CN107147910B - A kind of panoramic video hexagon method of sampling and device - Google Patents

Abstract

The invention discloses a kind of panoramic video hexagon method of sampling and devices, comprising: spherical surface is divided into N=6K annular surface with equally spaced weft, annular surface is successively denoted as O_i, i=0,1 ... N-1, from the angle of the centre of sphere, O_iIt arranges in accordance with the order from top to bottom；With equally spaced warp by annular surface O_iIt is divided evenly into T_iA region；The average value of the light in each region on the spherical surface being calculated, the sampled value as the corresponding sampled point in the region；Sampled value is aligned in a matrix.The method of sampling avoids sampling structure redundancy, so that sampling number reduces under conditions of reaching identical spatial resolution, the data volume after sampling reduces.

Description

A kind of panoramic video hexagon method of sampling and device

Technical field

The present invention relates to video coding techniques, more particularly to a kind of panoramic video hexagon method of sampling and device.

Background technique

Panoramic video refers to the scene that a point of observation surrounding is all in space, the institute that can be received by this point of observation There is light composition.Panoramic video can be abstracted into a spherical surface centered on point of observation.

When active computer handles panoramic video, discretized space inevitably is carried out to panoramic video and is adopted Sample.When carrying out discretized space sampling to panoramic video, need to guarantee certain space sampling densities, needed for reaching Clarity.Meanwhile it being needed in some way in view of the memory of computer is not suitable for the data of storage spherical structure again Sampled point is aligned in plane.

Spherical surface sampling process is usually divided into three steps: the first step by the existing spherical surface method of sampling, mapping, spherical Map For a region in plane；Second step, plane sampling, designs the distribution side of sampled point based on the plane domain after mapping Formula；Sampled point, is arranged as a rectangle by third step, arrangement.

The first step mapping during, be not often homalographic mapping, and the different zones of spherical surface before the mapping after The ratio of area is also different.During second step plane samples, sampled point is often designed in the mapped flat Face is uniformly distributed in region.The first step and second step combine, and will result in sampled point being unevenly distributed on spherical surface.Reason By upper, the sampling number of spherical surface equator should be more than sampling number near the two poles of the earth.Third step tends not to do in arrangement To the complete adjacent of content, it is unfavorable to come to Video coding compression strap.

Currently, there are three types of the common spherical surface method of samplings: the sampling of longitude and latitude figure, hexahedron sampling and pyramid sampling.

The sampling of longitude and latitude figure is as shown in Figure 1, any one point can use locating longitude θ and latitude on spherical surfaceDescription, θ ∈ [0,2 π),Then spherical Map can be arrivedThe next the ratio of width to height of coordinate system is the rectangle of 2:1.It is right This rectangle carries out uniform sampling.Although the longitude and latitude figure method of sampling has complete adjacency, sampled near the two poles of the earth of spherical surface Density is excessively high, can generate very big redundancy.

Hexahedron sampling as shown in Fig. 2, be six faces of its circumscribed regular hexahedron by spherical Map, to obtain six first A plane square, then uniform sampling is carried out to six plane squares, six squares are finally combined into one by some way A rectangle.Distribution of the sampled point on spherical surface of hexahedron sampling be also it is non-uniform, still remain biggish redundancy, and square The content of shape does not have complete adjacency yet.

Pyramid sampling is as shown in figure 3, be first a circumscribed positive rectangular pyramid by spherical Map, then by the every of the positive rectangular pyramid A side (isosceles triangle) is compressed along the direction of bottom edge vertical line, until apex angle becomes right angle.Bottom surface and four warps in this way The side for crossing Quadratic Map can just be combined into a square.Uniform sampling finally is carried out to this square.Pyramid sampling Distribution of the sampled point on spherical surface be also it is non-uniform, the content of rectangle does not have complete adjacency yet.Its degree of irregularity Between the sampling of longitude and latitude figure and hexahedron sampling, degree of redundancy is also between the sampling of longitude and latitude figure and hexahedron sampling.

In conclusion the major defect of the existing panoramic video method of sampling is there are biggish sampling structure redundancy and most End form at rectangle do not have content intact adjacency.Sampled point uneven distribution on spherical surface can not only bring sampling redundancy, Returning video quality evaluation makes troubles, because the corresponding area of spherical surface of final sampled point is different, each sampling Influence of the distortion of point to video quality is also different, and is handled calculating panoramic video by certain, such as compressed and decompressed When rear mass loss, it is necessary to consider that the importance of each sampled point is different.Rectangular video in terms of content non-conterminous Property is also unfavorable for the compressed encoding of Video coding.

Summary of the invention

In order to overcome the deficiencies of the above existing technologies, purpose of the present invention is to provide a kind of panoramic video hexagons to adopt Quadrat method and device, distribution of the sampled point on spherical surface can be made to meet, and equator sampling number is more, and the two poles of the earth nearby sample Points are seldom, so as to avoid sampling structure redundancy, so that sampling number subtracts under conditions of reaching identical spatial resolution It is few, moreover, the rectangle being combined into has good content adjacency, coding compression efficiency can be improved.

In view of the above and other objects, the present invention provides a kind of panoramic video hexagon method of sampling, include the following steps:

Spherical surface is divided into N=6K annular surface with equally spaced weft by step 1, and wherein K is positive integer；By what is obtained Annular surface is successively denoted as O_i, i=0,1 ... N-1, from the angle of the centre of sphere, by O_iIt arranges in accordance with the order from top to bottom；.

Step 2, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, wherein as 0≤i < K, T_i=8i+4, as K≤i < 3K, T_i=10K, as 3K≤i < N, T_i=T_N-1-i, wherein a warp is 0 ° of warp；Will To region be successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the increase of east longitude degree or west longitude The sequence of degree reduction arranges, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Step 3 calculates each region A on the spherical surface that step 2 obtains_i,jThe average value of interior light, as the area The sampled value S of the corresponding sampled point in domain_i,j；

Step 4, by sampled value S_i,jIt is arranged in a matrix, indicates S with x_i,jLocating column, y indicate S_i,jLocating row, Enable d_i=i-5K, h=K-d_i- 1, then

As 0≤i < K, 0≤j < 2i+1, y=i+K, x=11K-i+j-1；

As 0≤i < K, 2i+1≤j < 4i+2, y=3i+K-j+1, x=11K+i；

As 0≤i < K, 4i+2≤j < 6i+3, y=K-i-1, x=11K+5i-j+2；

As 0≤i < K, 6i+3≤j < 8i+4, y=K+j-7i-4, x=11K-i-1；

As K≤i < 5K, y=i-K, x=j；

As 0≤d_iWhen < K, 0≤j < 2h+1, y=d_i+ 2K, x=11K-h+j-1；

As 0≤d_iWhen < K, 2h+1≤j < 4h+2, y=j-3h-2+3K, x=12K-d_i-1；

As 0≤d_iWhen < K, 4h+2≤j < 6h+3, y=4K-d_i- 1, x=11K+5h-j+2；

As 0≤d_iWhen < K, 6h+3≤j < 8h+4, y=3K-j+7h+3, x=10K+d_i；

To 12K column × 4K row matrix, or by above-mentioned matrix transposition, obtain 4K column × 12K row matrix.

Further, in a kind of panoramic video hexagon method of sampling: in step 1, the multiple that N is 8, Or the multiple that N is 16.

Correspondingly, the present invention also provides a kind of panoramic video hexagon sampling apparatuses, comprising:

Spherical surface is divided into N=6K annular surface with equally spaced weft by spherical surface cutting unit, and K is positive integer；It will obtain Annular surface be successively denoted as O_i, i=0,1 ..., N-1, from the angle of the centre of sphere, O_iIt arranges in accordance with the order from top to bottom；

Annular surface cutting unit, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, wherein when 0 When≤i < K, T_i=8i+4, as K≤i < 3K, T_i=10K, as 3K≤i < N, T_i=T_N-1-i, wherein a warp is 0 ° Warp；Obtained region is successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to east longitude degree Increase or the sequence of west longitude degree reduction arranges, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Sampled value computing unit, each region A on the spherical surface being calculated_i,jThe average value of interior light, as this The sampled value S of the corresponding sampled point in region_i,j；

Arrangement units, by sampled value S_i,jIt is arranged in a matrix, indicates S with x_i,jLocating column, y indicate S_i,jLocating Row, enables d_i=i-5K, h=K-d_i- 1, then

As 0≤i < K, 0≤j < 2i+1, y=i+K, x=11K-i+j-1；

As 0≤i < K, 2i+1≤j < 4i+2, y=3i+K-j+1, x=11K+i；

As 0≤i < K, 4i+2≤j < 6i+3, y=K-i-1, x=11K+5i-j+2；

As 0≤i < K, 6i+3≤j < 8i+4, y=K+j-7i-4, x=11K-i-1；

As K≤i < 5K, y=i-K, x=j；

As 0≤d_iWhen < K, 0≤j < 2h+1, y=d_i+ 2K, x=11K-h+j-1；

As 0≤d_iWhen < K, 2h+1≤j < 4h+2, y=j-3h-2+3K, x=12K-d_i-1；

As 0≤d_iWhen < K, 4h+2≤j < 6h+3, y=4K-d_i- 1, x=11K+5h-j+2；

As 0≤d_iWhen < K, 6h+3≤j < 8h+4, y=3K-j+7h+3, x=10K+d_i；

To 12K column × 4K row matrix, or by above-mentioned matrix transposition, obtain 4K column × 12K row matrix.

Compared with prior art, a kind of panoramic video hexagon method of sampling of the present invention and device sampled point are on spherical surface It is more that distribution meets equator sampling number, and nearby sampling number is seldom at the two poles of the earth, thus reduce sampling structure redundancy, so that Under conditions of reaching identical spatial resolution, sampling number is reduced, moreover, the rectangle being combined into has good content adjacent Property, coding compression efficiency can be improved.

Detailed description of the invention

Fig. 1 is the schematic diagram of longitude and latitude figure sampling in the prior art spherical surface method of sampling；

Fig. 2 is the schematic diagram of hexahedron sampling in the prior art spherical surface method of sampling；

Fig. 3 is the lab diagram of pyramid sampling in the prior art spherical surface method of sampling；

Fig. 4 is a kind of step flow chart of the panoramic video hexagon method of sampling of the present invention；

Fig. 5 is a kind of system architecture diagram of panoramic video hexagon sampling apparatus of the present invention；

Fig. 6 show a wherein width figure for the panoramic video that one is indicated with the longitude and latitude figure that resolution ratio is 4096 × 2048 Picture；

Fig. 7 show resulting display image after applying the present invention.

Fig. 8 show resulting coded image after applying the present invention.

Fig. 9 show resulting coded image after applying the present invention.

Specific embodiment

Below by way of specific specific example and embodiments of the present invention are described with reference to the drawings, those skilled in the art can Understand further advantage and effect of the invention easily by content disclosed in the present specification.The present invention can also pass through other differences Specific example implemented or applied, details in this specification can also be based on different perspectives and applications, without departing substantially from Various modifications and change are carried out under spirit of the invention.

Fig. 4 is a kind of step flow chart of the panoramic video hexagon method of sampling of the present invention.As shown in figure 4, the present invention one The kind panoramic video hexagon method of sampling, includes the following steps:

The first step is required according to spatial resolution, and spherical surface is divided into N=6K annular surface with equally spaced weft, Middle K is positive integer, in the preferred embodiment of the invention, multiple that the multiple or N that N is 8 are 16；The annular surface that will be obtained Successively it is denoted as O_i, i=0,1 ... N-1, from the angle of the centre of sphere, by O_iIt arranges in accordance with the order from top to bottom；

Second step, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, wherein as 0≤i < K, T_i=8i+4, as K≤i < 3K, T_i=10K, as 3K≤i < N, T_i=T_N-1-i, wherein a warp is 0 ° of warp；Will To region be successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the increase of east longitude degree or west longitude The sequence of degree reduction arranges, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Third step calculates each region A on the spherical surface that step 2 obtains_i,jThe average value of interior light, as the area The sampled value S of the corresponding sampled point in domain_i,j。

4th step, by sampled value S_i,jIt is arranged in a matrix, indicates S with x_i,jLocating column, y indicate S_i,jLocating row, Enable d_i=i-5K, h=K-d_i- 1, then

As 0≤i < K, 0≤j < 2i+1, y=i+K, x=11K-i+j-1；

As 0≤i < K, 2i+1≤j < 4i+2, y=3i+K-j+1, x=11K+i；

As 0≤i < K, 4i+2≤j < 6i+3, y=K-i-1, x=11K+5i-j+2；

As 0≤i < K, 6i+3≤j < 8i+4, y=K+j-7i-4, x=11K-i-1；

As K≤i < 5K, y=i-K, x=j；

As 0≤d_iWhen < K, 0≤j < 2h+1, y=d_i+ 2K, x=11K-h+j-1；

As 0≤d_iWhen < K, 2h+1≤j < 4h+2, y=j-3h-2+3K, x=12K-d_i-1；

As 0≤d_iWhen < K, 4h+2≤j < 6h+3, y=4K-d_i- 1, x=11K+5h-j+2；

As 0≤d_iWhen < K, 6h+3≤j < 8h+4, y=3K-j+7h+3, x=10K+d_i；

To 12K column × 4K row matrix, or by above-mentioned matrix transposition, obtain 4K column × 12K row matrix.

Fig. 5 is a kind of system architecture diagram of panoramic video hexagon sampling apparatus of the present invention.As shown in figure 5, the present invention one Kind of panoramic video hexagon sampling apparatus, comprising: spherical surface cutting unit, annular surface cutting unit, sampled value computing unit with And arrangement units.

Spherical surface is divided into N=6K annular surface with equally spaced weft by the spherical surface cutting unit, and K is positive integer；It will Obtained annular surface is successively denoted as O_i, i=0,1 ..., N-1, from the angle of the centre of sphere, O_iIn accordance with the order from top to bottom Arrangement；

The annular surface cutting unit, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, In, as 0≤i < K, T_i=8i+4, as K≤i < 3K, T_i=10K, as 3K≤i < N, T_i=T_N-1-i, wherein a warp Line is 0 ° of warp；Obtained region is successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to east Increase through degree or the sequence of west longitude degree reduction arranges, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

The sampled value computing unit, each region A on the spherical surface being calculated_i,jThe average value of interior light is made For the sampled value S of the corresponding sampled point in the region_i,j；

The arrangement units, by sampled value S_i,jIt is arranged in a matrix, indicates S with x_i,jLocating column, y indicate S_i,jInstitute The row at place, enables d_i=i-5K, h=K-d_i- 1, then

As 0≤i < K, 0≤j < 2i+1, y=i+K, x=11K-i+j-1；

As 0≤i < K, 2i+1≤j < 4i+2, y=3i+K-j+1, x=11K+i；

As 0≤i < K, 4i+2≤j < 6i+3, y=K-i-1, x=11K+5i-j+2；

As 0≤i < K, 6i+3≤j < 8i+4, y=K+j-7i-4, x=11K-i-1；

As K≤i < 5K, y=i-K, x=j；

As 0≤d_iWhen < K, 0≤j < 2h+1, y=d_i+ 2K, x=11K-h+j-1；

As 0≤d_iWhen < K, 2h+1≤j < 4h+2, y=j-3h-2+3K, x=12K-d_i-1；

As 0≤d_iWhen < K, 4h+2≤j < 6h+3, y=4K-d_i- 1, x=11K+5h-j+2；

As 0≤d_iWhen < K, 6h+3≤j < 8h+4, y=3K-j+7h+3, x=10K+d_i；

To 12K column × 4K row matrix, or by above-mentioned matrix transposition, obtain 4K column × 12K row matrix.

Hereinafter the present invention will be further illustrated by specific embodiment:

Implementation example

It is illustrated in figure 6 a wherein width figure for the panoramic video that one is indicated with the longitude and latitude figure that resolution ratio is 4096 × 2048 Picture, in the specific embodiment of the invention, which is color video, and there are three components for tool.Assuming that used color point Amount is RGB, still indicates the color of each sampled point after sampling with RGB.Assuming that the requirement of spatial resolution is to three components It is identical, it is all N=1536, K=256, it is all to be quantified as 256 grades that quantified precision requirement, which is also identical to each component,.Then To each component, repeats following steps one and arrives step 4:

Spherical surface is divided into 1536=6*256 annular surface with equally spaced weft by the first step.The annular surface that will be obtained Successively it is denoted as O_i, i=0,1 ..., 1535, from the angle of the centre of sphere, O_iIt arranges in accordance with the order from top to bottom；

Second step, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, wherein when 0≤i < 256 When, T_i=8i+4, as 256≤i < 768, T_i=2560, as 768≤i < 1536, T_i=T_1535-i, wherein a warp is 0 ° of warp.Obtained region is successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to east longitude degree Number increases or the arrangement of the sequence of west longitude degree reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Third step calculates each region A on the spherical surface that step 2 obtains_i,jThe average value of interior light, and carry out 256 grades of quantizations, obtain the integer value between one 0 to 255, as the sampled value of the corresponding sampled point in the region, are denoted as S_i,j。

Sampled value is arranged in a matrix, indicates S with x by the 4th step_i,jLocating column, y indicate S_i,jLocating row enables d_i=i-1280, h=255-d_i, then

As 0≤i < 256,0≤j < 2i+1, y=i+256, x=2816-i+j-1；

As 0≤i < 256,2i+1≤j < 4i+2, y=3i+256-j+1, x=2816+i；

As 0≤i < 256,4i+2≤j < 6i+3, y=256-i-1, x=2816+5i-j+2；

As 0≤i < 256,6i+3≤j < 8i+4, y=256+j-7i-4, x=2816-i-1；

As 256≤i < 1280, y=i-256, x=j；

As 1280≤i < 1536,0≤j < 2h+1, y=d_i+ 512, x=2816-h+j-1；

As 1280≤i < 1536,2h+1≤j < 4h+2, y=j-3h-2+768, x=3072-d_i-1；

As 1280≤i < 1536,4h+2≤j < 6h+3, y=1024-d_i- 1, x=2816+5h-j+2；

As 1280≤i < 1536,6h+3≤j < 8h+4, y=768-j+7h+3, x=2560+d_i；

The matrix of × 1024 rows is arranged to one 3072, as shown in Figure 8；Or by above-mentioned matrix transposition, obtain one 1024 The matrix of × 3072 rows is arranged, as shown in Figure 9.

Resulting sampled point can also be arranged in the rectangular area of 1536 rows, 3072 column, and every a line is compiled Number it is placed on the 1536th position of every a line for the data of (i, 0), number is (i, k), k=1..T_i/ 2, sampled point data It is placed on the 1536+k position of the i-th row, number is (i, T_i- k), k=1..T_i/ 2-1, the data of sampled point be placed on the i-th row 1536-k position.255 fillings of the part that do not fill up in rectangular area, gained image are as shown in Figure 7.

In conclusion the distribution of a kind of panoramic video hexagon method of sampling of the present invention and device sampled point on spherical surface is full Sufficient equator sampling number is more, and nearby sampling number is seldom at the two poles of the earth, so as to avoid sampling structure redundancy, so that reaching phase Under conditions of same spatial resolution, sampling number is reduced, moreover, the rectangle being combined into has good content adjacency, can be mentioned Height coding compression efficiency.

The above-described embodiments merely illustrate the principles and effects of the present invention, and is not intended to limit the present invention.Any Without departing from the spirit and scope of the present invention, modifications and changes are made to the above embodiments by field technical staff.Therefore, The scope of the present invention, should be as listed in the claims.

Claims (3)

1. a kind of panoramic video hexagon method of sampling, includes the following steps:

Spherical surface is divided into N=6K annular surface with equally spaced weft by step 1, and wherein K is positive integer；The annular that will be obtained Curved surface is successively denoted as O_i, i=0,1 ... N-1, from the angle of the centre of sphere, by O_iIt arranges in accordance with the order from top to bottom；

Step 2, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, wherein as 0≤i < K, T_i= 8i+4, as K≤i < 3K, T_i=10K, as 3K≤i < N, T_i=T_N-1-i, wherein a warp is 0 ° of warp；By what is obtained Region is successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the increase of east longitude degree or west longitude degree The sequence of reduction arranges, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Step 3 calculates each region A on the spherical surface that step 2 obtains_i,jThe average value of interior light, as the region pair The sampled value S for the sampled point answered_i,j；

Step 4, by sampled value S_i,jIt is arranged in a matrix, indicates S with x_i,jLocating column, y indicate S_i,jLocating row, enables d_i =i-5K, h=K-d_i- 1, then

As 0≤i < K, 0≤j < 2i+1, y=i+K, x=11K-i+j-1；

As 0≤i < K, 2i+1≤j < 4i+2, y=3i+K-j+1, x=11K+i；

As 0≤i < K, 4i+2≤j < 6i+3, y=K-i-1, x=11K+5i-j+2；

As 0≤i < K, 6i+3≤j < 8i+4, y=K+j-7i-4, x=11K-i-1；

As K≤i < 5K, y=i-K, x=j；

As 0≤d_iWhen < K, 0≤j < 2h+1, y=d_i+ 2K, x=11K-h+j-1；

As 0≤d_iWhen < K, 2h+1≤j < 4h+2, y=j-3h-2+3K, x=12K-d_i-1；

As 0≤d_iWhen < K, 4h+2≤j < 6h+3, y=4K-d_i- 1, x=11K+5h-j+2；

As 0≤d_iWhen < K, 6h+3≤j < 8h+4, y=3K-j+7h+3, x=10K+d_i；

To 12K column × 4K row matrix, or by above-mentioned matrix transposition, obtain 4K column × 12K row matrix.

2. a kind of panoramic video hexagon method of sampling as described in claim 1, it is characterised in that: in step 1, N 8 Multiple or N be 16 multiple.

3. a kind of panoramic video hexagon sampling apparatus, comprising:

Spherical surface is divided into N=6K annular surface with equally spaced weft by spherical surface cutting unit, and K is positive integer；The ring that will be obtained Shape curved surface is successively denoted as O_i, i=0,1 ..., N-1, from the angle of the centre of sphere, O_iIt arranges in accordance with the order from top to bottom；

Annular surface cutting unit, with equally spaced warp by annular surface O_iIt is divided evenly into T_iA region, wherein as 0≤i When < K, T_i=8i+4, as K≤i < 3K, T_i=10K, as 3K≤i < N, T_i=T_N-1-i, wherein a warp is 0 ° of warp； Obtained region is successively denoted as A_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to east longitude degree increase or The sequence of west longitude degree reduction arranges, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Sampled value computing unit, each region A on the spherical surface being calculated_i,jThe average value of interior light, as the region The sampled value S of corresponding sampled point_i,j；

Arrangement units, by sampled value S_i,jIt is arranged in a matrix, indicates S with x_i,jLocating column, y indicate S_i,jLocating row enables d_i=i-5K, h=K-d_i- 1, then

As 0≤i < K, 0≤j < 2i+1, y=i+K, x=11K-i+j-1；

As 0≤i < K, 2i+1≤j < 4i+2, y=3i+K-j+1, x=11K+i；

As 0≤i < K, 4i+2≤j < 6i+3, y=K-i-1, x=11K+5i-j+2；

As 0≤i < K, 6i+3≤j < 8i+4, y=K+j-7i-4, x=11K-i-1；

As K≤i < 5K, y=i-K, x=j；

As 0≤d_iWhen < K, 0≤j < 2h+1, y=d_i+ 2K, x=11K-h+j-1；

As 0≤d_iWhen < K, 2h+1≤j < 4h+2, y=j-3h-2+3K, x=12K-d_i-1；

As 0≤d_iWhen < K, 4h+2≤j < 6h+3, y=4K-d_i- 1, x=11K+5h-j+2；

As 0≤d_iWhen < K, 6h+3≤j < 8h+4, y=3K-j+7h+3, x=10K+d_i；

To 12K column × 4K row matrix, or by above-mentioned matrix transposition, obtain 4K column × 12K row matrix.