CN107147910A - 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 device, including：Sphere is divided into N=6K annular surface with equally spaced parallel, annular surface is designated as O successively_i, i=0,1 ... N 1, from the angle of the centre of sphere, O_iArrange in accordance with the order from top to bottom；With equally spaced warp by annular surface O_iIt is divided evenly into T_iIndividual region；The average value of the light in each region on obtained sphere is calculated, the sampled value of the corresponding sampled point in the region is used as；Sampled value is aligned in a matrix.The method of sampling avoids sampling structure redundancy so that under conditions of identical spatial resolution is reached, sampling number reduction, 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 technique, more particularly to a kind of panoramic video hexagon method of sampling and device.

Background technology

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

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

The existing sphere method of sampling, is generally divided into three steps by sphere sampling process：The first step, 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 the 3rd step, arrangement.

The first step map during, be not often homalographic mapping, and sphere different zones before the mapping after The ratio of area is also different.During second step plane is sampled, 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 skewness of the sampled point on sphere.Reason By upper, the sampling number of sphere equator should be more than sampling number near the two poles of the earth.3rd step tends not to do in arrangement To the complete adjacent of content, come unfavorable to Video coding compression strap.

At present, there are three kinds of conventional sphere method of samplings：Longitude and latitude figure is sampled, hexahedron sampling and pyramid sampling.

Longitude and latitude figure is sampled as shown in figure 1, any one point can use residing longitude θ and latitude on sphereDescription, θ ∈ [0,2 π),Then spherical Map can be arrivedThe next the ratio of width to height of coordinate system is 2:1 rectangle.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 sphere Density is too high, can produce very big redundancy.

Hexahedron is sampled as shown in Fig. 2 first by six faces that spherical Map is its circumscribed regular hexahedron, so as to obtain six Individual plane square, then uniform sampling is carried out to six plane squares, six squares are finally combined into one by some way Individual rectangle.Distribution of the sampled point of hexahedron sampling on sphere is also uneven, still there is larger redundancy, and square The content of shape does not have complete adjacency yet.

Pyramid is sampled as shown in figure 3, be circumscribed positive rectangular pyramid first by spherical Map, then by the every of the positive rectangular pyramid Direction of the individual side (isosceles triangle) along base vertical line is compressed, until drift angle becomes at a right angle.So bottom surface and four warp A square can be just combined into by crossing the side of Quadratic Map.Uniform sampling finally is carried out to this square.Pyramid is sampled Distribution of the sampled point on sphere be also uneven, 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 summary, the major defect of the existing panoramic video method of sampling is the presence of larger sampling structure redundancy and most End form into rectangle do not have content intact adjacency.Sampled point uneven distribution on sphere, 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, therefore each sampling Influence of the distortion of point to video quality is also different, is handled calculating panoramic video by certain, such as compressed and decompressed During 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.

The content of the invention

To overcome the shortcomings of that above-mentioned prior art is present, the purpose of the present invention is that providing a kind of panoramic video hexagon adopts Quadrat method and device, it is many that it can make distribution of the sampled point on sphere meet equator sampling number, is sampled near the two poles of the earth Points are seldom, so as to avoid sampling structure redundancy so that under conditions of identical spatial resolution is reached, sampling number subtracts 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, comprise the following steps：

Step one, sphere is divided into N=6K annular surface with equally spaced parallel, wherein K is positive integer；By what is obtained Annular surface is designated as O successively_i, i=0,1 ... N-1, from the angle of the centre of sphere, by O_iArrange 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_iIndividual region, wherein, when 0<=i<During K, T_i=8i+4, works as K<=i<During 3K, T_i=10K, works as 3K<=i<During N, T_i=T_N-1-i, wherein a warp is 0 ° of warp；Will To region be designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the increase of the east longitude number of degrees or west longitude The order arrangement of number of degrees reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Each region A on step 3, the sphere that calculation procedure two is obtained_i,jThe average value of interior light, is used as the area The sampled value S of the corresponding sampled point in domain_i,j；

Step 4, by sampled value S_i,jA matrix is arranged in, S is represented with x_i,jResiding row, y represents S_i,jResiding row, Make d_i=i-5K, h=K-d_i- 1, then

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

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

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

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

As K≤i<During 5K, y=i-K, x=j；

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

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

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

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

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

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

Accordingly, the present invention also provides a kind of panoramic video hexagon sampling apparatus, including：

Sphere cutting unit, is divided into N=6K annular surface, K is positive integer with equally spaced parallel by sphere；It will obtain Annular surface be designated as O successively_i, i=0,1 ..., N-1, from the angle of the centre of sphere, O_iArrange 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_iIndividual region, wherein, when 0 <=i<During K, T_i=8i+4, works as K<=i<During 3K, T_i=10K, works as 3K<=i<During N, T_i=T_N-1-i, wherein a warp is 0 ° Warp；Obtained region is designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the east longitude number of degrees The order arrangement of increase or west longitude number of degrees reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Sampled value computing unit, calculates each region A on obtained sphere_i,jThe average value of interior light, is used as this The sampled value S of the corresponding sampled point in region_i,j；

Arrangement units, by sampled value S_i,jA matrix is arranged in, S is represented with x_i,jResiding row, y represents S_i,jResiding OK, d is made_i=i-5K, h=K-d_i- 1, then

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

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

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

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

As K≤i<During 5K, y=i-K, x=j；

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

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

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

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

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

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

Brief description of the drawings

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

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

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

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

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

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

Fig. 7 show the display image using gained after the present invention.

Fig. 8 show the coded image using gained after the present invention.

Fig. 9 show the coded image using gained after the present invention.

Embodiment

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

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

The first step, is required according to spatial resolution, and sphere is divided into N=6K annular surface with equally spaced parallel, its Middle K is positive integer, in the preferred embodiment of the invention, and N is 8 multiple, or the multiple that N is 16；By obtained annular surface O is designated as successively_i, i=0,1 ... N-1, from the angle of the centre of sphere, by O_iArrange 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_iIndividual region, wherein, when 0<=i<During K, T_i=8i+4, works as K<=i<During 3K, T_i=10K, works as 3K<=i<During N, T_i=T_N-1-i, wherein a warp is 0 ° of warp；Will To region be designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the increase of the east longitude number of degrees or west longitude The order arrangement of number of degrees reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Each region A on 3rd step, the sphere that calculation procedure two is obtained_i,jThe average value of interior light, is used as the area The sampled value S of the corresponding sampled point in domain_i,j。

4th step, by sampled value S_i,jA matrix is arranged in, S is represented with x_i,jResiding row, y represents S_i,jResiding row, Make d_i=i-5K, h=K-d_i- 1, then

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

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

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

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

As K≤i<During 5K, y=i-K, x=j；

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

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

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

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

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

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

The sphere cutting unit, is divided into N=6K annular surface, K is positive integer with equally spaced parallel by sphere；Will Obtained annular surface is designated as O successively_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_iIndividual region, its In, when 0<=i<During K, T_i=8i+4, works as K<=i<During 3K, T_i=10K, works as 3K<=i<During N, T_i=T_N-1-i, wherein a warp Line is 0 ° of warp；Obtained region is designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to east Increase through the number of degrees or the order of west longitude number of degrees reduction is arranged, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

The sampled value computing unit, calculates each region A on obtained sphere_i,jThe average value of interior light, makees For the sampled value S of the corresponding sampled point in the region_i,j；

The arrangement units, by sampled value S_i,jA matrix is arranged in, S is represented with x_i,jResiding row, y represents S_i,jInstitute The row at place, makes d_i=i-5K, h=K-d_i- 1, then

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

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

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

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

As K≤i<During 5K, y=i-K, x=j；

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

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

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

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

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

The present invention will be further illustrated by specific embodiment below：

Implementation example

It is illustrated in figure 6 a wherein width figure for a panoramic video represented with the longitude and latitude figure that resolution ratio is 4096 × 2048 Picture, in the specific embodiment of the invention, the panoramic video is color video, with three components.Assume used color point Amount is RGB, still represents the color of each sampled point after sampling with RGB.Assuming that the requirement of spatial resolution is to three components Identical, is all N=1536, K=256, quantified precision requirement is also identical to each component, is all to be quantified as 256 grades.Then To each component, step one is repeated below to step 4：

The first step, is divided into 1536=6*256 annular surface with equally spaced parallel by sphere.By obtained annular surface O is designated as successively_i, i=0,1 ..., 1535, from the angle of the centre of sphere, O_iArrange 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_iIndividual region, wherein, when 0<=i<256 When, T_i=8i+4, when 256<=i<When 768, T_i=2560, when 768<=i<When 1536, T_i=T_1535-i, wherein a warp is 0 ° of warp.Obtained region is designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to east longitude degree Number increase or the order arrangement of west longitude number of degrees reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Each region A on 3rd step, the sphere that calculation procedure two is obtained_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 designated as S_i,j。

4th step, is arranged in a matrix by sampled value, S is represented with x_i,jResiding row, y represents S_i,jResiding row, order d_i=i-1280, h=255-d_i, then

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

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

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

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

As 256≤i<When 1280, y=i-256, x=j；

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

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

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

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

To one 3072 row × 1024 rows matrix, 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.

The sampled point of gained can also be arranged in 1536 rows, 3072 row rectangular areas in, and by per a line compile Number the 1536th position of every a line is placed on for the data of (i, 0), numbering is (i, k), k=1..T_i/ 2, sampled point data The 1536+k position of the i-th row is placed on, numbering 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.The part do not filled up in rectangular area is filled with 255, and gained image is as shown in Figure 7.

In summary, the distribution of a kind of panoramic video hexagon method of sampling of the invention and device sampled point on sphere is expired 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 carried Height coding compression efficiency.

The above-described embodiments merely illustrate the principles and effects of the present invention, not for the limitation present invention.Any Art personnel can be modified above-described embodiment and changed under the spirit and scope without prejudice to the present invention.Therefore, The scope of the present invention, should be as listed by claims.

Claims (3)

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

Step one, sphere is divided into N=6K annular surface with equally spaced parallel, wherein K is positive integer；By obtained annular Curved surface is designated as O successively_i, i=0,1 ... N-1, from the angle of the centre of sphere, by O_iArrange 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_iIndividual region, wherein, when 0<=i<During K, T_i= 8i+4, works as K<=i<During 3K, T_i=10K, works as 3K<=i<During N, T_i=T_N-1-i, wherein a warp is 0 ° of warp；By what is obtained Region is designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the increase of the east longitude number of degrees or the west longitude number of degrees The order arrangement of reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Each region A on step 3, the sphere that calculation procedure two is obtained_i,jThe average value of interior light, is used as the region pair The sampled value S for the sampled point answered_i,j；

Step 4, by sampled value S_i,jA matrix is arranged in, S is represented with x_i,jResiding row, y represents S_i,jResiding row, makes d_i =i-5K, h=K-d_i- 1, then

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

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

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

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

As K≤i<During 5K, y=i-K, x=j；

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

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

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

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

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

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

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

Sphere cutting unit, is divided into N=6K annular surface, K is positive integer with equally spaced parallel by sphere；By obtained ring Shape curved surface is designated as O successively_i, i=0,1 ..., N-1, from the angle of the centre of sphere, O_iArrange 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_iIndividual region, wherein, when 0<=i <During K, T_i=8i+4, works as K<=i<During 3K, T_i=10K, works as 3K<=i<During N, T_i=T_N-1-i, wherein a warp is 0 ° of warp； Obtained region is designated as A successively_i,j, j=0,1 ... T_i- 1, from the angle of the centre of sphere, A_i,jAccording to the east longitude number of degrees increase or The order arrangement of west longitude number of degrees reduction, wherein A_i,0And A_i,Ti-1Adjacent and cut-off rule is 0 ° of warp；

Sampled value computing unit, calculates each region A on obtained sphere_i,jThe average value of interior light, is used as the region The sampled value S of corresponding sampled point_i,j；

Arrangement units, by sampled value S_i,jA matrix is arranged in, S is represented with x_i,jResiding row, y represents S_i,jResiding row, order d_i=i-5K, h=K-d_i- 1, then

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

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

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

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

As K≤i<During 5K, y=i-K, x=j；

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

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

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

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

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