CN115474048A - Rapid color image compression method based on split quaternion model - Google Patents
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Abstract
The invention provides a rapid color image compression method based on a split quaternion model. The method mainly comprises the following steps: s1: constructing a split quaternion model, and representing an original color image by using the split quaternion model to obtain a split quaternion matrix; s2: calculating a real expression matrix of the split quaternion matrix and singular value decomposition of the real expression matrix; s3: calculating the singular value decomposition of the split quaternion matrix according to the singular value decomposition of the real expression matrix, namely the singular value decomposition of the original color image; s4: substituting a given threshold value and a singular value norm of a split quaternion matrix into a threshold value function to obtain the dimension reduction degree; s5: and under the constraint of a threshold function, storing the effective decomposition matrix of the optimal approximate matrix according to the dimension reduction degree, and completing image compression. The invention can keep the internal relation among the three primary color channels of the color image, greatly reduces the calculation complexity, and displays the good performance of the method by the corresponding calculation time, the PSNR value and the SSIM value.
Description
Technical Field
The invention relates to the technical field of image processing, in particular to a rapid color image compression method based on a split quaternion model.
Background
Images are one of the most important information carriers in the field of visual information. The increasing use of digital images raises storage and transmission issues. Image compression is a common processing method, and the purpose is to transform and combine image source data to be processed according to a certain rule, so as to represent the image with the least bit number and restore the image quality as good as possible, so that the image compression meets the requirements of the predetermined application occasions. The image data can be compressed mainly for the following reasons: (1) The method comprises the following steps that certain correlation exists between pixels forming an image in both the row direction and the column direction, namely, original image data are highly correlated, and the purpose of compressing the data can be achieved by extracting or reducing the correlation by using a certain coding method; (2) From the information theory, the data describing the image information source consists of two parts of effective information quantity and redundant quantity, and the removal of the redundant quantity can save the expenses in transmission and storage without damaging the effective information quantity of the image information source; (3) Many occasions allow some distortion in image coding and are also an important reason that images can be compressed. Since the beginning of image compression research in 1948, grayscale image compression methods and techniques have been greatly developed. However, due to the rapid application of color image technology to real-life and network technologies, researchers have turned their attention to color image compression, see documents Wu p, xie k, yu h, zheng y, yu w, a new processing algorithm in color image compression, advances in FCCS,1, aisc.2012; 159-471 ] and "Singh s.k., kumar s., a frame to design SVD base color image compression, third UKSim European Symposium on Computer Modeling and Simulation, IEEE.2009;235-240. Compared with a gray image, a color image can greatly improve the capacity and the fidelity of information, and can be decomposed into a plurality of color channels, so that the compression of the color image is more challenging than the compression of a single gray image.
With regard to color images, there are various technologies in the literature, YCbCr, RGB, YIQ, HSI, etc., where RGB is obviously the most popular technology, since this channel format is able to express colors most naturally in the real world. Each of the three R, G, B channels is highly correlated with the other two. One of the typical algorithms for color image compression is multi-channel compression. For example: documents Wu p, xie k, yu h, zheng y, yu w, a new preprocessing algorithm used in color image compression, advances in FCCS,1, aisc.2012; 159.. The essence of this approach is to process grayscale images, but the disadvantage of this process is that the link between color channels is ignored. That is, based on the original gray image processing method and technology, the conventional color image compression processing is to divide a color image into three R, G, and B gray images, and then combine the processing results of the three gray images to construct the original color image. Obviously, the technology forces the three-dimensional combined components to be separated, and ignores the internal relation among the components.
Quaternion-based algorithms have become very popular in recent years. The pixels of the color image may be considered as quaternions. Thus, a color image can be considered as a quaternion matrix. If one color pixel is processed in its entirety, the spectral relationship between the color channels will go through the entire operation and process. But we must note that this type of algorithm requires more operations. The equivalence isomorphism of any split quaternion matrix and a 2m multiplied by 2n real matrix enables the algorithm to be simple, which is also caused by a special four-dimensional algebraic structure of the split quaternion, and the traditional quaternion does not have the real equivalence isomorphism relationship.
Compression algorithms using Singular Value Decomposition (SVD) are becoming more popular. For example: document "single s.k., kumar s., a frame to design novel SVD based color image compression, third UKSim European Symposium on Computer Modeling and Simulation, ieee.2009;235-240 ], the RGB image is converted into YCbCr format, and then SVD compression is performed on the luminance, blue and red components, respectively. Documents Andrews h.c., patterson c.l., single value compositions and digital image processing, ieee transformations on optics, speech and Signal processing, 1976;1 (24): 26-53.Doi 10.1109/TASSP.1976.1162766.), an SVD and DCT based image compression technique is proposed. The literature "Aharon M., elad M., bruckstein A., K-SVD: an algorithm for designing over complete dictionary for spark reproduction. IEEE Transactions on Signal processing.2006;11 (54) 4311-4322. Doi; 19 And (4) dividing the whole image into a plurality of sub-blocks with equal size, and performing SVD on each sub-block in 270-282. Doi. Documents Shih y.t., chien c.s., chuang c.y., an adaptive parametric block-based single value composition for image de-noise and compression, appl.math.com.2012; 218, 10370-10385, an algorithm is proposed to automatically select the appropriate number of singular values on each sub-block by studying the relationship between the required PSNR values and the number of singular values. Therefore, processing the image compression problem based on the matrix singular value theory and algorithm is also a research hotspot, and particularly after the matrix singular value theory and the algorithm are combined with the four-dimensional matrix algebraic theory, not only is the mathematical model explanation of the image compression algorithm given, but also the internal relation among color channels is reserved to a certain extent.
A color image can be represented by a pure virtual quaternion matrix, and then some color image problems can be converted into algebraic structure problems of the quaternion matrix, so that some quaternion matrix algorithms are evolved. For example: quaternion toolbox (short for QTFM), complex structure algorithm of quaternion problem, real structure algorithm of quaternion problem and the like. However, in this representation, the three-dimensional color image problem is transformed into a four-dimensional quaternion matrix problem, and each step of processing the problem must ensure the algebraic structure of the quaternion matrix, and although the internal relation among the R, G and B components of the color image is preserved, the computational complexity of the problem is greatly increased.
Disclosure of Invention
The present invention is directed to solve one of the above technical problems, and provides a fast color image compression method based on a split quaternion model, which can maintain the internal relationship among the three primary color channels R, G, and B of a color image, and greatly reduce the computational complexity.
In order to achieve the purpose, the invention adopts the technical scheme that:
a fast color image compression method based on a split quaternion model comprises the following steps:
s1: constructing a split quaternion model, and representing an original color image by using the split quaternion model to obtain a split quaternion matrix A;
s2: calculating a real expression matrix of the split quaternion matrix A and singular value decomposition of the real expression matrix;
s3: calculating the singular value decomposition of the split quaternion matrix A according to the singular value decomposition of the real expression matrix, namely the singular value decomposition of the original color image;
s4: giving a threshold value, and substituting the given threshold value and the singular value norm of the split quaternion matrix A into a threshold function to obtain the dimensionality reduction degree;
s5: and under the constraint of a threshold function, selecting the optimal approximate matrix of the original color image, and storing the effective decomposition matrix of the optimal approximate matrix according to the dimension reduction degree to complete image compression.
In some embodiments of the present invention, in step S1, when the real part of the split quaternion model is zero, the split quaternion model is expressed as:
q(x,y)＝r(x,y)i+g(x,y)j+b(x,y)k (1)
in the formula, r (x, y), g (x, y), b (x, y) respectively represent red, green and blue of (x, y) point in the color image, i.e. x rows and y columns of pixel points in the three primary color matrix, and i, j, k are three imaginary part units of the split quaternion q.
In some embodiments of the present invention, in step S1, the original color image has m rows and n columns, and the original color image is represented by a split quaternion model as:
in the formula, A is a split quaternion matrix of an original color image, and R, G and B are real number matrixes which respectively represent three color matrixes of red, green and blue;
representing a split quaternion ring H s M × n order matrix above.
In some embodiments of the present invention, in step S2, the specific step of calculating the real representation matrix of the split quaternary matrix is:
for arbitrarily split quaternion matrices
Wherein A is s Is a real matrix of m × n order, s =1,2,3,4, a real representation matrix a of a split quaternion matrix a σ Is defined as follows:
and the real representation matrix satisfies the following equation:
(A+D) σ ＝A σ +D σ ,(AC) σ ＝A σ C σ ,(aA) σ ＝aA σ ,(A H ) σ ＝(A σ ) T (4)
in the formula (I), the compound is shown in the specification,
is an arbitrary m x n order split quaternion matrix,
is an arbitrary n x p-order split quaternion matrix, a is a real number,
an i-conjugate transpose representing the split quaternion matrix a,
representing a real matrix A s The transposed matrix of (2);
the real representation matrix of the split quaternion matrix is calculated by equation (3).
In some embodiments of the present invention, in step S2, the specific step of calculating the singular value decomposition of the real expression matrix is:
for arbitrarily split quaternion matrices
Let m be more than or equal to n, and an orthogonal matrix exists through the singular value decomposition theory of the real expression matrix
And a real matrix
Such that:
in the formula (I), the compound is shown in the specification,
u i ∈R 2m ,i＝1,2,...,2n，
τ 1 ≥τ 2 ≥…≥τ 2n ≥0，τ 1 ,τ 2 ,…,τ 2n is a real representation matrix A σ The singular value of (a) is,
v j ∈R 2n ,j＝1,2,…,2n；
calculating a real expression matrix A by equation (5) σ Singular value decomposition of (c).
In some embodiments of the present invention, in step S3, the specific steps of obtaining the singular value decomposition of the split quaternion matrix are:
calculating the non-zero singular value sigma of the split quaternion matrix A according to the following formula t ：
In the formula (I), the compound is shown in the specification,
p is a real representation matrix A σ The number of non-zero singular values of (c);
by constructive form of singular values and τ 1 ≥τ 2 ≥…≥τ p Is more than or equal to 0, and the | sigma is obtained 1 |≥|σ 2 |≥…≥|σ r |≥0；
The left and right singular vectors of the split quaternion matrix A are calculated as follows:
in the formula, mu s Is a left singular vector, v, of a split quaternion matrix A s Is a right singular vector of the split quaternion matrix a, s =1,2, \ 8230;, n;
satisfy the requirement of
k＝1,2,…,2n；
In combination with equation (6) and equation (7), there is a unique diagonally split quaternion matrix Σ r ＝diag(σ 1 ,σ 2 ,…,σ r ),|σ 1 |≥|σ 2 |≥…≥|σ r | ≧ 0, such that:
in the formula (I), the compound is shown in the specification,
U H U＝I n ,V H V＝I n ；
then sigma 1 ,…,σ r Singular values referred to as split quaternion matrix A;
according to the singular value decomposition of the split quaternion matrix A, the component product form of the singular value decomposition of the split quaternion matrix A is obtained as follows:
in the formula, mu t Is the left singular vector, i.e., the column of U; v is t Is the right singular vector, i.e., the column of V;
in some embodiments of the present invention, in step S4, the specific steps of obtaining the dimensionality reduction degree are:
given a threshold, substituting the given threshold and the singular value norm of the split quaternion matrix a into the following threshold function:
in the formula (I), the compound is shown in the specification,
the method comprises the steps that an optimal approximate matrix of a split quaternion matrix A is obtained, alpha = s-1 is the singular value number of the split quaternion matrix A, and s is the dimensionality reduction degree;
the dimensionality reduction is calculated by equation (10).
In some embodiments of the present invention, in step S5, the specific step of storing the effective decomposition matrix of the optimal approximation matrix according to the dimension reduction degree is: according to the dimension reduction degree, retaining alpha = s-1 singular values, decomposing the singular values of the m multiplied by n original color image into an m multiplied by alpha order matrix, an alpha multiplied by alpha order matrix and an n multiplied by alpha order matrix, wherein alpha is far smaller than the minimum value of { m, n }, and at the moment, storing the m multiplied by alpha order matrix, the alpha multiplied by alpha order matrix and the n multiplied by alpha order matrix can complete image compression.
In some embodiments of the present invention, the specific steps of reconstructing the optimal approximation matrix are: extracting a decomposition matrix m × α order matrix, an α × α order matrix, and an n × α order matrix of the compressed image in step S5, and multiplying the three decomposition matrices so as to satisfy matrix multiplication m × n = (m × α) × (α × α) × (n × α) H And reconstructing the optimal approximate matrix, and obtaining an approximate image of the original color image after the optimal approximate matrix is imaged, thereby completing image compression reconstruction.
Compared with the prior art, the invention has the beneficial effects that:
the method comprises the steps of firstly utilizing a split quaternion matrix theory to process a color image problem, and converting the three-dimensional color image compression problem into a singular value decomposition problem of a split quaternion matrix through equivalent transformation on an actual problem. In the process of compressing the color image, the traditional image processing method is not used, the original color image is not required to be divided into three gray images for processing, and the internal relation among three primary color channels is kept to a certain extent. Different from the problem of processing color images by utilizing a quaternion algebra technology, the algebraic structure of a split quaternion matrix is not required to be guaranteed in the matrix decomposition process, the problem of a real matrix of 2m multiplied by 2n is completely simplified, the problem of a four-dimensional space is converted into the problem of a two-dimensional space, and the calculation complexity is greatly reduced.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings needed to be used in the embodiments or the prior art descriptions will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive exercise.
Fig. 1 is a flowchart of a fast color image compression method based on a split quaternion model according to the present invention.
Fig. 2a is a CPU time comparison diagram of the split quaternion model-based fast color image compression method provided by the present invention and the existing 5 SVD algorithms.
Fig. 2b is an error comparison diagram of the split quaternion model-based fast color image compression method provided by the present invention and the existing 5 SVD algorithms.
Fig. 3a is a Lena original color image.
Fig. 3b is an original color image of Baboon.
FIG. 3c is an original color image of Peppers.
Fig. 3d is an airplan original color image.
Fig. 4a is a compressed image of a color image "Lena" preserved by the fast color image compression method based on the split quaternion model provided by the present invention.
Fig. 4b is a compressed image of the SVD algorithm preserving 10 singular values of the color image "Lena" based on the prior quaternion toolbox.
Fig. 4c is a compressed image of the SVD algorithm preserving 10 singular values of the color image "Lena" based on the existing quaternion matrix.
Fig. 4d is a compressed image of the SVD algorithm preserving 10 singular values of the color image "Lena" based on the complex-equivalent matrix of the prior quaternion matrix.
FIG. 4e is a block [ R; g; b ] holds a compressed image of 10 singular values of the color image "Lena".
Fig. 4f is a compressed image of the SVD algorithm preserving 10 singular values of the color image "Lena" based on the existing three color spaces.
Fig. 5a is a compressed image of a color image "Lena" with 30 singular values preserved based on the split quaternion model-based fast color image compression method provided by the present invention.
Fig. 5b is a compressed image of the SVD color image compression method based on the prior quaternion toolbox, which preserves 30 singular values of the color image "Lena".
Fig. 5c is a compressed image of a color image "Lena" with 30 singular values retained by the SVD color image compression method based on the existing quaternion matrix.
Fig. 5d is a compressed image of the SVD color image compression method based on the complex equivalent matrix of the prior quaternion matrix, which preserves 30 singular values of the color image "Lena".
FIG. 5e is a block [ R; g; b ] the SVD color image compression method preserves the compressed image of 30 singular values of the color image "Lena".
Fig. 5f is a compressed image of a color image "Lena" with 30 singular values preserved based on the existing SVD color image compression method of three color spaces.
Fig. 6a is a compressed image of a color image "Lena" with 50 singular values preserved based on the split quaternion model-based fast color image compression method provided by the present invention.
Fig. 6b is a compressed image of the SVD algorithm preserving 50 singular values of the color image "Lena" based on the prior quaternion toolbox.
Fig. 6c is a compressed image of a prior art quaternion matrix-based SVD algorithm preserving 50 singular values of a color image "Lena".
Fig. 6d is a compressed image of the SVD algorithm preserving 50 singular values of the color image "Lena" based on the complex-equivalent matrix of the prior quaternion matrix.
FIG. 6e is a block [ R; g; b ] holds a compressed image of 50 singular values of the color image "Lena".
Fig. 6f is a compressed image of the SVD algorithm preserving 50 singular values of the color image "Lena" based on the existing three color spaces.
Fig. 7a is a CPU time chart required for reconstructing four color images, i.e., 3a, 3b, 3c, and 3d, by selecting 30 singular values in the split quaternion model-based fast color image compression method and the existing 5 SVD algorithms.
Fig. 7b is a CPU time diagram for reconstructing four color images 3a, 3b, 3c, and 3d by selecting 60 singular values according to the split quaternion model-based fast color image compression method and the existing 5 SVD algorithms.
Fig. 7c is a CPU time diagram for reconstructing four color images 3a, 3b, 3c, and 3d by selecting 90 singular values according to the split quaternion model-based fast color image compression method and the existing 5 SVD algorithms.
Detailed Description
In order to make the technical problems, technical solutions and advantageous effects to be solved by the present invention more clearly apparent, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a rapid color image compression method based on a split quaternion model, which can keep the internal relation among three primary color channels of R, G and B of a color image in the color image compression process and greatly reduce the calculation complexity.
The steps of the method are detailed below, and the flow chart refers to fig. 1.
S1: and constructing a split quaternion model, and representing the original color image by using the split quaternion model to obtain a split quaternion matrix A.
Specifically, when the real part of the split quaternion model is zero, the split quaternion model is expressed as:
q(x,y)＝r(x,y)i+g(x,y)j+b(x,y)k (1)
in the formula, r (x, y), g (x, y), b (x, y) respectively represent red, green and blue of (x, y) point in the color image, i.e. x rows and y columns of pixel points in the three primary color matrix, and i, j, k are three imaginary parts of the split quaternion q.
Specifically, the original color image has m rows and n columns, and is represented by a split quaternion model as:
wherein A is the split quaternion matrix of the original color image, R, G, BReal number matrixes respectively representing three middle color matrixes of red, green and blue;
representing a split quaternion ring H s M × n order matrix above.
S2: and calculating a real representation matrix of the split quaternion matrix A and singular value decomposition of the real representation matrix.
Specifically, the specific steps of calculating the real representation matrix of the split quaternion matrix are as follows:
for arbitrarily split quaternion matrices
Wherein A is s Is a real matrix of m × n order, s =1,2,3,4, a real representation matrix a of a split quaternion matrix a σ Is defined as:
and the real representation matrix satisfies the following equation:
(A+D) σ ＝A σ +D σ ,(AC) σ ＝A σ C σ ,(aA) σ ＝aA σ ,(A H ) σ ＝(A σ ) T (4)
in the formula (I), the compound is shown in the specification,
is an arbitrary m x n order split quaternion matrix,
is an arbitrary n x p-th order split quaternion matrix, a e R is a real number,
an i-conjugate transpose representing the split quaternion matrix a,
representing a real matrix A s The transposed matrix of (2);
the real representation matrix of the split quaternion matrix is calculated by equation (3).
It should be noted that the advantage of using the split quaternion model for color images is that the split quaternion matrix algebraic structure problem is completely isomorphic to the 2m × 2n real matrix problem described above. Thus, the real representation matrix is a ring
Para-ring R 2m×2n Is isomorphic.
Specifically, the specific steps of calculating the singular value decomposition of the real expression matrix are as follows:
for arbitrarily split quaternion matrices
Let m be more than or equal to n (without loss of generality), an orthogonal matrix exists through the singular value decomposition theory of the real representation matrix
And a real matrix
Such that:
in the formula (I), the compound is shown in the specification,
u i ∈R 2m ,i＝1,2,...,2n，
τ 1 ≥τ 2 ≥…≥τ 2n ≥0，τ 1 ,τ 2 ,…,τ 2n is a real representation matrix A σ The singular value of (a) is,
v j ∈R 2n ,j＝1,2,…,2n；
calculating a real expression matrix A by equation (5) σ Singular value decomposition of (c).
S3: the singular value decomposition of the split quaternion matrix a, i.e. the singular value decomposition of the original color image, is calculated from the singular values of the real representation matrix.
The specific steps for obtaining the singular value decomposition of the split quaternion matrix are as follows:
calculating the non-zero singular value sigma of the split quaternion matrix A according to the following formula t ：
In the formula (I), the compound is shown in the specification,
p is a real representation matrix A σ The number of non-zero singular values of (c);
by constructive form of singular values and τ 1 ≥τ 2 ≥…≥τ p Is more than or equal to 0, and the | sigma is obtained 1 |≥|σ 2 |≥…≥|σ r |≥0；
The left and right singular vectors of the split quaternion matrix A are calculated as follows:
in the formula, mu s Is a left singular vector, v, of a split quaternion matrix A s Is a right singular vector of the split quaternion matrix a, s =1,2, \ 8230;, n;
satisfy the requirement of
k＝1,2,…,2n；
Combining equations (6) andequation (7) there is a unique diagonally split quaternion matrix ∑ r ＝diag(σ 1 ,σ 2 ,…,σ r ),|σ 1 |≥|σ 2 |≥…≥|σ r | ≧ 0, such that:
in the formula (I), the compound is shown in the specification,
U H U＝I n ,V H V＝I n ；
then sigma 1 ,…,σ r Singular values referred to as split quaternion matrix a;
according to the singular value of the split quaternion matrix A, the component product form of the singular value decomposition of the split quaternion matrix A is obtained as follows:
in the formula, mu t Is the left singular vector, i.e., the column of U; v is t Is the right singular vector, i.e., the column of V;
s4: and giving a threshold value, and substituting the given threshold value and the singular value norm of the split quaternion matrix A into a threshold function to obtain the dimensionality reduction degree.
The specific steps for obtaining the dimensionality reduction degree are as follows:
given a threshold, substituting the given threshold and the singular value norm of the split quaternion matrix a into the following threshold function:
in the formula (I), the compound is shown in the specification,
the method comprises the steps that an optimal approximate matrix of a split quaternion matrix A is obtained, alpha = s-1 is the singular value number of the split quaternion matrix A, and s is the dimensionality reduction degree;
the dimensionality reduction is calculated by equation (10).
It should be noted that the dimensionality reduction can also be determined by using a graph sorted from large to small singular value norms, i.e. the inflection point of the curve in the graph.
S5: and under the constraint of a threshold function, selecting the optimal approximate matrix of the original color image, and storing the effective decomposition matrix of the optimal approximate matrix according to the dimension reduction degree to finish image compression.
The specific steps of storing the effective decomposition matrix of the optimal approximate matrix according to the dimension reduction degree are as follows: according to the dimension reduction degree, the singular values of the m multiplied by n optimal approximate matrix are decomposed into an m multiplied by alpha order matrix, an alpha multiplied by alpha order matrix and an n multiplied by alpha order matrix, wherein alpha is far smaller than the minimum value of { m, n }, and at the moment, the m multiplied by alpha order matrix, the alpha multiplied by alpha order matrix and the n multiplied by alpha order matrix are stored to complete image compression.
In an embodiment, with continued reference to fig. 1, the method for compressing a fast color image based on a split quaternion model further includes the following steps: extracting a decomposition matrix m × α order matrix, an α × α order matrix, and an n × α order matrix of the compressed image in step S5, and multiplying the three decomposition matrices so as to satisfy matrix multiplication m × n = (m × α) × (α × α) × (n × α) H And reconstructing the optimal approximate matrix, and imaging the optimal approximate matrix to obtain an approximate image of the original color image to complete image compression reconstruction.
Through the optimal approximate matrix reconstruction, the compressed image can be reconstructed back to the original color image to be displayed.
In the field of image processing, visual fidelity can be measured by a number of numerical parameters. For example: original color image A' and compressed image (i.e., best approximation image A) α ') structural similarity index (hereinafter simply referred to as: SSIM) or peak signal-to-noise ratio (hereinafter: PSNR).
Several different quantization standards (including CPU time, PSNR, SSIM) are used below to determine the performance of the split quaternion model-based fast color image compression method of the present invention compared to the existing 5 color image compression methods. The existing 5 color image compression methods are respectively as follows: (1) an SVD color image compression method of a quaternion tool kit; (2) A real-guarantee structure SVD color image compression method of a quaternion matrix; (3) An SVD color image compression method of an equivalent complex matrix of a quaternion matrix; (4) a real matrix block [ R; g; b ] SVD color image compression method; and (5) an SVD color image compression method of three color spaces.
The structural similarity index is defined as:
in the formula, A α 'is the best approximation image of the original color image A', mu A' Is the local mean, σ, of the original color image A A' Is the standard deviation of the original color image a',
is the best approximate image A α ' a local mean value of a local mean value,
is the best approximation image A α The standard deviation of the' is that of,
is the original color image A' and the best approximation image A α The covariance of `; c 1 ＝(0.01×L) 2 Is the regularization constant of luminance, C 2 ＝(0.03×L) 2 Is the regularization constant for contrast, and L is a specified dynamic range value.
The peak signal-to-noise ratio is defined as:
in the formula, A α ' is originalThe best approximation of the starting color image a ', a' (: 1) = R, a '(: 2) = G, a' (: 3) = B.
Example 1: to simulate the pixel values of a color image matrix, matrix A 1 、A 2 、A 3 、A 4 The definition is as follows:
A 1 ＝zeros(m,n)，A 2 ＝255×rand(m,n)，A 3 ＝255×rand(m,n)，
A 4 ＝255×rand(m,n)，m＝n＝25:25:500。
comparing the CPU time and the error of the fast color image compression method based on the split quaternion model and the existing 5 color image compression methods. Referring to fig. 2a and 2b, in the drawings, SVDSQ represents a fast color image compression method based on a split quaternion model, QSVD of QTFM represents an SVD color image compression method of a quaternion tool box, structure-prediction method represents an SVD color image compression method of a real-guarantee Structure of a quaternion matrix, and SVD of a complex matrix represents an SVD color image compression method of an equivalent complex matrix of the quaternion matrix; SVD of [ R; g; b]Represents a real matrix block [ R; g; b is]The SVD color image compression method of (1); SVD in three color space represents the SVD color image compression method. Obviously, the running time of the SVD color image compression method of the quaternion toolbox and the SVD color image compression method of the solid structure of the quaternion matrix is slow, the running time of the other methods is relatively fast, and the 500-order matrix SVD decomposition is completely finished in less than 1 second. In terms of errors, the errors of the fast color image compression method based on the split quaternion model are kept at 3 x 10 relative to other methods -15 The following is not only time-advantageous but also has good performance. In addition, the color image compression problem is processed based on a four-dimensional algebraic model (such as a quaternion model and a split quaternion model), so that the internal relation of three color spaces can be better ensured, and the performance is superior.
Example 2: fig. 3a-3d show four classical color images from a CVGUGR (english: computer Vision Group-University of Granada) image database. Each color image can be represented as a purely imaginary split quaternion matrix or as a purely imaginary quaternion matrix. The color image given in fig. 3a-3d is compressed using the split quaternion model based fast color image compression method of the present invention and 5 existing compression methods in example 1.
Fig. 4a-4f, 5a-5f, and 6a-6f show the best compressed approximation images for the split quaternion model based fast color image compression method of the present invention and 5 prior art compression methods, corresponding to the 10, 30, and 50 singular values of fig. 3 a. It is obvious that when 10 singular values are selected for color image compression, the fast color image compression method based on the split quaternion model of the present invention can more completely retain the eye information of the image. When 30 and 50 singular values are selected for color image compression, the split quaternion model-based rapid color image compression method can more completely and clearly retain the background information of the image.
Fig. 7a-7c show the CPU time required for the split quaternion model based fast color image compression method of the present invention and 5 existing compression methods to compress and reconstruct four color images, selecting a certain number of singular values. Of these, fig. 7a selects 30 singular values, fig. 7b selects 60 singular values, and fig. 7c selects 90 singular values. Obviously, the fast color image compression method based on the split quaternion model has obvious time advantage.
The fidelity of the compressed image after the color image compression is carried out by the rapid color image compression method based on the split quaternion model and 5 existing compression methods is shown in the table 1 and the table 2. Table 1 shows PSNR data of the 6 compression methods, and table 2 shows SSIM data of the 6 compression methods. In table 1 and table 2, PSNR1 and SSIM1 show the peak signal-to-noise ratio and the structural similarity index of the fast color image compression method based on the split quaternion model according to the present invention; PSNR2 and SSIM2 show the performance of the SVD color image compression method of the quaternion toolbox; PSNR3 and SSIM3 show the performance of the SVD color image compression method with the real-guarantee structure of the quaternion matrix; PSNR4 and SSIM4 show the performance of the SVD color image compression method of the equivalent complex matrix of the quaternion matrix; PSNR5, SSIM5 show the real matrix block [ R; g; b ] performance of the SVD color image compression method; PSNR6, SSIM6 show the performance of the SVD color image compression method for three color spaces. Compared with the traditional color image compression algorithm, the color image processing process based on the four-dimensional algebraic model can store more image information, and the corresponding PSNR and SSIM values are relatively higher under the same compression condition. In addition, the three color image compression methods based on quaternion singular value decomposition only have difference in time, and the performances are the same.
TABLE 1
TABLE 2
As can be seen from tables 1 and 2, it is clear that the fast color image compression method based on the split quaternion model is significantly superior to other image compression methods in terms of visual fidelity. The compression speed is affected by the computational complexity and the size of the storage space. The invention is based on the fast color image compression method of the split quaternion model, and the unique algebraic structure of the split quaternion ensures the internal connection of three color spaces, and simultaneously, the unique 2m multiplied by 2n real number matrix represents, thereby greatly improving the speed of the algorithm, and the performance of the algorithm is the best of 6 compression methods.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (9)
1. A fast color image compression method based on a split quaternion model is characterized by comprising the following steps:
s1: constructing a split quaternion model, and representing an original color image by using the split quaternion model to obtain a split quaternion matrix A;
s2: calculating a real expression matrix of the split quaternion matrix A and singular value decomposition of the real expression matrix;
s3: calculating the singular value decomposition of the split quaternion matrix A according to the singular value decomposition of the real expression matrix, namely the singular value decomposition of the original color image;
s4: giving a threshold value, substituting the given threshold value and the singular value norm of the split quaternion matrix A into a threshold function to obtain the dimensionality reduction degree;
s5: and under the constraint of a threshold function, selecting the optimal approximate matrix of the original color image, and storing the effective decomposition matrix of the optimal approximate matrix according to the dimension reduction degree to complete image compression.
2. The split quaternion model-based fast color image compression method as claimed in claim 1, wherein in step S1, when the real part of the split quaternion model is zero, the split quaternion model is expressed as:
q(x,y)＝r(x,y)i+g(x,y)j+b(x,y)k (1)
in the formula, r (x, y), g (x, y), b (x, y) respectively represent red, green and blue of (x, y) point in the color image, i.e. x rows and y columns of pixel points in the three primary color matrix, and i, j, k are three imaginary part units of the split quaternion q.
3. The split quaternion model-based fast color image compression method as claimed in claim 2, wherein in step S1, the original color image has m rows and n columns, and the original color image is represented by the split quaternion model as:
in the formula, A is a split quaternion matrix of an original color image, and R, G and B are real number matrixes which respectively represent three color matrixes of red, green and blue;
representing a split quaternion ring H s M × n order matrix above.
4. The split quaternion model-based fast color image compression method as claimed in claim 3, wherein in step S2, the specific step of calculating the real representation matrix of the split quaternion matrix is:
for arbitrarily split quaternion matrices
Wherein A is s Is a real matrix of m × n order, s =1,2,3,4, a real representation matrix a of a split quaternion matrix a σ Is defined as:
and the real representation matrix satisfies the following equation:
(A+D) σ ＝A σ +D σ ,(AC) σ ＝A σ C σ ,(aA) σ ＝aA σ ,(A H ) σ ＝(A σ ) T (4)
in the formula (I), the compound is shown in the specification,
is an arbitrary m x n order split quaternion matrix,
is an arbitrary n x p-order split quaternion matrix, a is a real number,
an i-conjugate transpose representing the split quaternion matrix a,
representing a real matrix A s The transposed matrix of (2);
the real representation matrix of the split quaternion matrix is calculated by equation (3).
5. The split quaternion model-based fast color image compression method as claimed in claim 4, wherein in step S2, the specific step of calculating the singular value decomposition of the real representation matrix is:
for arbitrarily split quaternion matrices
Let m be more than or equal to n, and an orthogonal matrix exists through the singular value decomposition theory of the real expression matrix
And a real matrix
Such that:
in the formula (I), the compound is shown in the specification,
u i ∈R 2m ,i＝1,2,...,2n，
τ 1 ≥τ 2 ≥…≥τ 2n ≥0，τ 1 ,τ 2 ,…,τ 2n is a real representation matrix A σ The singular value of (a) is,
v j ∈R 2n ,j＝1,2,...,2n；
calculating a real expression matrix A by equation (5) σ Singular value decomposition of (c).
6. The split quaternion model-based fast color image compression method as claimed in claim 5, wherein the step S3 of obtaining the singular value decomposition of the split quaternion matrix comprises the specific steps of:
calculating the non-zero singular value sigma of the split quaternion matrix A according to the following formula t ：
Wherein t =1,2, \ 8230;, r,
p is a real representation matrix A σ The number of non-zero singular values of (c);
by constructive form of singular values and τ 1 ≥τ 2 ≥…≥τ p Is more than or equal to 0, and the | sigma is obtained 1 |≥|σ 2 |≥…≥|σ r |≥0；
The left and right singular vectors of the split quaternion matrix A are calculated as follows:
in the formula, mu s Is a left singular vector, v, of a split quaternion matrix A s Is a right singular vector of the split quaternion matrix A，s＝1,2,…,n；
Satisfy the requirement of
k＝1,2,…,2n；
In combination with equation (6) and equation (7), there is a unique diagonally split quaternion matrix Σ r ＝diag(σ 1 ,σ 2 ,…,σ r ),|σ 1 |≥|σ 2 |≥…≥|σ r | ≧ 0, such that:
in the formula (I), the compound is shown in the specification,
U H U＝I n ,V H V＝I n ；
then sigma 1 ,…,σ r Singular values referred to as split quaternion matrix a;
according to the singular value decomposition of the split quaternion matrix A, the component product form of the singular value decomposition of the split quaternion matrix A is obtained as follows:
in the formula, mu t Is the left singular vector, i.e., the column of U; v is t Is the right singular vector, i.e., the column of V;
7. the method for compressing a fast color image based on a split quaternion model as claimed in claim 6, wherein the step S4 of obtaining the dimensionality reduction degree comprises the specific steps of:
given a threshold, substituting the given threshold and the singular value norm of the split quaternion matrix a into the following threshold function:
in the formula (I), the compound is shown in the specification,
the method comprises the steps that an optimal approximate matrix of a split quaternion matrix A is obtained, alpha = s-1 is the singular value number of the split quaternion matrix A, and s is the dimensionality reduction degree;
the dimensionality reduction is calculated by equation (10).
8. The split quaternion model-based fast color image compression method as claimed in claim 7, wherein the specific step of storing the effective decomposition matrix of the optimal approximation matrix according to the dimension reduction degree is: in step S5, according to the dimension reduction degree, α = S-1 singular values are retained, and then the m × n best approximate matrix singular value is decomposed into three decomposition matrices, i.e., an m × α order matrix, an α × α order matrix, and an n × α order matrix, where α is much smaller than the minimum value of { m, n }, and at this time, the image compression can be completed by storing the three decomposition matrices, i.e., the m × α order matrix, the α × α order matrix, and the n × α order matrix.
9. The split quaternion model based fast color image compression method of claim 8 further comprising the steps of: extracting a decomposition matrix m × α order matrix, an α × α order matrix, and an n × α order matrix of the compressed image in step S5, and multiplying the three decomposition matrices so as to satisfy matrix multiplication m × n = (m × α) × (α × α) × (n × α) H And reconstructing the optimal approximate matrix, and imaging the optimal approximate matrix to obtain an approximate image of the original color image to complete image compression reconstruction.
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