CN105760351B - Integer transformation and sparse decomposition algorithm based on moment function for image processing - Google Patents

Integer transformation and sparse decomposition algorithm based on moment function for image processing Download PDF

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CN105760351B
CN105760351B CN201610102138.1A CN201610102138A CN105760351B CN 105760351 B CN105760351 B CN 105760351B CN 201610102138 A CN201610102138 A CN 201610102138A CN 105760351 B CN105760351 B CN 105760351B
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moment
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moment function
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orthogonal
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郭芬红
熊昌镇
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North China University of Technology
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Abstract

The invention relates to an integer transformation and sparse decomposition algorithm based on a moment function for image processing, which comprises the following steps: constructing a discrete orthogonal moment function by utilizing a discrete orthogonality and compression replication method; step two, taking a discrete value of the discrete orthogonal moment function and multiplying the discrete value by a common factor of the moment function to obtain integer transformation; and step three, obtaining a sparse decomposition algorithm of integer transformation by adopting a recursive decomposition method. The polynomial degree of the discrete orthogonal moment function constructed by the invention is fixed, so that the defect of unstable calculation of high-degree polynomial moment can be effectively avoided; and then, because the constructed moment function is discrete and orthogonal, orthogonal integer transformation can be obtained through simple common factor multiplication, a fast algorithm of a matrix is obtained by adopting a sparse decomposition mode, and decomposed coefficients are integers, so that the fast algorithm only comprising integer addition and shift operation is obtained.

Description

The integer transform and sparse decomposition algorithm based on moment function for image procossing
Technical field
The present invention relates to a kind of integer transforms and sparse decomposition algorithm based on moment function for image procossing, specifically relate to And a kind of integer transform and sparse decomposition algorithm based on Discrete Orthogonal moment function.
Background technology
Moment function has a wide range of applications in image analysis.Square transformation is reversible, and each rank square is independent of one another, is had most Small redundancy.Moment function includes continuous orthogonal moment function and Discrete Orthogonal moment function, wherein continuous orthogonal moment can generally lead to Simple inverse transformation form is crossed to solve the problems, such as signal reconstruction.But continuous orthogonal moment active computer carries out numerical computations When need to carry out sliding-model control, this sliding-model control influences whether the orthogonality of kernel function, to further influence figure As the performance of square, therefore its application in image analysis is very limited.And discrete orthogonal moments are due to that need not integrate Approximation processing avoids coordinate space conversion to square image itself in this way because of the conversion without carrying out coordinate space Caused negative effect, this makes Discrete Orthogonal moment function have preferably application in image analysis compared with continuous moment function Foreground.Increase however as the number of discrete orthogonal polynomials, the stable type of algorithm can not also ensure.It is existing with moment function Transformation is mostly floating type transformation, and computation complexity is high, cannot be satisfied wanting for the real-time signal analysis such as image and Video coding It asks.
Invention content
In view of the above-mentioned deficiency of the prior art, the integer transform that the present invention is provided to image procossings based on moment function and Sparse decomposition algorithm, it is intended that solve computational accuracy error greatly simultaneously by using integer transform and efficiency is low asks Topic.To achieve the goals above, the present invention adopts the following technical scheme that：
The integer transform and sparse decomposition algorithm based on moment function for image procossing comprising：
Step 1 constructs a kind of Discrete Orthogonal moment function using Discrete Orthogonal and compression clone method；
Step 2, quantizing to the Discrete Orthogonal moment function of step 1 construction, it is whole to be obtained multiplied by the common factor with moment function Transformation of variables；
Step 3 obtains the sparse decomposition of integer transform using recurrence decomposition method.
Further, in step 1, the constitution step of the Discrete Orthogonal moment function is as follows：
It is basic moment function that step 11, which takes preceding 2 multinomials of Discrete Orthogonal Chebyshev polynomials,；
Step 12 constructs two moment functions using Discrete Orthogonal and generates member；
Step 13 generates member by two moment functions constructed in step 12 and generates 4 square letters through N/2 compression shift copies Number, the m powers that wherein N is 2, m are the integer more than 2；
4 moment functions that step 14 constructs step 13 generate 8 moment functions by N/4 compression translations, with this iteration Go down, until obtaining N number of moment function.
It is that fundamental polynomials are as follows that step 11, which takes preceding 2 multinomials of Discrete Orthogonal Chebyshev polynomials,：
t0(x, N)=1,
t1(x, N)=2x+1-N
Wherein x=0,1 ..., N-1, N are positive integer.
First 1 order polynomial moment function that step 12 constructs generates member t2(x, N) is divided into two sections, just symmetrical with intermediate point The t of function2(x, N), formula is as follows：
Wherein a is coefficient to be solved, then the following solution of accounting equation：
Finding out undetermined coefficient a is
Step 13 further comprises：
Step (1) enablesT is substituted into instead of N2(x, N) obtains t4The left-half of (x, N), right half part value are 0, obtain t4 (x, N), formula is as follows：
Step (2) will translate t4The left-half of (x, N) moves to right half part, and left-half takes 0, obtains t5(x, N), Its formula is as follows：
Step (3) is obtained using same method by t3T can be obtained through overcompression shift copy in (x, N)6(x, N) and t7(x, N) moment function
Step 2 further comprises：
0,1 is taken respectively to the dependent variable x of each moment function ..., when N-1, the row vector that a line length is N is obtained, to row Vector carries out the reduction of fractions to a common denominator and is multiplied by denominator and obtains integer row vector, combined by the row vector of N number of moment function just obtained it is orthogonal INTEGER MATRICES.
In step 2, the integer transform takes 4 ranks, that is, works as N=4, x=0,1, and when 2,3, it is as follows to obtain 4 moment functions：
t0(x, 4)=1
t1(x, 4)=2x-3
Enable t2(x, 4)=2 × t2(x, 4), t3(x, 4)=4 × t3(x, 4) obtains 4 rank integer orthogonal matrixes and is denoted as T4
In step 2, the integer transform takes 8 ranks, that is, works as N=8, x=0,1, when 2,3,4,5,6,7, obtains 8 square letters Number is as follows：
t0(x, 8)=1
t1(x, 8)=2x-7
Enable t2(x, 8)=2 × t2(x, 8), t3(x, 8)=8 × t3(x, 8), t4(x, 8)=2 × t4(x, 8), t5(x, 8)= 2×t5(x, 8), t6(x, 8)=4 × t6(x, 8), t7(x, 8)=4 × t7(x, 8) obtains 8 rank integer orthogonal matrixes and is denoted as T8：
Beneficial effects of the present invention：The present invention can solve computational accuracy error simultaneously greatly using integer transform and efficiency is low The problem of, its main feature is that replace floating number transformation matrix, multiplication of integers that can be replaced with addition and subtraction and displacement with integer transform matrix, Therefore conversion process can realize that operand is greatly reduced by addition and subtraction and displacement completely.
Description of the drawings
Fig. 1 is the integer transform and sparse decomposition algorithm flow based on moment function according to the present invention for image procossing Figure.
Specific implementation mode
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete Site preparation describes, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.It is based on Embodiment in the present invention, those of ordinary skill in the art are obtained every other without creative efforts Embodiment shall fall within the protection scope of the present invention.
Present invention is disclosed the integer transforms and sparse decomposition algorithm based on moment function for image procossing comprising such as Lower step：
Step 1 constructs a kind of Discrete Orthogonal moment function using Discrete Orthogonal and compression clone method；
Step 2, quantizing to the Discrete Orthogonal moment function of step 1 construction, it is whole to be obtained multiplied by the common factor with moment function Transformation of variables；
Step 3 obtains the sparse decomposition of integer transform using recurrence decomposition method.
In step 1, the construction algorithm step of Discrete Orthogonal moment function is specific as follows：
It is basic moment function that step 11, which takes preceding 2 multinomials of Discrete Orthogonal Chebyshev polynomials,；
Step 12 constructs two moment functions using Discrete Orthogonal and generates member；
Step 13 generates member by two moment functions constructed in step 12 and generates 4 square letters through N/2 compression shift copies Number, the m powers that wherein N is 2, m are the integer more than 2；
4 moment functions that step 14 constructs step 13 generate 8 moment functions by N/4 compression translations, with this iteration Go down, until obtaining N number of moment function.
Specifically, it is fundamental polynomials that step 11, which takes preceding 2 multinomials of Discrete Orthogonal Chebyshev polynomials,
t0(x, N)=1,
t1(x, N)=2x+1-N
Wherein x=0,1 ..., N-1, N are positive integer.
Specifically, first 1 order polynomial moment function that step 12 constructs generates member t2(x, N) is divided into two sections, with centre The t of the positive symmetric function of point2(x, N), formula is as follows：
Wherein a is coefficient to be solved, then the following solution of accounting equation：
Finding out undetermined coefficient a is
Second 1 order polynomial moment function that step 12 constructs generates member with the t of intermediate point antisymmetric function3(x, N), Formula is as follows：
Coefficient to be solved wherein b, then the following solution of accounting equation, finds out undetermined coefficient b
It can easily be proven that t0(x,N),t1(x,N),t2(x,N),t3(x, N) is Discrete Orthogonal.
Step 13 includes：
Step (1) enablesT is substituted into instead of N2(x, N) obtains t4The left-half of (x, N), right half part value are 0, obtain t4 (x, N), formula is as follows：
Step (2) will translate t4The left-half of (x, N) moves to right half part, and left-half takes 0, obtains t5(x, N), Its formula is as follows：
Step (3) is obtained using same method by t3T can be obtained through overcompression shift copy in (x, N)6(x, N) and t7(x, N) moment function
Step 2 is to quantize to obtain multiplied by with the common factor of moment function to the Discrete Orthogonal moment function that step 1 constructs Integer transform.It further comprises：0,1 is taken respectively to the dependent variable x of each moment function ..., when N-1, it is N to obtain a line length Row vector, the reduction of fractions to a common denominator is carried out to row vector and is multiplied by denominator and obtains integer row vector, is combined by the row vector of N number of moment function Orthogonal integer matrix is just obtained.
In a specific embodiment of step 2, the integer transform construction embodiment by taking 4 ranks as an example is as follows：
(1) work as N=4, x=0,1, when 2,3, it is as follows to obtain 4 moment functions：
t0(x, 4)=1,
t1(x, 4)=2x-3
Enable t2(x, 4)=2 × t2(x, 4), t3(x, 4)=4 × t3(x, 4) obtains 4 rank integer orthogonal matrixes and is denoted as T4
In another specific embodiment of step 2, the integer transform construction embodiment by taking 4 ranks as an example is as follows：
(2) work as N=8, x=0,1, when 2,3,4,5,6,7, obtain 8 moment functions
t0(x, 8)=1
t1(x, 8)=2x-7
Enable t2(x, 8)=2 × t2(x, 8), t3(x, 8)=8 × t3(x, 8), t4(x, 8)=2 × t4(x, 8), t5(x, 8)= 2×t5(x, 8), t6(x, 8)=4 × t6(x, 8), t7(x, 8)=4 × t7(x, 8) obtains 8 rank integer orthogonal matrixes and is denoted as T8：
Specifically, for N (N >=8) rank orthogonal integer matrix T in step 3NIt is decomposed into following recursive form：
WhereinFor obtaining by the Discrete Orthogonal moment function that constructsRank INTEGER MATRICES,ForRank full 0 matrix, T are Transfer matrix, it is a sparse matrix, the element for enabling the i-th row jth in T (i, j) representing matrix T arrange, sparse matrix T by T (2, 1),T(4,1),T(4,2),6 coefficients to be asked, T (1,1)=1,T (2,2)=1,T (3,2)=1, T (5+2k, 3+k)=1,All it is that 0 element is constituted with remaining, 6 coefficients to be asked T can be passed throughNWithRelation derivation come out.
Can using sparse decomposition as relational matrix withProduct, decompose T always in this way2, just obtain N rank orthogonal integers Matrix TNSparse decomposition algorithm.
Enumerate the sparse decomposition algorithm that step 3 obtains integer transform using recurrence decomposition method below one is specific real Apply example.
INTEGER MATRICES T8Matrix sparse decomposition is following form
04For 4 rank full 0 matrixes, T4For 4 rank INTEGER MATRICESs, T4Further sparse decomposition is
Result in the sparse decomposition algorithm of 8 ranks and 4 rank INTEGER MATRICESs.
It is the description of this invention above, under the premise of not departing from design spirit of the present invention, this field ordinary skill skill The various modifications and replacement that art personnel make technical scheme of the present invention should all fall into the guarantor that the claim of the present invention determines It protects in range.

Claims (7)

1. the integer transform and sparse decomposition algorithm based on moment function for image procossing, which is characterized in that including：
Step 1 constructs a kind of Discrete Orthogonal moment function using Discrete Orthogonal and compression clone method；
Step 2 quantizes to the Discrete Orthogonal moment function of step 1 construction and obtains integer change multiplied by the common factor with moment function It changes；
Step 3 obtains the sparse decomposition of integer transform using recurrence decomposition method；
In step 1, the constitution step of the Discrete Orthogonal moment function is as follows：
It is basic moment function that step 11, which takes preceding 2 multinomials of Discrete Orthogonal Chebyshev polynomials,；
Step 12 constructs two moment functions using Discrete Orthogonal and generates member；
Step 13 generates member by two moment functions constructed in step 12 and generates 4 moment functions through N/2 compression shift copies, The m powers that middle N is 2, m are the integer more than 2；
4 moment functions that step 14 constructs step 13 generate 8 moment functions by N/4 compression translations, and with this, iteration continues, directly Until obtaining N number of moment function.
2. the integer transform and sparse decomposition algorithm based on moment function according to claim 1 for image procossing, It is characterized in that, it is that fundamental polynomials are as follows that step 11, which takes preceding 2 multinomials of Discrete Orthogonal Chebyshev polynomials,：
t0(x, N)=1,
t1(x, N)=2x+1-N
Wherein x=0,1 ..., N-1, N are positive integer.
3. the integer transform and sparse decomposition algorithm based on moment function according to claim 1 for image procossing, It is characterized in that, first 1 order polynomial moment function that step 12 constructs generates member t2(x, N) is divided into two sections, with intermediate point face Claim the t of function2(x, N), formula is as follows：
Wherein a is coefficient to be solved, then the following solution of accounting equation：
Finding out undetermined coefficient a is
Second 1 order polynomial moment function that step 12 constructs generates member with the t of intermediate point antisymmetric function3(x, N), formula It is as follows：
Coefficient to be solved wherein b, then the following solution of accounting equation, finds out undetermined coefficient b
4. according to any claim in claim 1-3 for integer transform of the image procossing based on moment function and Sparse decomposition algorithm, which is characterized in that step 13 includes：
Step (1) enablesT is substituted into instead of N2(x, N) obtains t4The left-half of (x, N), right half part value are 0, obtain t4(x, N), formula is as follows：
Step (2) will translate t4The left-half of (x, N) moves to right half part, and left-half takes 0, obtains t5(x, N), it is public Formula is as follows：
Step (3) is obtained using same method by t3T can be obtained through overcompression shift copy in (x, N)6(x, N) and t7(x, N) square Function
5. the integer transform and sparse decomposition algorithm based on moment function according to claim 1 for image procossing, It is characterized in that, step 2 further comprises：0,1 is taken respectively to the dependent variable x of each moment function ..., when N-1, it is long to obtain a line Degree is the row vector of N, and being multiplied by denominator to the row vector progress reduction of fractions to a common denominator obtains integer row vector, is combined by the row vector of N number of moment function Orthogonal integer matrix has just been obtained together.
6. the integer transform and sparse decomposition algorithm based on moment function according to claim 1 for image procossing, It is characterized in that, in step 2, the integer transform takes 4 ranks, that is, works as N=4, x=0,1, and when 2,3, it is as follows to obtain 4 moment functions：
t0(x, 4)=1,
t1(x, 4)=2x-3
Enable t2(x, 4)=2 × t2(x, 4), t3(x, 4)=4 × t3(x, 4) obtains 4 rank integer orthogonal matrixes and is denoted as T4
7. the integer transform and sparse decomposition algorithm based on moment function according to claim 1 for image procossing, It is characterized in that, in step 2, the integer transform takes 8 ranks, that is, works as N=8, x=0,1, when 2,3,4,5,6,7, obtains 8 square letters Number is as follows：
t0(x, 8)=1
t1(x, 8)=2x-7
Enable t2(x, 8)=2 × t2(x, 8), t3(x, 8)=8 × t3(x, 8), t4(x, 8)=2 × t4(x, 8), t5(x, 8)=2 × t5 (, 8), t6(x, 8)=4 × t6(x, 8), t7(x, 8)=4 × t7(x, 8) obtains 8 rank integer orthogonal matrixes and is denoted as T8：
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