CN100452880C - Integral discrete cosine transform method in use for encoding video - Google Patents

Abstract

The present invention provides an integral discrete cosine transform method for video encoding, which belongs to the technical field of video transmission. The method is that firstly, every element in an integral discrete cosine transform core is disassembled in an equivalent mode, so N matrixes are obtained. The N matrixes are grouped and added, so M auxiliary transform cores are obtained. According to the M auxiliary transform cores, M auxiliary transforms are calculated, and according to the low-to-high order of a suffix, M auxiliary transform results are combined. A transformed result DX<T> matrix of a first processing unit is used as a Y matrix of a second processing unit. The steps are repeated, so an integral discrete cosine transform coefficient is obtained. The method of the present invention has the advantages that by using the cutting operation of integral DCT transform, before the cutting operation is carried out, partial computation redundancy is removed, so the whole bit wide of an adder in PU is decreased, and hardware resources are saved.

Description

A Method of Integer Discrete Cosine Transform for Video Coding

technical field

The invention relates to an integer discrete cosine transform method for video coding, belonging to the technical field of video transmission.

Background technique

In the prior art, a paper titled "Development of integer cosine transforms by the principle of dyadic symmetry" (Proc.Inst.Elect.Eng., Partl, vol.136, Aug.1989, pp.276-282.) is published An integer discrete cosine transform method for video coding that can be used for video coding is provided. Its principle is: the mathematical formula of discrete cosine transform is F=D×D ^T , wherein D, X is a matrix of N×N, and D ^T is a matrix of D Transpose, D is called the transformation kernel of this transformation. For integer discrete cosine transform, the elements of D are all integers, and the size and value of D are not unique. For example, the transformation kernel used in the benchmark grade of the H.264 video codec standard is $\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 1& -2& 2& -1\end{array}\right],$ In the digital audio and video codec standard (hereinafter referred to as AVS), the transform kernel is $\left[\begin{array}{cccccccc}8& 8& 8& 8& 8& 8& 8& 8\\ 10& 9& 6& 2& -2& -6& -9& -10\\ 10& 4& -4& -10& -10& -4& 4& 10\\ 9& -2& -10& -6& 6& 10& 2& -9\\ 8& -8& -8& 8& 8& -8& -8& 8\\ 6& -10& 2& 9& -9& -2& 10& -6\\ 4& -10& 10& -4& -4& 10& -10& 4\\ 2& -6& 9& -10& 10& -9& 6& -2\end{array}\right].$

The traditional method for quickly realizing the above-mentioned integer discrete cosine transform is to use a butterfly computing structure in the form of a flow graph to calculate the 1-dimensional integer discrete cosine transform (hereinafter referred to as DCT), and then use two processing units with the same structure ( ^hereinafter referred to as PU) in series to realize the whole two-dimensional integer DCT transformation, which is ^called the method of decomposing rows and columns ^. The calculation completed by one PU is DX ^T , that is, the one-dimensional integer DCT transformation, and the entire transformation can be realized by connecting two PUs in series. Therefore, the core calculation of the integer discrete cosine transform is the PU. The mathematical operations performed by the PU include: matrix transposition and matrix multiplication. For the first PU, the input matrix can be input according to the transposed matrix, while for the second A PU requires an arithmetic unit dedicated to calculating the transpose.

In the above-mentioned traditional DCT algorithm, taking the transformation kernel adopted by AVS as an example, the implementation flow diagram based on the butterfly structure is shown in Figure 2:

x0~x7 is a column of data of X ^T , and the whole flow graph realizes the result X0~X7 of multiplying a column of data of X and the transformation kernel, respectively, the 8 columns of data of X ^T are calculated through the above flow graph, and a PU can be obtained The calculation result of DX ^T . This structure is beneficial to realize by hardware, because the multiplication operation is transformed into the multiplication of small integers, and the multiplication of small integers can be realized by shift operation.

The disadvantage of the existing traditional butterfly structure is that since the integer transform kernel elements are all integers, while the standard discrete cosine transform kernel is an orthogonal matrix, the absolute value of the elements is all less than 1, so the multiplication with the integer kernel will have a large Data gain, in order to offset this gain, an additional truncation operation is required. Since the data is represented in binary, the truncation operation specifically refers to directly discarding several bits of the binary representation of the data. Such an operation is equivalent to performing a certain power of 2 on the data. division. The result of the truncation operation is that the lower bits of the data binary representation are discarded. As shown in Figure 1, each PU is followed by a right shift operation. The first PU shifts a bit to the right, and the second PU shifts to the right by b bits. Taking the transformation kernel used by AVS as an example, the values of a and b satisfy a+b=9, because the integer transformation kernel is the same as the standard floating-point number Compared with the transformation kernel, the gain is 2 to the 9th power, and this value is different due to different transformation kernels. Due to the right shift operation, the lower bits of the calculation result are discarded, so all calculations related to these lower bits in the PU are redundant, and these redundant calculations lead to waste of hardware resources. However, due to the traditional butterfly structure Inherent in nature, these redundancies cannot be extracted separately, which is not good for the effective use of hardware resources.

Contents of the invention

The purpose of the present invention is to propose an integer discrete cosine transform method for video coding, to address the shortcomings of the prior art, to directly remove part of the calculation redundancy in the PU calculation process, so as to save hardware resources.

The integer discrete cosine transform method that the present invention proposes for video coding comprises the following steps:

(1) Each element in the integer discrete cosine transform kernel is equivalently split to obtain N matrices, D ₀ , D ₁ , ... D _N-1 , then the expression D of the original integer discrete cosine transform kernel D =(D ₀ +D ₁ +...+D _N-1 );

(2) adding the above N matrices into groups to obtain M sub-transformation kernels H ₀ , H ₁ , H _M-1 , wherein each sub-transformation kernel is the sum of a set of n matrices, 1<n<N;

(3) According to the above M sub-transform kernels, calculate M sub-transformations H ₀ X ^T , H ₁ X ^T , H ₂ X ^T ... H _M-1 X ^T , and perform M sub-transformations in the order of subscripts from low to high The results are combined to obtain the transformation result of the first processing unit, that is, DX ^T =(H ₀ X ^T +H ₁ X ^T +…+H _M-1 X ^T ) ^T , where X is the brightness block matrix of the video frame, and X ^T is the transpose of X;

(4) the transformation result DX ^T matrix of the above-mentioned first processing unit is used as the Y matrix of the second processing unit, repeat steps (1)～(3), obtain DY ^T , then the integer discrete cosine transform coefficient is: F= DXD ^T ＝(H ₀ +H ₁ +…H _M-1 )X(H ₀ +H ₁ +…+H _M-1 ) ^T ＝(H ₀ +H ₁ +…+H _M-1 )(H ₀ X ^T +H ₁ X ^T +…+H _M-1 X ^T ) ^T .

In the above method, the method for equivalently splitting each element in the integer discrete cosine transform kernel includes the following steps:

(1) According to the binary representation of the integer element in the transformation kernel, it is split into the sum of multiple powers of 2;

(2) According to the order from low power to high power, the i-th power split items of each element in the integer discrete cosine transform kernel are combined into a matrix Di, where 0≤i≤N-1.

In the above method, the method of merging the M sub-transformations is:

(1) For each of the M sub-transformations, a common factor of 2 to the power of j (0≤j≤N-1) is proposed;

(2) Merge H ₀ X ^T and H ₁ X ^T , set the common factor proposed from H ₁ X ^T to be 2 ^j1 , if j1≤a, then only merge the parts higher than j1 in the binary representation of matrix elements , if j1>a, then only the part of the binary representation of the matrix elements higher than a is merged, and the merged result is recorded as X _Temp , where a is determined by the integer discrete cosine transform kernel to be performed;

(3) Merge the above-mentioned merged result X _Temp and H ₂ X ^T according to the method of the above-mentioned step (2), and the merged result is recorded as X _Temp ;

(4) Steps (2) and (3) are repeated, and all sub-transformations are merged one by one in order to complete the merger of M sub-transformations.

The integer discrete cosine transform method for video coding proposed by the present invention has the characteristics and advantages of adopting a method completely different from the traditional fast algorithm, starting from the integer transform kernel, and splitting the transform kernel whose elements are small integers into For several sub-transformation cores, since each sub-transformation core can propose a common factor of a power of 2, the truncation operation of the integer DCT transformation can be used to reduce the overall bit width of the adder in the PU and save hardware resources. Compared with the traditional butterfly structure, the method of the present invention can separately separate a part of the calculation redundancy in the integer discrete cosine transform, and remove these calculation redundancy before the truncation operation, thereby reducing the overall bit width of the adder of the PU , saving hardware resources, compared with the existing butterfly structure, the method of the present invention adopts programmable gate array (FPGA) or customizable integrated circuit (ASIC) to realize, and hardware resources can save more than 10%.

Description of drawings

FIG. 1 is a flowchart of an existing integer DCT using a decomposed structure.

FIG. 2 is a flow graph structure diagram of a traditional DCT used to realize AVS.

FIG. 3 is a structural block diagram of a conventional register file used to implement a transpose operation.

Fig. 4 is a partial flow graph structure diagram for realizing DCT by adopting the method of the present invention.

FIG. 5 is a part of the flowchart structure diagram shown in FIG. 2 .

Fig. 6 is a hardware structure diagram for realizing the DCT of the present invention.

Fig. 7 and Fig. 8 are comparison diagrams of the effect of the adder resource adopted by the method of the present invention and the existing method.

Detailed ways

The integer discrete cosine transform method for video coding proposed by the present invention first performs equivalent splitting of each element in the integer discrete cosine transform kernel to obtain N matrices, D ₀ , D ₁ , ... D _N-1 , then the expression D=(D ₀ +D ₁ +…+D _M-1 ) of the original integer discrete cosine transform kernel D; the above N matrices are grouped and added to obtain M sub-transform kernels H ₀ , H ₁ , H _{M -1} , where each sub-transformation kernel is the sum of a set of N matrices; according to the above M sub-transformation kernels, calculate M sub-transformations H ₀ X ^T , H ₁ X ^T , H ₂ X ^T ... H _M-1 X ^T , and Merge the M sub-transformation results in order of subscripts from low to high to obtain the transformation result of the first processing unit, that is, DX ^T =(H ₀ X ^T +H ₁ X ^T +...+H _M-1 X ^T ) ^T , where X is the brightness block matrix of the video frame, and X ^T is the transposition of X; the transformation result DX ^T matrix of the first processing unit is used as the Y matrix of the second processing unit, and the above steps are repeated to obtain DY ^T , then the integer discrete cosine transform coefficients are: F＝DXD ^T ＝(H ₀ +H ₁ +…H _M-1 )X(H ₀ +H ₁ +…+H _M-1 ) ^T ＝(H ₀ +H ₁ +…+H _M-1 )(H ₀ X ^T +H ₀ X ^T +…+H _M-1 X ^T ) ^T .

In the above method, the method for equivalently splitting each element in the integer discrete cosine transform kernel may include the following steps: (1) according to the binary representation of the integer element in the transform kernel, split into a sum of multiple powers of 2 ; (2) According to the order from the low power to the high power, the i-th power split items of each element in the integer discrete cosine transform kernel are combined into a matrix Di, where 0≤i≤N-1.

In the above method, the method for merging the M sub-transformations may be: (1) for each sub-transformation in the M sub-transformations, respectively propose a common factor of 2 to the power of j (0≤j≤N-1); (2 ) merge H ₀ X ^T and H ₁ X ^T , set the common factor proposed from H ₁ X ^T to be 2 ^j1 , if j1≤a, then only merge the parts higher than j1 in the binary representation of matrix elements, if j1>a, then only the part of the binary representation of the matrix elements that is higher than a is combined, and the combined result is recorded as X _Temp , where a is determined by the integer discrete cosine transform kernel to be performed; (3) the above combined result X _Temp and H ₂ X ^T is merged according to the method of the above step (2), and the merged result is recorded as X _Temp ; (4) Steps (2) and (3) are repeated, and all sub-transformations are merged one by one in order to complete the merging of M sub-transformations.

The invention belongs to the field of image coding and image processing, and is especially applied to the rapid realization of integer discrete cosine transform.

In the present invention, the DCT transformation takes the 8-point one-dimensional DCT transformation as the basic unit, and adopts the decomposed structure of rows and columns to realize the two-dimensional DCT transformation. The mathematical expression of row and column decomposition is F=DXD ^T ＝D(DX ^T ) ^T , that is, DX ^T is a PU, and the function completed by a PU is to transpose the input matrix and multiply the matrix D to the left, through two The serialization of PUs can realize two-dimensional DCT transformation through one-dimensional DCT transformation, as shown in Figure 1. Although the content of the mathematical operations of the two PUs is the same, their transpose operations are different. The transpose operation of the first PU can be input according to the transposed matrix, while the second PU must have a dedicated The transposition operation unit, the transposition operation unit adopted in the present invention is realized through the register file shown in FIG. 3 in the prior art.

The present invention is aimed at the integer DCT transformation, and the transformation kernel of the integer DCT transformation is composed of integers, which is convenient for hardware realization by using an adder and a shifter.

For any integer transformation kernel, especially a small integer transformation kernel, the present invention splits each element thereof equivalently. Additive sums of powers, such as element 6, can be split into 2 ¹ +2 ² . According to the order from the low power to the high power, the i-th power split items of the transformation kernel elements are combined into a matrix Di, then Di only contains 2 ⁱ and 0 elements. According to the above operation, N matrices D ₀ , D ₁ , ... D _N-1 are obtained, and these N matrices are grouped and added again to obtain M new sub-transformation kernels H ₀ , H ₁ , H _M-1 , where The elements of each transform kernel are the sum of the correspondingly positioned elements of the matrices that make up the set.

The obtained M sub-transformation kernels H ₀ , H ₁ , and H _M-1 are called sub-transformation kernels in the present invention. Then the result of the whole transformation is the sum of the results of these M sub-transformations, namely

F＝DXD ^T ＝(H ₀ +H ₁ +…+H _M-1 )X(H ₀ +H ₁ +…+H _M-1 ) ^T

＝(H ₀ +H ₁ +…+H _M-1 )(H ₀ X ^T +H ₁ X ^T +…+H _M-1 X ^T ) ^T

Wherein for H ₀ X ^T , H ₁ X ^T ... H _M-1 X ^T , the present invention defines sub-transformation.

For each sub-transformation, a common factor of 2 to the power of j (0≤j≤N-1) can be proposed. This common factor can reduce the absolute value of the transformation kernel elements, so that the elements of the sub-transformation kernels are all in a small In this way, the calculation of the sub-transform can be greatly simplified, and the input bit width of the adder can be reduced to a certain extent by using the truncation operation. The method of reducing the input bit width is described as follows:

Suppose the truncation operation is to truncate the lowest a bit of the resulting binary representation.

The integer DCT algorithm proposed by the present invention is realized by adopting the decomposed structure of ranks ^{and columns} . After the results of each sub-transformation _{are calculated} , the results need to be combined _. X ^T merge, and repeat this process until all sub-transforms have been merged.

Since H ₁ X ^T can propose a common factor 2 ^j1 , when adding the corresponding elements of the two sub-transformation results, the addition of the low j1 bits of the binary representation of the elements will not overflow. If j1≤a, then it can be Directly discard the lower j1 bits of the result of H ₀ X ^T without performing the addition operation. If j1>a, then directly discard the lower a bits of the result of H ₀ X ^T without performing the addition operation, and record the combined result as X _temp .

Merge the first two sub-transforms as described above, and record the result of the merge as X _Temp . Then merge X _Temp with the sub-transformation H ₂ X ^T , the sub-transformation H ₂ X ^T can come up with the common factor 2 ^j2 , follow the same process as above to get the combined result, which is still recorded as X _Temp .

Then merge X _Temp with the remaining sub-transforms in turn until all sub-transforms are merged to get the final result H ₀ X ^T +H ₁ X ^T +…+H _M-1 X ^T , that is, DX ^T , record is the Y matrix.

If the structure of row and column decomposition is used for forward transformation, then the calculation result DX ^T of the first PU above needs to be re-inputted as a Y matrix to the second PU with the same structure to calculate DY ^T . After the calculation of the second PU, the Complete a complete integer DCT transform.

As shown in Fig. 4, it is the partial realization structure described by the flow diagram of the method proposed by the present invention, and Fig. 5 shows the realization form of the same part of the traditional butterfly operation, and it can be seen that the two are compared, and the present invention proposes The increase of the bit width in the direction of the algorithm data flow is smaller than that of the traditional butterfly algorithm. Fig. 4 and Fig. 5 only provide the flow graph form of a part of transformation coefficients as a comparison, because the algorithm proposed by the present invention is quite different from the structure of the traditional butterfly algorithm, and it is not convenient to describe in the form of a flow graph, otherwise it will be It is messy and not conducive to explaining the difference in algorithms.

An embodiment of the inventive method is introduced below:

Taking the integer transform kernel used in the AVS video coding standard as an example to illustrate the fast integer DCT implementation algorithm based on transform kernel bit-plane decomposition proposed by the present invention.

$\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 9</span>}& \mathrm{<span\; class="notranslate">\; 6</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}\\ \mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 10</span>}\\ \mathrm{<span\; class="notranslate">\; 9</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}& \mathrm{<span\; class="notranslate">\; 6</span>}& \mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 6</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 9</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}\\ \mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}& \mathrm{<span\; class="notranslate">\; 9</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}& \mathrm{<span\; class="notranslate">\; 10</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}& \mathrm{<span\; class="notranslate">\; 6</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]<span\; class="notranslate">\; .</span>$

The elements of the integer transformation kernel are all integers whose absolute value is less than 16, and the fourth and above digits of the binary representation are all zeros, so the transformation kernel can be split into four matrices (the elements 6 and -6 are special, and their respective There can be two different split methods: 6=4+2=8-2, -6=-4-2=-8+2, after practical comparison, it is found that 6=8-2, -6=-8+ 2 This form of splitting is more favorable)

$\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ $\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]$

$\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ $\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]$

Among them, the above two matrices can be combined as

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$

The next two matrices can be combined as

${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="h"\; filenames="C20061001216100141.png"\; state="\{\{state\}\}">h</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}\\ \mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 4</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 8</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]$

Such a PU is decomposed into DX ^T ＝H ₀ X ^T +H ₁ X ^T , where H ₁ can also be written as ${\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="C20061001216100141.png"\; state="\{\{state\}\}">h</figure-callout>}}_1=4\&times;\left[\begin{array}{cccccccc}2& 2& 2& 2& 2& 2& 2& 2\\ 2& 2& 2& 0& 0& -2& -2& -2\\ 2& 1& -1& -2& -2& -1& 1& 2\\ 2& 0& -2& -2& 2& 2& 0& -2\\ 2& -2& -2& 2& 2& -2& -2& 2\\ 2& -2& 0& 2& -2& 0& 2& -2\\ 1& -2& 2& -1& -1& 2& -2& 1\\ 0& -2& 2& -2& 2& -2& 2& 0\end{array}\right],$ The two sub-transform cores of such a PU only contain four integers -2, -1, 1, and 2. Due to the similarity between the coefficients, the repeatability of the operation will increase, and the multiplexing degree of the adder can be fully utilized. .

According to the left-right symmetry of the integer DCT transformation kernel, the transformation kernels of the two sub-transformations H ₀ X ^T and H ₁ X ^T can be simplified as

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="h"\; filenames="C20061001216100141.png"\; state="\{\{state\}\}">h</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]$

Then the input of the even-numbered rows of the two sub-transformations is the same, and the input of the odd-numbered rows is the same, so the adder multiplexing can also be realized between the even-numbered rows and between the odd-numbered rows of the two sub-transformations, with the odd-numbered behavior of H ₀ X ^T and H ₁ X ^T For example, H ₀ X ^T can be decomposed into odd row transformation H _0e X _p ^T and even row transformation H _0o X _m ^T , where the transformation kernel of H _0e X _p ^T corresponds to the odd row elements of H ₀ , H _0o X _m ^T The transformation kernel of corresponds to the even-numbered row elements of H ₀ , and the results of the two transformations correspond to the odd-numbered and even-numbered row results of H ₀ X ^T , and the same conclusion is drawn for H ₁ X ^T. As described above, since H _0e X _p ^T and H _1e X _p ^T have the same input, they can be multiplexed by the adder.

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="i"\; filenames="C20061001216100141.png"\; state="\{\{state\}\}">i</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="i"\; filenames="C20061001216100122.png,C20061001216100142.png"\; state="\{\{state\}\}">i</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="3"\; label="i"\; filenames="C20061001216100122.png"\; state="\{\{state\}\}">i</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="4"\; label="i"\; filenames="C20061001216100122.png,C20061001216100152.png"\; state="\{\{state\}\}">i</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 4</span>}}\end{array}\right]<span\; class="notranslate">\; ,</span>$

Observing H _0e and H _1e , it can be found that the calculation redundancy of these two transformation kernels is very large, and X _p ^T [0]±X _p ^T [3] and X _p ^T [1]±X _p ^T [ 2], the calculation results of H _0e X _p ^T and H _1e X _p ^T can be obtained by adding and subtracting the combination of these four calculation results.

same for

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; o</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; o</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right]<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; \&times;</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$

$x_m^T=\left[\begin{array}{c}{x}_{i0}-x_{i7}\\ x_{i1}-x_{i6}\\ x_{i2}-x_{i5}\\ x_{i3}-{\mathrm{<figure-callout\; id="4"\; label="x\; i"\; filenames="C20061001216100122.png,C20061001216100152.png"\; state="\{\{state\}\}">x</figure-callout>}}_i\end{array}\right],$ The calculation redundancy of these two transformation cores can also be used to compress the number of adders used, and since H _1e can propose a common factor of 4, H _1o can propose a common factor of 8, which is useful for reducing The input bit width of the adder is very useful, because in the DCT transformation of video coding, if it is based on row-column decomposition, after the first PU, the lowest 3 bits of the transformation result will be truncated, and since H _1e can propose a common factor of 4, H _1o A common factor of 8 can be proposed, so when combining sub-transform calculation results, 3-bit truncation can be considered together, and the addition of the lower 2 bits of H _0e X _p ^T +H _1e X _p ^T will not overflow, so this can be ignored directly Two bits, reduce the input bit width of the adder, the addition of the lower 3 bits of H _0o X _m ^T +H _1o X _m ^T will not overflow, so these three bits can be directly ignored to reduce the input bit width of the adder. Also notice that the elements of the first row and the third row of H _0e are all zero, so no additional adder is needed for the combination of these two rows when the sub-transforms are merged.

Through the series connection of the two PUs described above, the method of the present invention can be completed to realize fast and lossless integer DCT transformation.

Adopt the algorithm based on transform nuclear bit plane decomposition that mentions in the present invention, realize the integer DCT transform that meets AVS standard, can obtain the main hardware resources that the proposed algorithm of the present invention consumes statistically:

name number 8bit adder 8 9bit adder 8 10bit adder 10 11bit adder 10 12bit adder 2

And the main hardware resources consumed by traditional butterfly operations:

name number 8bit adder 8 9bit adder 8 10bit adder 2 11bit adder 6 12bit adder 4 13bit adder 6

14bit adder 15bit adder 4

Below by Fig. 7 and Fig. 8 illustrate the present invention method compared with prior art, advantage and effect that have: Fig. 7 is a butterfly algorithm, Fig. 8 is the method of the present invention, and horizontal axis represents the input bit width of adder, The vertical axis represents the number used, the first PU of the integer DCT transform conforming to the AVS standard has a 3-bit truncation, and there is a 7-bit truncation for the second PU. It can be seen by comparison that the number of adders used by the two algorithms is the same (38), but the adders used in the algorithm based on bit-plane decomposition are concentrated in 8, 9, 10, 11 inputs, while the butterfly operation It varies from 8 to 15, so the input bit width of the algorithm based on bit-plane decomposition is generally low. Therefore, if the DCT module is implemented by FPGA or ASIC, the algorithm based on bit-plane decomposition will save more resources. Using FPGA implementation, taking Xilinx VirtexIV as an example, the algorithm based on bit-plane decomposition will save more than 10% of resources.

For different integer transformation kernels, bit-plane decomposition algorithms can be used, but for different integer transformation kernels, the decomposed bit-plane matrix and the bit-plane grouping with the maximum adder multiplexing degree may be different, but they are all as far as possible Extract large coefficients, reduce the range of elements of the sub-transform kernel, use truncation to reduce the bit width of the adder, and adjust the bit plane coefficients to make the multiplexing of the adder as large as possible, and reduce the resource consumption from these two aspects overhead.

Claims (1)

1. An integer discrete cosine transform method for video coding, characterized in that the method comprises the following steps: (1) Each element in the integer discrete cosine transform kernel is equivalently split to obtain N matrices, D ₀ , D ₁ , ... D _N-1 , then the original integer discrete cosine transform kernel D The expression D=(D ₀ +D ₁ +...+D _N-1 ), wherein the method for equivalently splitting each element in the integer discrete cosine transform kernel includes the following steps: (1-1) According to the binary representation of the integer elements in the transformation kernel, it is split into the sum of multiple powers of 2; (1-2) According to the order from the low power to the high power, the splitting items of the i power of each element in the integer discrete cosine transform kernel are combined into a matrix Di, where 0≤i≤N-1; (2) adding the above N matrices into groups to obtain M sub-transformation kernels H ₀ , H ₁ , H _M-1 ; (3) According to the above M sub-transform kernels, calculate M sub-transforms H ₀ X ^T , H ₁ X ^T , H ₂ X ^T ... H _M-1 X ^T , and perform M sub-transformations in order of subscripts from low to high The sub-transformation results are combined to obtain the transformation result of the first processing unit, that is, DX ^T =(H ₀ X ^T +H ₁ X ^T +...+H _M-1 X ^T ) ^T , where X is the brightness of the video frame Block matrix, X ^T is the transposition of X, wherein the method for merging M sub-transformation results includes the following steps: (3-1) For each of the M sub-transformations, a common factor of 2 to the power of j is proposed, where 0≤j≤N-1; (3-2) Merge H ₀ X ^T and H ₁ X ^T , set the common factor proposed from H ₁ X ^T to be 2 ^j1 , if j1≤a, then only the bit higher than j1 in the binary representation of matrix elements Partial merging, if j1>a, then only the part of the binary representation of the matrix elements higher than a is merged, and the result of the merging is recorded as X _Temp , where a is determined by the integer discrete cosine transform kernel to be performed; (3-3) Merge the above-mentioned merged result X _Temp and H ₂ X ^T according to the method of the above-mentioned step (3-2), and record the merged result as X _Temp ; (3-4) Repeating steps (3-2) and (3-3), merging all sub-transformations one by one in turn, completing the merging of M sub-transformations; (4) make above-mentioned conversion result D X ^T matrix=Y, with Y matrix as the X matrix in the step (3), repeat steps (1)～(3), obtain DY ^T , then integer discrete cosine transform coefficient is: F= DXD ^T ＝(H ₀ +H ₁ +...H _M-1 )X(H ₀ +H ₁ +...+H _M-1 ) ^T ＝(H ₀ +H ₁ +...+H _{M -1} )(H ₀ X ^T +H ₁ X ^T +...+H _M-1 X ^T ) ^T .

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