Δ with a subscript “1” and a superscript “*” is expressed as “Δ ₁ ^* ”. For example, a symbol Σ with a subscript “i = 0” and a superscript “m” is expressed as “Σ _{i = 0} ^m ”.

[Adaptive quantizer]
FIG. 1 is a diagram illustrating an example of the configuration of the quantization device 1. The quantization device 1 is an information processing device such as a personal computer, a tablet terminal, a smartphone terminal, or a server device. The quantizing device 1 expresses a histogram of an input signal representing a value that can take a range expressed by the first number of levels (number of gradations) with a second number of levels smaller than the first level number. Quantization processing (adaptive quantization) that is approximated by a histogram of an output signal representing a value that can take a range is executed.

  The quantization device 1 includes a histogram generation unit 11, an element number storage unit 12, a class number storage unit 13, an approximate error determination unit 14, an approximate error storage unit 15, an initialization unit 16, and a composite element storage unit. 17, minimum value determination unit 18, minimum storage unit 19, upper end tracking unit 20, upper end storage unit 21, update determination unit 22, element update unit 23, index conversion unit 24, and switching unit 25 Is provided.

  A histogram generation unit 11, an approximation error determination unit 14, an initialization unit 16, a minimum value determination unit 18, an upper end tracking unit 20, an update determination unit 22, an element update unit 23, an index conversion unit 24, Part or all of the switching unit 25 is a software function unit that functions when a processor such as a CPU (Central Processing Unit) executes a program stored in the storage unit. Some or all of these functional units may be hardware functional units such as LSI (Large Scale Integration) and ASIC (Application Specific Integrated Circuit).

  Some or all of the element number storage unit 12, the class number storage unit 13, the approximate error storage unit 15, the composite element storage unit 17, the minimum storage unit 19, and the upper end storage unit 21 may be, for example, a magnetic hard disk It is configured using a storage device having a nonvolatile storage medium (non-temporary recording medium) such as a device or a semiconductor storage device. Each storage unit may include, for example, a volatile storage medium such as a RAM (Random Access Memory) or a register.

The histogram generation unit 11 acquires an input signal. The histogram generator 11 generates a histogram of the input signal. The histogram generation unit 11 stores the histogram of the input signal and the level number K of the input signal in the element number storage unit 12.
The element number storage unit 12 stores a histogram of input signals and a level number K of input signals.
The class number storage unit 13 stores a quantization class number M.

The approximate error determination unit 14 acquires the histogram of the input signal and the number K of input signal levels from the element number storage unit 12. The approximate error determination unit 14 acquires the quantization class number M from the class number storage unit 13. The approximation error determination unit 14 calculates an approximation error when the section in the histogram is approximated with a representative value.
The approximation error storage unit 15 stores an approximation error when a section in the histogram is approximated with a representative value.

  Hereinafter, K represents the number of levels of the input signal (hereinafter referred to as “number of input levels”). M (M <K) indicates the number of levels of the output signal based on the quantized input signal (hereinafter referred to as “quantization level number”). A histogram (frequency) of the pixel value k is expressed as h [k] (k = 0,..., K−1).

The initialization unit 16 acquires the histogram h [k] (k = 0,..., K−1) from the element number storage unit 12. The initialization unit 16 converts the histogram h [k] (k = 0,..., K−1) of the input signal values into G ⁽ⁿ⁾ = {g ⁽ⁿ⁾ [˜k] | ˜k = 0,. ˜K ⁽ⁿ⁾ −1}. n is an index representing the loop count (update count) of an algorithm described later. The initialization unit 16 sets the initial value of the composite element g ⁽ⁿ⁾ [˜k] of the histogram h [k] (k = 0,..., K−1) as G ⁽⁰⁾ = {g ⁽⁰⁾ [˜ k] | ˜k = 0,..., ˜K ⁽⁰⁾ −1}.

  The composite element storage unit 17 stores composite elements of the histogram h [k] (k = 0,..., K−1). Hereinafter, a set of composite elements that are approximated with a representative value is referred to as a “composite quantization class”.

  The minimum value determination unit 18 is configured to approximate the composite quantum in the case where the maximum element in the transition destination composite element and the minimum element in the transition source composite element are approximated by the centroid of the same section. Approximation error of the optimization class The minimum value determination unit 18 treats the approximation error value stored in the approximation error storage unit 15 as a necessary approximation error value.

  The minimum value determination unit 18 determines an optimal composite quantization class of the composite element. The minimum value determination unit 18 causes the minimum value storage unit 19 to store an approximation error corresponding to the optimal composite quantization class of the composite element. The minimum value storage unit 19 stores an approximation error corresponding to the optimal composite quantization class of the composite element.

  The upper end tracking unit 20 causes the upper end storage unit 21 to store the index of the composite element corresponding to the upper end of the composite quantization class representing the optimal composite quantization class of the composite element calculated by the minimum value determining unit 18.

  The update determination unit 22 acquires the number of elements in all composite elements corresponding to the upper end of the optimal composite quantization class. When the number of all elements is 1, the update determination unit 22 transmits the composite element corresponding to the upper end of the optimal composite quantization class to the element update unit 23 via the switching unit 25. If any of the number of elements is not 1, the update determination unit 22 transmits a trigger signal to the index conversion unit 24 via the switching unit 25.

  The element update unit 23 acquires a composite element corresponding to the upper end of the optimal composite quantization class. The element update unit 23 decomposes a composite element having two or more elements into an upper subset and a lower subset. The element update unit 23 divides the composite element having the number of elements of 2 or more so that the difference in frequency values between the upper subset and the lower subset is as small as possible. The element update unit 23 causes the composite element storage unit 17 to store the upper subset and the lower subset obtained by the division as new composite elements.

  The element update unit 23 causes the composite element storage unit 17 to store the upper subset obtained by the division as a candidate for the upper end of the composite quantization class. The element update unit 23 stores the lower subset obtained by the division in the composite element storage unit 17 as a candidate for the lower end of the composite quantization class.

When acquiring the trigger signal from the update determination unit 22, the index conversion unit 24 acquires the indexes of M composite elements corresponding to the upper end of the composite quantization class stored in the upper end storage unit 21. The index conversion unit 24 outputs the element index for the element included in each composite element as information representing the upper end of the optimal quantization class.
The switching unit 25 switches the signal path in accordance with the control by the update determination unit 22.

[Quantizer design]
When the image signal (luminance signal) is 8 bits, the possible range of the pixel value k is an integer from 0 to “number of input levels−1”, that is, a value from 0 to 255. Consider a case in which an image signal having an input level number K is quantized into an image signal having a quantization level number. The parameters to be obtained are M parameters that satisfy Expression (1).

Here, L _m is the upper end of the m-th section in the histogram and is defined by Expression (2).

Here, delta _m represents the section width of the m section of the histogram. Since each gradation after quantization needs to include at least one gradation before quantization, L _m (0 ≦ m ≦ M−2) is m ≦ L _m ≦ K− ( M-m). Since L _M-1 is fixed to L _M-1 = K-1, for L _m , m = M-1 may be omitted and m ≦ M-2 may be considered.

e (L _m − (Δ _m −1), L _m ) is expressed by Equation (3). Equation (3) is an approximation error when the histogram section [L _m − (Δ _m −1), L _m ] is approximated by the representative value c (L _m − (Δ _m −1), L _m ). .

Furthermore, ^ c (L _m − (Δ _m −1), L _m ) is an integer obtained by rounding off the real value c (L _m − (Δ _m −1), L _m ) obtained by the equation (4). Value.

c (L _m − (Δ _m −1), L _m ) represents the position of the center of gravity in the section [L _m − (Δ _m −1), L _m ] of the histogram. Hereinafter, the section [L _m − (Δ _m −1), L _m ] of the histogram is referred to as “mth quantization class”.

The M parameter _{(Δ 1, ..., Δ M} ) Possible combinations of, for increases exponentially with M, optimal combination from among the ^{_{(Δ 1 *, ..., Δ}} M *) in a round robin Searching is difficult computationally. Therefore, focusing on the fact that the quantization error e (L _m − (Δ _m −1), L _m ) of the m-th quantization class depends on the upper end L _m of the same class and the section width Δ _m of the same class, The optimal solution is calculated as follows. Further, at that time, the calculation amount is reduced based on the non-significant elements of the histogram.

[Basic solution]
First, a histogram of the interval _{[0, L m] (m} + 1) approximation error sum during the divided ^{_{Σ i = 0 m e (L}} i - (Δ i -1), L i) the minimum value _{of, S} m ( L _m ). In other words, the optimal ^{_{Δm, ..., Σ i = 0}} m e in the case of using the _{Δ 0 (L i - (Δ} i -1), L i) is a minimum for. Here, focusing on the fact that e (L _m − (Δ _m −1), L _m ) depends on the section width Δm of the same class as the upper end L _m of the m th quantization class, S _m (L _m ) is , S _m−1 (L _m −Δ _m ), and is expressed by equation (5).

Note that m = 1,..., M-1. Further, L _m = m,..., K− (M−m). Range of delta _m is as shown in equation (6). Because it is _{L m-1 = L m -Δ} m, the range of _L m - [delta _{m is, m-1 ≦ L m -Δ} m ≦ K- (M-m + 1) become. Therefore, the range of delta _m, using the given L _m, represented by the formula (6).

Further, considering that Δ _m ≧ 1, Equation (7) is obtained.

Here, the calculated S _m (L _m ) is stored and used in the calculation of S _{m + 1} (L _{m + 1} ). Further, as an expression of the value of delta _m that minimizes the right-hand side delta _m ^(Lm) of (13), each of _{L m (= m, ...,} k-M + m) with respect to _{^ L m-1 [L m} ] = L _m −Δ _m ^(Lm) is stored. The minimization problem of Equation (1) is expressed as Equation (8).

Δ _M−1 that minimizes Equation (8) is expressed as Δ _M−1 ^{* as} shown in Equation (9).

Using this ΔM -1 * , the optimum value of the upper end of the M-2 class is obtained as LM -2 * = LM -1− ΔM -1 * = K−1−ΔM -1 *. . Optimal value of the upper end of the M-3 class, the optimal solution to the L M-2 *, ^ L M-3 because it is stored as [L M-2 *], with reference to the appropriate value, L M −3 * = ^ L M−3 [L M−2 ]. As a result, the section width of the M-2 class is obtained as ΔM -2 * = LM -2 * -LM -3 * .

Hereinafter, the same reference process ^{_{L M-4 * = ^ L}} M-4 [L M-3 *], ..., L 0 * = ^ L 0 [L 1 *] repeated as the upper end of each obtained class using the ^{_{values, Δ M-3 * = L}} M-3 * -L M-4 *, ..., Δ 0 * = L 0 * calculated as _{-L -1} ^*. Note that L ₋₁ = −1.

  The visual system is less sensitive to changes in high luminance pixel values than changes in low luminance pixel values. Therefore, when quantization is performed in consideration of such visual characteristics, the following is performed. Hereinafter, w [k] represents a weighting factor for the pixel value k (k = 0,..., K−1). The weighting coefficient w [k] is determined in advance. For example, if the weight of high luminance (large k) is set to a value smaller than the weight of low luminance (small k), the visual characteristic for the difference (luminance difference) between high luminance and low luminance can be incorporated into the quantization process. It becomes possible. Using the weighting coefficient w [k], the frequency h [k] for the pixel value k is corrected as ˜h [k] = w [k] h [k], and the corrected histogram ˜h [k] is corrected. Then, the above-described quantization process is performed.

[Computation volume reduction based on class integration and division]
The initialization unit 16 converts the histogram h [k] (k = 0,..., K−1) of the input signal values into G ⁽ⁿ⁾ = {g ⁽ⁿ⁾ [˜k] | ˜k = 0,. ˜K ⁽ⁿ⁾ −1}. Here, the composite element g ⁽ⁿ⁾ [˜k] is grouped from the 1st [˜k] ⁽ⁿ⁾ element to the u [˜k] ⁽ⁿ⁾ element of the histogram as the 1st to kth groups. This is a set of histogram elements. g ⁽ⁿ⁾ [˜k] is as follows: g ⁽ⁿ⁾ [˜k] = {h [k] | k = l [˜k] ⁽ⁿ⁾ ,..., u [˜k] ⁽ⁿ⁾ } expressed.

In addition, the number of elements of the histogram in the ˜kth group is expressed as “|| g ⁽ⁿ⁾ [˜k] || = u [˜k] ⁽ⁿ⁾ −1 [˜k] ⁽ⁿ⁾ +1”. Note that l [0] ⁽ⁿ⁾ = 0 and u [˜K] ⁽ⁿ⁾ = K−1. The l [˜k] ⁽ⁿ⁾ element (lowermost element) in the composite element g ⁽ⁿ⁾ [˜k] is referred to as “lowest element index”. In addition, the u [˜k] ⁽ⁿ⁾ element (uppermost element) in the composite element g ⁽ⁿ⁾ [˜k] is referred to as “uppermost element index”.

[Quantization for grouped histograms]
The set of composite elements ^{{g (n) [~k]} | ~k = 0, ..., L ~m} a (~m + 1) is divided into subsets, approximating the histogram representative value for each subset think of. The complex quantization class approximation is evaluated using the lower limit value of the approximation error in the corresponding histogram section.

  At this time, the parameters to be obtained are ˜M parameters that satisfy Expression (10).

Here, L to _m are the upper ends of the ˜m-th subsets of the set of composite elements, and are represented by Expression (11).

Here, Δ _{~ m} represents the number of composite elements of the ~ mth subset of composite elements. _{~e (L ~m - (Δ ~m} -1), L ~m) is a value obtained by equation (12). _{~e (L ~m - (Δ ~m} -1), L ~m) is the _{L ~m} - _{(Δ ~m} -1) th maximum element and the _{L ~m} th complex element within the composite element This is an approximation error in the case of approximating the section in the histogram having the smallest element in the both ends as the center of gravity of the section.

The value of Expression (12) is the value in the (L to _m − (Δ to _m −1)) th composite element with respect to the subset [L to _m −Δ to _m −1), L to _m ] of the composite element. Introduced as the lower limit value of the approximation error when approximating with the centroid of the same section for the section in the histogram whose end points are any one of the elements and any element in the L-th to _m- th composite elements It is the value. Hereinafter, the subset [L to _m − (Δ to _m −1), L to _m ] of the composite element is referred to as “the m-th composite quantization class”.

The minimum value of the sum Σ _{i = 0 to} ^{m to} e (L _i − (Δ _i −1), L _i ) of the lower limit values of the approximation errors is denoted as S to _m (L to _m ). In other words, the optimal ^{_{Δ~m, ..., Σ i = 0}} ~m ~e when using _{Δ 0 (L i - (Δ} i -1), L i) is a minimum for. _{Here, ~e (L ~m - (Δ} ~m -1), L ~m) is focused on the fact that depends on the ~m upper _{L ~m} and section width delta _~m the same class of composite quantization class Then, _Sm ( _Lm ) is represented by Formula (13) using _Sm-1 ( _Lm- ( _DELTA ) _m ).

Note that ~ m = 1, ..., ~ M-1. Moreover, it is L- _m = -m, ..., -K- (-M--m).

The range of Δ to _m is as follows. Because it is _{L ~m-1 = L ~m -Δ} ~m, the range of _{L ~m} - [delta _{~m is, ~m-1 ≦ L ~m -Δ} ~m ≦ ~K- (~M-~ m + 1). For this reason, the range of (DELTA) _-m is represented by Formula (14) using the given L- _m .

Further, considering that Δ _˜m ≧ 1, Equation (15) is obtained.

Here, the calculated S 1 to _m (L 1 to _{m 2} ) are stored and used in the calculation of S _{1 to m + 1} (L _{1 to m + 1} ). Further, Δ _{~ m} ⁽ L _{~ m} ⁾ that minimizes the right side of the expression (13) is Δ _{~ m} ⁽ L _{~ m} ⁾ , and for each of L _{~ m} (= ~ m, ..., ~ ^K- ~ M + ~ m). ^ L to _m−1 [L to _m ] = L to _m −Δ to _m ⁽ L to _m ⁾ are stored. The minimization problem of equation (1) is represented by equation (16).

Hereinafter, Δ to _M−1 that minimizes the expression (16) will be referred to as “Δ to _M−1 ^* ”. _Δ˜M−1 ^* is expressed by Expression (17).

Optimal value of the upper end of the ~M-2 complex quantization classes, L ~M-2 * = L ~M-1 -Δ ~M-1 * = ~K-1-Δ ~M-1 * and determined. Optimal value of the upper end of the ~M-3 complex quantization class is stored as the optimum solution ^ L ~M-3 [L ~M -2 *] for L ~M-2 *. The optimum value at the upper end of the ˜M-3th complex quantization class is determined as L 1 −M−3 * = ^ L 3 −M 3 [L 1 −M−2 * ] based on the stored value. The section width of the ˜M-2 composite quantization class is determined as Δ˜M−2 * = L˜M−2 * −L˜M−3 * .

Hereinafter, like reference _{processing ^{L ~M-4 * = ^ L}} ~M-4 * [L ~M-3 *], ..., L 0 * = ^ L 0 * repeatedly as _[L ^{1 *],} obtained Using the upper end value of each composite quantization class, _Δ˜M−3 ^* = _L˜M−3 ^* _−L˜M−4 ^* ,..., Δ ₀ ^* = L ₀ ^* −L ₋₁ ^* . Note that L ₋₁ ^* = − 1.

The termination condition is determined for g ⁽ⁿ⁾ [L _{~ m} ^* ] (~ m = 0, ..., ~ M) obtained by the above processing. The termination condition is that all the composite elements obtained as the optimal solution are composed of only a single element, that is, || g ⁽ⁿ⁾ [L _{~ m} ^* ] || = 1 (~ m = 0, ..., ~ M).

When the above condition is not satisfied, that is, when there is a composite element satisfying || g ⁽ⁿ⁾ [L to _m ^* ] ||> 1, the same element is divided, and the composite element obtained by the division A quantization process similar to the above is performed on the set. This iteration is repeated until the above termination condition is satisfied. A method for dividing the composite element will be described later.

[How to configure composite elements]
FIG. 2 is a diagram showing a processing procedure for determining the lower limit and the upper limit of elements in the composite element in the m-th stage. The frequency value sum in the composite element is equalized within a possible range by the processing procedure shown in FIG. The procedure of the process shown in FIG. 2 will be described below. An initial value 0 is assigned to the index of the composite element (step S101).

Variables used in subsequent processing are initialized to 0 (step S102).
L [0] in the m-th stage is set to l [0] = m (step S103). This is because the selectable histogram elements in the m-th stage are m ≦ k ≦ m−M + K.
Steps S105 to S110 are repeated for the section of the histogram indicated by the inequality “m ≦ k ≦ m−M + K” (step S104).

The cumulative frequency value sum of the histogram sections is stored in the variable s (step S105). When the cumulative frequency value sum of the variable s exceeds the threshold θ (step S106), the index k at that time is set as the upper limit of the composite element (step S107).
The composite element index ~ k is incremented (step S108).
The index .about.k is stored as the lower limit value of the next composite element (step S109).
A variable s for storing the cumulative frequency value sum of the histogram sections is initialized to 0 (step S110). The threshold value θ used above is expressed by equation (18). The threshold value θ shown in Expression (18) is the maximum integer that does not exceed the real number “Σ _{k = 0} ^K−1 h [k] / ˜K (n)”.

  Alternatively, as another configuration example of the composite element, it is possible to give a fixed value B as the number of elements so that the number of elements in the composite element becomes equal. The trellis transition diagram for the composite element is set so as to satisfy the following condition regarding the path between the composite elements.

[How to divide composite elements]
When the composite element corresponding to the upper end of the optimal composite quantization class is composed of a plurality of elements, the composite element is divided into two subsets. The specific division is performed so that the sum of the frequency values in the composite element after the division is as uniform as possible with respect to the composite element to be divided. Hereinafter, a subset including a large element index among two subsets obtained by division is referred to as an “upper subset”. Hereinafter, a subset including a small element index among two subsets obtained by division is referred to as a “lower subset”.

  The lower subset is added as a candidate for the lower end of the composite quantization class to the lower subset obtained by dividing the composite element corresponding to the upper end of the optimal (m-1) th composite quantization class. At this time, the condition of the composite element that can be set as the upper end of the composite quantization class is the composite quantization class in which the highest element index in the composite element that is the upper end of the composite quantization class corresponds to the lower subset described above. This is a value that is larger than the lowest element index in the composite element that is the lower end of.

  A new transition path is added to the transitionable composite element in the m-th stage (including the upper subset and the lower subset generated by the above division). In addition, the cost value for the added transition path is also calculated.

  For the upper subset obtained by dividing the composite element corresponding to the upper end of the optimal (m−1) -th composite quantization class, the same subset is added as a candidate for the upper end of the composite quantization class. At this time, the condition of the composite element that can be set as the upper end of the composite quantization class is the composite quantization class in which the highest element index in the composite element that is the upper end of the composite quantization class corresponds to the above-mentioned upper subset. This is a value that is larger than the lowest element index in the composite element that is the lower end of.

  When the upper subset obtained by dividing the composite element corresponding to the upper end of the optimal m-th composite quantization class is used as the upper end of the composite quantization class with the same subset as the lower end, the approximation of the composite quantization class is used. The error is newly acquired from the reference table. On the other hand, when a composite element other than the above is used as the upper end of the same composite quantization class, the approximation error of the same composite quantization class causes the composite element corresponding to the upper end of the composite quantization class before division of the same subset to be the same. It is the same as the approximation error in the case of the lower end of the conversion class. This is because the composite element corresponding to the upper end of the composite quantization class before division of the same subset is the same as the composite element corresponding to the upper end of the same subset.

  When the composite element on the optimal path in the m-th stage is composed of a plurality of elements, and the composite element is divided into a lower subset and an upper subset, the m-th stage from the upper subset in the m-1 stage A new transition path is added to the upper subset after division in. Except for the added transition path, the transition cost is the same as the transition cost from the composite element before the division at the (m−1) -th stage to the composite element before the division at the m-th stage. Note that if the composite element on the optimal path in the m-th stage is composed of a single element and no division is performed on the composite element, no new transition path is added.

  The composite element in the m-th stage that can be transitioned from the composite element in the (m-1) th stage is a value in which the highest element index of the composite element in the m-th stage is larger than the lowest element index of the composite element in the (m-1) th stage. Is a composite element.

  As another division method, it is also possible to divide so that the number of elements in the composite element becomes equal. When there is a transitionable path between elements in the composite element, a path between the composite elements is provided. The transition cost of the path between the composite elements is the minimum value of the transition cost of the path between the elements in the composite element.

[Adaptive quantization]
FIG. 3 is a flowchart showing the adaptive quantization process.
The histogram generation unit 11 acquires an input signal. The histogram generator 11 generates a histogram h [K] (k = 0,..., K−1) of input signal values (step S201).
The approximate error determination unit 14 acquires a histogram of input signal values. The approximate error determination unit 14 calculates an approximate error when the section in the histogram is approximated with the representative value based on the equation (3). The approximate error determination unit 14 stores the calculated approximate error in the reference table of the approximate error storage unit 15 (step S202).

The initialization unit 16 acquires the histogram generated in step S201. The initialization unit 16 divides the histogram into composite elements. That is, the initialization unit 16 converts the composite element of the acquired histogram h [k] (k = 0,..., K−1) to G ⁽⁰⁾ = {g ^{(0 ) [~k] | ~k = 0} , ..., defined as ^{~K (0)} -1} ^(step S203).

  The minimum value determination unit 18 acquires a histogram and the upper and lower ends of each composite element. The minimum value determination unit 18 calculates an approximation error when the section in the histogram is approximated with the representative value based on the equation (12). The minimum value determination unit 18 stores the calculated approximation error in the reference table of the minimum value storage unit 19 as an approximation error of each composite quantization class. The minimum value determination unit 18 approximates an error in the case of approximating the section in the histogram having the maximum element in the transition destination composite element and the minimum element in the transition source composite element as both end points with the center of gravity of the same section. Set as. The minimum value determination unit 18 determines the approximation error by referring to the reference table generated in step S202 (step S204).

  The minimum value determination unit 18 determines an optimal composite quantization class of the composite element (step S205). Get the number of elements in all composite elements corresponding to the top of the optimal composite quantization class. The minimum value determination unit 18 determines whether or not the number of all elements is 1 (step S206).

  The minimum value determination unit 18 acquires a composite element corresponding to the upper end of the optimal composite quantization class. The minimum value determination unit 18 decomposes a composite element having two or more elements into an upper subset and a lower subset. The minimum value determination unit 18 divides the composite element having the number of elements of 2 or more so that the difference between the frequency values of the upper subset and the lower subset is as small as possible (step S207).

The element updating unit 23 repeats Step S209 and Step S210 for the quantized classes m = 1,..., M−1 (Step S208).
The element updating unit 23 uses the lower subset as a candidate for the lower end of the composite quantization class with respect to the lower subset obtained by dividing the composite element corresponding to the upper end of the optimal (m−1) -th composite quantization class. to add. At this time, the condition of the composite element that can be set as the upper end of the composite quantization class is the composite quantization class in which the highest element index in the composite element that is the upper end of the composite quantization class corresponds to the lower subset described above. This is a value that is larger than the lowest element index in the composite element that is the lower end of (step S209).

  The element updating unit 23 adds the above subset as a candidate for the top end of the composite quantization class to the top subset obtained by dividing the composite element corresponding to the top end of the optimal (m-1) -th composite quantization class. To do. At this time, the condition of the composite element that can be set as the upper end of the composite quantization class is the composite quantization class in which the highest element index in the composite element that is the upper end of the composite quantization class corresponds to the above-mentioned upper subset. This is a value that is larger than the lowest element index in the composite element that is the lower end of.

  When the upper subset obtained by dividing the composite element corresponding to the upper end of the optimal m-th composite quantization class is used as the upper end of the composite quantization class having the same subset as the lower end, the element update unit 23 An approximate error of the composite quantization class is newly acquired from the reference table. On the other hand, when a composite element other than the above is used as the upper end of the same composite quantization class, the approximation error of the same composite quantization class causes the composite element corresponding to the upper end of the composite quantization class before division of the same subset to be the same. This is the same as the approximation error when the lower end of the conversion class is set (step S210).

  The element update unit 23 determines whether the quantized class m is less than M (step S211). If the class m after quantization is M, the process returns to step S208. If the quantized class m is less than M, the process returns to step S205.

  The index conversion unit 24 outputs element indexes of M composite elements corresponding to the upper end of the optimal composite quantization class. Since each composite element includes only one element, the composite element can be uniquely identified (step S212).

FIG. 4 is a flowchart showing details of step S205 shown in FIG. In FIG. 4, ~ m is represented as m. ~ M is written as M. ~ K is expressed as k. The L ₀ is referred to as L_0. L to _m are expressed as L_m.

The minimum value determination unit 18 acquires a histogram of the input signal (step S301).
Minimum value determining unit 18 initializes the _{L 0} to 0 (step S302).
The minimum value determining unit 18 initializes S ₀ (i) to 0 for i = 1,..., K− to M + 1 (step S303).
The minimum value determination unit 18 repeats Steps S305 to S313 for the levels after quantization ˜m = 1,..., M−1 (Step S304).

The minimum value determination unit 18 repeats Steps S306 to S312 for the upper ends L _{~ m} = ~ m, ..., ~ K- (~ M- ~ m) -1. _L˜m represent the upper end of the section of the histogram corresponding to the ˜mth quantization level (step S305).
The minimum value determination unit 18 repeats Steps S307 to S309 for the section widths Δ _{~ m} = 1, ..., L _{~ m-} (~ m-1) -1. _Δ˜m indicate the section width of the section of the histogram corresponding to the mth quantization level (step S306).

The minimum value determination unit 18 _calculates an approximation error and a maximum element index in the _L.sup.m = (. _{DELTA..about.m-} 1) th composite element, a minimum element index in the _L.sup.mth composite element, and an approximation error. Get the stored table. The minimum value determination unit 18 uses a lower limit value _~ e (L _~ m-(Δ _{~ m} -1) of approximation errors for a subset [L _~ m-(Δ _{~ m} -1), L _{~ m} ]] of composite elements, _Lm ] is calculated based on the equation (12) (step S307).

The minimum value determination unit 18 acquires S to _m−1 (L to _m −Δ to _m ) and to e (L to _m − (Δ to _m −1), L to _m ). Minimum value determining unit _{18, S ~m-1 (L ~m} -Δ ~m) + ~e (L ~m - (Δ ~m-1), L ~m) is calculated (step S308).
The minimum value determining unit 18 determines whether or not the section widths Δ to _m are less than L _m − (m−1) (step S309). The minimum value determination unit 18 returns the process to step S306 when the section widths Δ to _m are less than L _m − (m−1).

When the section widths Δ to _m are L _m − (m−1), the minimum value determining unit 18 performs S to _m−1 (L for Δ to _m = 1,..., L _m − (m−1). _{~m -Δ ~m) + ~e (L} ~m - (Δ ~m -1), calculates the minimum value of _{L ~m),} is stored in a variable _{S ~m (L} ~m) _(step S310) .

The minimum value determining unit 18 is a value of Δ to _m that minimizes S to _m−1 (L to _m −Δ to _m ) + to e (L to _m − (Δ to _m−1 ), L to _m ). ^Is defined as Δ _{~ m} ^(L _{~ m} ⁾ . The minimum value determination unit 18 stores, for each of L to _m , ^ L to _m −1 [L to _m ] = L to _m −Δ to _m ⁽ L to _m ⁾ (step S311).

The minimum value determination unit 18 determines whether or not the upper ends L to _m are less than ~ K- (~ M- ~ m) (step S312). The minimum value determination unit 18 returns the process to step S305 when the upper ends L to _m are less than ~ K- (~ M- ~ m).
When the upper ends L to _m are ˜K− (˜M− to m), the minimum value determination unit 18 determines whether or not the level ˜m after quantization is less than ˜M (step S313).

The upper end tracking unit 20 (upper end determination unit) calculates the optimization interval width of the first to (M-1) th complex quantization classes according to the equation (17) when the level after quantization ˜m is less than ˜M, and the variable Store in _Δ˜M−1 ^* . The upper end tracking unit 20 calculates the optimum values ˜k−1−Δ to _M−1 ^* of the upper end of the ˜M−2 complex quantization class, and stores them in the variables L to _M−2 ^* (step S314).
The upper end tracking unit 20 repeats step S316 with the increment of ~ m set to -1 for the quantized level ~ m = ~ M-2, ..., 0 (step S315).

The upper end tracking unit 20 calculates the upper end values L to _m = ^ L to _m [L to _{m + 1} ^* ] of the composite quantization class (step S316).
The upper end tracking unit 20 determines whether or not the level ˜m after quantization is 0 or more. The upper end tracking part 20 returns a process to step S315, when the level -m after quantization is 0 or more. The upper end tracking unit 20 ends step S205 illustrated in FIG. 3 when the level after quantization ˜m is −1.

  As described above, the quantization apparatus 1 uses the second level that is smaller than the first level number as the histogram of the input signal that represents a value that can take a range represented by the first level number (input level number). It approximates with the histogram of the output signal showing the value which can take the range represented by a number (quantization level number). The quantization device 1 includes a histogram generation unit 11, a minimum value determination unit 18, and an upper end tracking unit 20. The histogram generation unit 11 configures (generates) a composite element that is a set of elements in the histogram of the input signal. The minimum value determination unit 18 uses an approximation error when approximating one or more values included in a plurality of composite elements as a representative value, and uses the element in the lowest composite element of the class that is a set of composite elements as the lower end. , By defining the lower limit of the approximation error for the histogram section of the input signal composed of the elements in the uppermost composite element of the class as the upper end, limiting the class boundary candidates that are class boundaries to composite elements, A quantized value that minimizes the cumulative value of the approximation error up to the class boundary and the minimum value of the cumulative value of the approximation error are stored in the memory. When selecting the class boundary, the upper end tracking unit 20 calls the stored minimum value from the memory, and uses the minimum value called from the memory for calculating the approximation error in the selection of the class boundary. When class boundary selection processing that selects a class boundary that minimizes the sum of approximation errors is performed, and the composite element selected as the class boundary includes multiple elements, the selected composite element is Dividing into subsets, the upper and lower subsets obtained by the division are newly added to the subset composite element, class boundary selection processing is performed, and the number of elements of the composite element selected as the class boundary is one. Repeat until.

  As a result, the quantization device 1 can reduce the amount of calculation required for quantization even when the conversion error that occurs when performing gradation conversion of a signal is minimized.

The quantization method uses the maximum level element of the lowest composite element as the lower end and the lowest level element of the highest composite element as the upper end as an approximation error when approximating multiple composite elements with representative values. The method further includes the step of defining as an approximation error for a section of the histogram of the input signal to be constructed (generated).

  In the quantization method, when a composite element selected as a class boundary is divided into an upper subset and a lower subset, the composite element at the class boundary of the upper class of the upper subset and the upper subset class is divided. By calculating the approximation error for the set of composite elements determined by the obtained upper subset, and reusing the approximation error for the composite element before splitting and the set of other composite elements in the upper subset And a step of calculating an approximation error for a set of composite elements determined by the upper subset and other composite elements based on a result of performing the selecting process.

  In the quantization method, when a composite element selected as a class boundary is divided into an upper subset and a lower subset, an approximation error for the set of composite elements determined by the lower subset and other composite elements is calculated, and the class in the subsequent stage is calculated. The method further includes a step of performing a process of selecting a boundary.

  In the quantization method, the set of composite elements includes a composite element in which a minimum level value of a value that is an element of the lowest composite element is smaller than a maximum level value of a value that is an element of the highest composite element Is further included.

  You may make it implement | achieve at least one part of the quantization apparatus in embodiment mentioned above with a computer. In that case, a program for realizing this function may be recorded on a computer-readable recording medium, and the program recorded on this recording medium may be read into a computer system and executed. Here, the “computer system” includes an OS and hardware such as peripheral devices. The “computer-readable recording medium” refers to a storage device such as a flexible medium, a magneto-optical disk, a portable medium such as a ROM and a CD-ROM, and a hard disk incorporated in a computer system. Furthermore, the “computer-readable recording medium” dynamically holds a program for a short time like a communication line when transmitting a program via a network such as the Internet or a communication line such as a telephone line. In this case, a volatile memory inside a computer system serving as a server or a client in that case may be included and a program held for a certain period of time. Further, the program may be a program for realizing a part of the above-described functions, and may be a program capable of realizing the functions described above in combination with a program already recorded in a computer system. You may implement | achieve using programmable logic devices, such as FPGA (Field Programmable Gate Array).

  The embodiment of the present invention has been described in detail with reference to the drawings. However, the specific configuration is not limited to this embodiment, and includes designs and the like that do not depart from the gist of the present invention.

DESCRIPTION OF SYMBOLS 1 ... Quantizer, 11 ... Histogram generation part, 12 ... Element number memory | storage part, 13 ... Class number memory | storage part, 14 ... Approximation error determination part, 15 ... Approximation error memory | storage part, 16 ... Initialization part, 17 ... Composite element Storage unit, 18 ... minimum value determination unit, 19 ... minimum value storage unit, 20 ... upper end tracking unit, 21 ... upper end storage unit, 22 ... update determination unit, 23 ... element update unit, 24 ... index conversion unit, 25 ... switch Part

Claims (7)

A histogram of an input signal representing a value that can take a range expressed by the first number of levels is represented by an output signal that represents a value that can take a range expressed by the second number of levels less than the first number of levels. A quantization method in a quantization device approximating with a histogram,
Constructing a composite element that is a set of elements in a histogram of the input signal;
An approximation error when approximating one or more values included in a plurality of the composite elements with a representative value is defined as an element in the composite element at the lowest end of the class that is a set of the composite elements, and By defining the lower limit value of the approximation error for the histogram section of the input signal configured with the element in the uppermost end of the composite element as the upper end, class boundary candidates that are boundaries of the class are limited to the composite element. Storing a quantized value that minimizes a cumulative value of the approximation error up to the class boundary and a minimum value of the cumulative value of the approximation error in a memory;
When selecting the class boundary, the stored minimum value is called from the memory, and the minimum value called from the memory is used for the calculation of the approximation error in the selection of the class boundary. Performing a class boundary selection process for selecting the class boundary that minimizes the sum of the approximation errors for the composite element;
When the composite element selected as the class boundary includes a plurality of elements, the selected composite element is divided into an upper subset and a lower subset, and the upper subset and the lower part obtained by the division are divided Adding a set to a new subset composite element, performing the class boundary selection process, repeating until the number of elements of the composite element selected as the class boundary becomes one;
Including a quantization method. As the approximation error when approximating a plurality of the composite elements with representative values, the element of the maximum level value of the composite element at the bottom end is the lower end, and the element of the minimum level value of the composite element at the top end is the upper end The quantization method according to claim 1, further comprising: defining the approximation error with respect to a section of a histogram of the input signal configured as follows. When the composite element selected as the class boundary is divided into the upper subset and the lower subset, the composite element of the class boundary of the upper class of the upper subset and the class of the upper subset Calculating the approximation error for the set of composite elements determined by the upper subset obtained by dividing;
Based on the result of performing the process of selecting the class boundary in the subsequent stage by reusing the approximation error for the composite element before division and the set of other composite elements into the upper subset. Calculating the approximation error for the set of composite elements determined by the other composite elements;
The quantization method according to claim 1, further comprising: When the composite element selected as the class boundary is divided into the upper subset and the lower subset, the approximation error for the set of composite elements determined by the lower subset and the other composite elements is calculated, The quantization method according to claim 1, further comprising: performing a process of selecting the class boundary in the subsequent stage. The set of composite elements is configured by the composite elements having a minimum level value that is an element of the composite element at the lowest end smaller than a maximum level value of a value that is an element of the composite element at the top end. The quantization method according to claim 1, further comprising a step. A histogram of an input signal representing a value that can take a range expressed by the first number of levels is represented by an output signal that represents a value that can take a range expressed by the second number of levels less than the first number of levels. A quantization device approximating with a histogram,
A histogram generator that constitutes a composite element that is a set of elements in the histogram of the input signal;
An approximation error when approximating one or more values included in a plurality of the composite elements with a representative value is defined as an element in the composite element at the lowest end of the class that is a set of the composite elements, and By defining the lower limit value of the approximation error for the histogram section of the input signal configured with the element in the uppermost end of the composite element as the upper end, class boundary candidates that are boundaries of the class are limited to the composite element. A minimum value determination unit that stores a quantized value that minimizes a cumulative value of the approximation error up to the class boundary and a minimum value of the cumulative value of the approximation error in a memory;
When selecting the class boundary, the stored minimum value is called from the memory, and the minimum value called from the memory is used for the calculation of the approximation error in the selection of the class boundary. A class boundary selection process for selecting the class boundary that minimizes the sum of the approximation errors is performed on the composite element, and when the composite element selected as the class boundary includes a plurality of elements, the selected The composite element is divided into an upper subset and a lower subset, the upper subset and the lower subset obtained by the division are newly added to the subset composite element, the class boundary selection process is performed, and the class boundary is selected. An upper end tracking unit that repeats until the number of elements of the composite element selected as
A quantization apparatus comprising: A histogram of an input signal representing a value that can take a range expressed by the first number of levels is represented by an output signal that represents a value that can take a range expressed by the second number of levels less than the first number of levels. To the computer of the quantizer that approximates with a histogram,
Configuring a composite element that is a set of elements in a histogram of the input signal;
An approximation error when approximating one or more values included in a plurality of the composite elements with a representative value is defined as an element in the composite element at the lowest end of the class that is a set of the composite elements, and By defining the lower limit value of the approximation error for the histogram section of the input signal configured with the element in the uppermost end of the composite element as the upper end, class boundary candidates that are boundaries of the class are limited to the composite element. A procedure for storing a quantized value for minimizing a cumulative value of the approximation error up to the class boundary and a minimum value of the cumulative value of the approximation error in a memory;
When selecting the class boundary, the stored minimum value is called from the memory, and the minimum value called from the memory is used for the calculation of the approximation error in the selection of the class boundary. Performing a class boundary selection process for selecting the class boundary that minimizes the sum of the approximation errors for the composite element;
When the composite element selected as the class boundary includes a plurality of elements, the selected composite element is divided into an upper subset and a lower subset, and the upper subset and the lower part obtained by the division are divided Adding a new set to a subset composite element, performing the class boundary selection process, and repeating until the number of elements of the composite element selected as the class boundary becomes one;
Quantization program to execute.

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