JP2001268598A - Method for compressing vector data - Google Patents

Abstract

PROBLEM TO BE SOLVED: To solve the problem that calculation becomes difficult since calculation quantity becomes enormous when data quantity becomes large when singular value analysis and eigen value analysis are simply applied to compression of vector data. SOLUTION: N pieces of inputted M-dimensional vector data are divided and processed into plural groups (S1), a singular value analysis processing is performed for each group (S2), a basis vector is selected and processed from singular values obtained in each group and specified allowable error values (S3) and the selected basis vector weight (w) are stored (S4). These processings are repeated until the number of groups becomes 1 (S5).

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for efficiently compressing a large amount of time-series data such as a three-dimensional moving image, and relates to three-dimensional video communication, three-dimensional video recording, three-dimensional video distribution, The present invention relates to a method for compressing vector data handled in a dimensional TV game or the like.

[0002]

2. Description of the Related Art With the recent development of image processing technology, it has become possible to take pictures with a plurality of cameras and extract the three-dimensional shape and movement of a subject. For example, Nara
yanan. Rander and Kanade, C
onstructing Virtual World
s Using Dense Stereo, Proc
eatings of Sixth IEEE Int
international, Conference on
Computer Vision (ICCV'98),
Bombay India, January 199
8, pp. 3-10.

If this technology is established, images of dances, plays, sports, and the like are converted into three-dimensional CG (computer graphics) data, transmitted, and stored, and images viewed from a desired position are reproduced by CG technology. You can enjoy it.

However, the problem here is that the data amount becomes large, and an effective data compression method is the key to a three-dimensional video service. Since CG data is typically represented as a set of polyhedrons, it can be said that it is necessary to efficiently compress temporal changes X _i (t) in vertex coordinates.

[0005] Since a two-dimensional image is represented by dense pixel values arranged regularly, orthogonal transformation such as discrete cosine transformation is effective, but the vertices of the polyhedron are not included in the three-dimensional space. The application is difficult because of the uniform and irregular arrangement.

As a general data compression technique, a method using singular value analysis and eigenvalue analysis is known. In this method, N M-dimensional data,

[0007]

## EQU1 ## X _ij = X _j (t _i ), i = 1,..., M, j = 1, 2,..., N (1) are compressed as follows.

According to the teaching of linear algebra, among the N vectors, there are at most M linearly independent vectors. Further, for example, if N points are points on one rigid body, the degree of freedom is at most three. In this case, if three base vectors can be selected, each vector can be represented by a linear sum thereof, and significant data compression can be performed.

It is known from the mode analysis theory that vibration can be approximated by the sum of several vibration modes even for an object that vibrates flexibly, and it is considered that the movement of the coordinates of the polygon vertices can be expressed by a small number of basis vectors. A technique for obtaining this basis vector is singular value analysis.

[0010] Further, it can be obtained by eigenvalue analysis when N = M.
Recipes in C, Teukolsky et al.,
Translated by Dankei et al., Technical Review Company (hereinafter referred to as Document A).

Since it is considered that many objects can be approximated by a rigid body or a vibrating body, it can be said that by using singular value analysis, time-varying data of polygon vertices can be significantly compressed.

[0012]

However, applying the singular value analysis and eigenvalue analysis involves the following problems. That is, since the calculation amount is O (n ³ ) for N vectors, the calculation amount becomes enormous as N increases, and the calculation becomes difficult.

An object of the present invention is to solve the above-mentioned problem and to provide a vector data compression method capable of realizing data compression by singular value analysis even for a large amount of vector data with a small calculation amount.

[0014]

According to the present invention, attention is paid to the fact that the amount of calculation for singular value analysis and eigenvalue analysis is O (n ³ ), data is divided, and the processing result is processed hierarchically. By doing so, the amount of calculation is significantly reduced.

(Principle Description of the Invention) (a) Singular Value Division Considering the data compression of N M-dimensional vectors as shown in the above equation (1). 1 out of N vectors
Let F be the number of next independent items, that is, the degree of freedom. Since it is assumed that the number of data is large, N> M. Further, when F> = M, data cannot be compressed by singular value analysis, so that singular value analysis is effective, that is, N>
Assume F.

When a singular value decomposition is applied to an M × N matrix A having X _ij as elements, a product of three matrices,

[0017]

A = VΛU (2) Here, V is an M × N, U is an N × N orthonormal matrix, Λ is an N × N diagonal matrix, and a diagonal element λ _i
Are arranged in descending order. It is known that if there is no calculation error, λ _i = 0 for i> F. Therefore, from equation (2) above,

[0018]

Equation 3] it can be seen that can be calculated as _{X ij = V i1 λ 1 U} 1j + ... + V iF λ F U Fj. That is,

[0019]

{V _ij }, i = 1,..., M j = 1,..., F {λ _i }, i = 1,..., F {U _ij }, i = 1 _,. ,..., N, the original data can be reproduced. That is,

[0020]

## EQU5 ## If MF + F + NF <MN, data compression can be performed. Vector to be recorded

[0021]

## _EQU6 ## v _i = (V _1i , V _2i ,..., V _Mi ) ^t i = 1,
2,..., F are called basis vectors. Also, the corresponding λ _i
It is referred to as the singular values of V _i.

(B) Dividing process N vectors are divided into L = Lo groups by an arbitrary method. For simplicity, assume that the number of vectors belonging to each group is a fixed number n,

[0023]

[Mathematical formula-see original document] Let N = Lo * n. Also assume that n> M. For each group,
Perform singular value decomposition. The resulting number of basis vectors is at most F in each group, and the total number N ₁ is at most L × F
It is. The amount of calculation is O (Ln ³ ).

The N ₁ basis vectors are again divided, and
L to create _one of the group, the number of vectors belonging to each group is made to be in the order of n. The number of groups

[0025]

## EQU8 ## L ₁ <L (F / n). That is, the number of groups is reduced to about (F / n).

Now, a singular value analysis is performed on each group. Since the degree of freedom of each group is at most F, the number of base vectors for each group is at most F, and the total number N ₂ is at most L ₁ F.

When this process is repeated until L _i = 1, F base vectors are finally obtained. The original data can be reproduced by the linear sum of the F base vectors.

The number of groups L _j at each stage is exponentially

[0029]

(9) Since L _j <L _j1 (F / n), the total number of groups is at most O (L log
L). On the other hand, the computational complexity of the singular value analysis for one group is O (n ³ ).

[0030]

O (n ³ L logL) = O (Nn ² log
L) Assuming that N = 10000 and n = 100, the conventional method requires about 10 ¹² calculations, but according to the present invention,

[0031]

## EQU11 ## 100 × 100 × 10000 × log (10
0) = approximately 2 × 10 ⁸ can be realized, and the calculation amount can be reduced 1000 times or more.

(C) Error In the above description, it is assumed that λ _{i in} equation (2) is 0 except for F. However, in reality, it takes a small value other than 0 due to various factors such as calculation accuracy, noise, and nonlinearity of a signal source. Therefore, to satisfy the given tolerance, F
It is practical to determine In fact, the sum of the squares of the singular values is

[0033]

[Number 12] ^{_{aF = λ 1 2 + ... +}} λ F 2 and when, the sum of the squared error has been known to be aN · aF. Therefore, to express with approximation accuracy T,

[0034]

## EQU13 ## F may be selected so that T> 1−aF / aN.

As described above, when approximating the basis vector further, it is necessary to consider the propagation of the error. The approximated base vector is

[0036]

It is assumed that v _i ′ = v _i + e _i . Since the difference between the reproduced vector X ′ _j and the original vector X _j is λ _i V _ij e _i , if the sum of the squares of j is obtained, the square error E _i becomes

[0037]

(Equation 15) And propagates with the weight of the square of the singular value λ _i .

Results of singular value analysis performed on {Vi}
｛V_k ⁽²⁾｝, ｛Λ_k ⁽²⁾｝, ｛U_k ^{( 2)}｝. Each V_k
⁽²⁾Against

[0039]

Equation 16] _{^{(λ k (2)) 2}} (U ki) 2 is a square error which occurs V _i when not adopt the V _k ^(2). Therefore, the sum of square errors X _ij to generate through the V _i from the above equation (3),

[0040]

## EQU17 ## λ _i ² (λ _k ⁽²⁾ ) ² (U _ki ) ² By taking the sum of this i, the sum of square errors generated in X _ij when v _k ⁽²⁾ is not adopted is Because

[0041]

(18) It can be evaluated that w _k ⁽²⁾ = Σλ _i ² (λ _k ⁽²⁾ ) ² (U _ki ) ² (4) It can be seen that V _k ⁽²⁾ should be taken in order of W _k ⁽²⁾ until the sum becomes larger than the specified value.

Similarly, in the m-th loop,

[0043]

[Number 19] to calculate the ^{_{w k (m) = Σ i}} w i (m1) (λ k (m)) 2 (U ki) 2 ... (5) and the weights w. This is sorted in descending order, and the order is k _j , j = 1,..., N _ml , where

[0044]

Equation 20] _{^{1-Σ j Nm w kj (}} m) / Σ j w k (m) <T ... (6) and precision be determined the N _m so is ensured.

(Data Compression Method of the Invention) As described above, in the present invention, input vector data is divided into several groups, singular value decomposition is performed for each group, and necessary base vectors are selected. The resulting base vectors are repeated to perform group division, singular value analysis, and selection of base vectors. As described in the principle explanation, the hierarchical processing greatly reduces the amount of calculation as compared with the conventional method. The approximation accuracy can be controlled by selecting the basis vector in consideration of the propagation of the error based on the singular value, and is characterized by the following method.

In a vector data compression method for compressing vector data by singular value analysis or eigenvalue analysis, the vector data is divided into N-dimensional vector data into data for which singular value analysis is valid and data for which eigenvalue analysis is valid. Performing the eigenvalue analysis of the data for which the eigenvalue analysis is valid, dividing the data for which the singular value analysis is valid into a plurality of groups, performing the singular value analysis on the vector data of each group, A required eigenvector is selected from a result of the analysis or the eigenvalue analysis, and the division, the analysis, and the selection are repeatedly performed on the selected eigenvector.

Further, in the vector data compression method, a vector for reproducing the propagation of the error in the iterative processing and an original vector based on the singular value obtained by the singular value analysis or the eigenvalue obtained by the eigenvalue analysis. Is evaluated based on the sum of the square errors of the differences, and eigenvector selection processing is performed based on the evaluation.

Further, in a vector data compression method for compressing vector data by singular value analysis or eigenvalue analysis, the weighting is initialized to 1 and the input N M-dimensional vector data is converted into about n vector data. 1
A data division processing procedure for dividing the data into two groups, a singular value analysis processing for performing a singular value analysis on each of the groups, and a base vector based on the singular values obtained in each of the groups and a specified allowable error value. , A basis vector selection process, a basis vector storage process for storing the selected basis vectors and weights in the evaluation of the square error, and a stored basis vector. Is a repetitive processing procedure in which each of the above-mentioned processing procedures is repeated until becomes 1.

[0049]

FIG. 1 is a processing procedure showing an embodiment of the present invention. In the figure, the input N M-dimensional vector data initializes the weight w to 1. The data division processing S1 divides the input vector data so that about n vector data become one group. The vector may be selected in any manner, for example, in order from the beginning of the data.

In the singular value analysis processing S2, singular value analysis is performed on each group.

In the basis vector selection processing S3, a basis vector is selected from the singular values obtained in each group and the specified allowable error value in accordance with the above equation (6).

In the basis vector storage processing S4, the basis vectors selected in the processing S3 and the weight w shown in the above equation (5) are stored.

The repetition process S5 is performed on the stored base vectors until the number of groups becomes one.
Is repeated, and the process is terminated when the number of groups is one.

As is apparent from this processing procedure, it is possible to compress a large amount of vector data based on singular value analysis with a significantly smaller calculation amount than the conventional method.

[0055]

As described above, in the vector data compression method according to the present invention, the input data is divided into several groups, and each group is subjected to singular value analysis or eigenvalue analysis. A base vector is selected, and the above processing is repeatedly performed on the selected base vector. As a result, a large amount of vector data can be compressed based on the singular value analysis with a significantly smaller calculation amount than the conventional method.

[Brief description of the drawings]

FIG. 1 is a processing procedure diagram showing an embodiment of the present invention.

[Explanation of symbols]

 S1: Data division processing S2: Singular value analysis processing S3: Base vector selection processing S4: Base vector storage processing S5: Iterative processing

 ──────────────────────────────────────────────────続 き Continuing on the front page (72) Inventor Mikito Notomi 2-3-1 Otemachi, Chiyoda-ku, Tokyo Inside Nippon Telegraph and Telephone Corporation (72) Inventor Yuki Minami 2-chome Otemachi, Chiyoda-ku, Tokyo No.1 Nippon Telegraph and Telephone Corporation F-term (reference) 5B057 CA13 CA17 CB13 CB17 CG06 DB03 DC07 5C059 MB05 MB18 NN32 PP12 PP13 5C061 AA20 AB04 AB08 5J064 AA02 BC29 BD03

Claims (3)

[Claims] 1. A vector data compression method for compressing vector data by singular value analysis or eigenvalue analysis, wherein the N-dimensional vector data is divided into data for which singular value analysis is valid and data for which eigenvalue analysis is valid. Processing, performing eigenvalue analysis processing on the data for which the eigenvalue analysis is valid, dividing the data for which the singular value analysis is valid into a plurality of groups, performing singular value analysis processing on vector data for each group, A necessary eigenvector is selected from a singular value analysis or an eigenvalue analysis result, and the divided processing is performed on the selected eigenvector.
A vector data compression method characterized by repeatedly executing an analysis process and a selection process. 2. The vector data compression method according to claim 1, wherein a vector from which error propagation in the iterative processing is reproduced based on a singular value obtained in the singular value analysis or an eigenvalue obtained by the eigenvalue analysis. A method for compressing vector data, wherein the evaluation is performed based on the sum of the square errors of the differences between the original vectors, and the eigenvectors are selected based on the evaluation. 3. A method for compressing vector data by compressing vector data by singular value analysis or eigenvalue analysis, wherein weights are initialized to 1 and N input M-dimensional vector data are converted into about n vector data. A data division processing procedure for dividing the data into one group, a singular value analysis processing procedure for performing a singular value analysis on each of the groups, and a basis based on the singular values obtained in each of the groups and a specified allowable error value A basis vector selecting process for selecting a vector, the selected basis vectors, a basis vector storing process for storing the weighted weights in the evaluation of the square error, and the stored basis vectors, A repetition processing procedure for repeating the above-described processing procedures until the number becomes 1; Shrinkage method.