CN103106643A - Method for recovering quaternion field color image in low sampling frequency - Google Patents

Abstract

The invention discloses a method for recovering a quaternion field color image in a low sampling frequency, and belongs to the technical field of digital picture processing. Firstly, a color image is expressed in a quaternion field, namely, values of an R component, a G component and a B component of the color image are respectively placed in three imaginary parts of a quaternion; and then downsampling is conducted to an original whole color image by utilizing an even stochastic matrix, and part of the sampled color image is obtained; furthermore, part of the color image is stored or conveyed; and finally the original color image is recovered through the part of the sampled color image by utilizing a real number expression form of the quaternion and quaternion matrix filling theory. The method for recovering the quaternion field color image in the low sampling frequency improves precision of recovering of the color image in the low sampling frequency and meanwhile simplifies complexity of sampling and storage content of data by utilizing a characteristic that a quaternion algorithm is good in binding force.

Description

A kind of method of quaternion field color image restoration under low sampling rate

Technical field

The present invention relates to the method for quaternion field color image restoration under a kind of low sampling rate, belong to the digital image processing techniques fields.

Background technology

Hypercomplex number q is a kind of special supercomplex, is the extend type of plural number.Hypercomplex number q is comprised of 1 real part and 3 imaginary parts:

q＝R(q)+I(q)i+J(q)j+K(q)k(1)

Wherein, R (q), I (q), J (q),

The expression real number field, i, j, k are three imaginary units, satisfy following character:

i ²＝j ²＝k ²＝ijk＝-1(2)

ij＝-ji＝k,jk＝-kj＝i,ki＝-ik＝j(3)

Conjugation and the mould of hypercomplex number are defined as respectively:

$\stackrel{\‾}{q}=R\left(q\right)-I\left(q\right)i-J\left(q\right)j-K\left(q\right)---\left(4\right)$

$\left|q\right|=\sqrt{R^2\left(q\right)+I^2\left(q\right)+J^2\left(q\right)+K^2\left(q\right)}---\left(5\right)$

The real number expression-form of hypercomplex number q is:

$q^R=\left[\begin{array}{cccc}R\left(q\right)& -I\left(q\right)& -J\left(q\right)& -K\left(q\right)\\ I\left(q\right)& R\left(q\right)& -K\left(q\right)& J\left(q\right)\\ J\left(q\right)& K\left(q\right)& R\left(q\right)& -I\left(q\right)\\ K\left(q\right)& -J\left(q\right)& I\left(q\right)& R\left(q\right)\end{array}\right]---\left(6\right)$

For Quaternion Matrix A,

Wherein,

Represent that respectively size is the Quaternion Matrix of m * n, n * s:

(A+B) ^R＝A ^R+B ^R,

(αA) ^R＝αA ^R,(7)

(AC) ^R＝A ^RC ^R

The norm of Quaternion Matrix X:

1) nuclear norm is defined as:

${\left|\right|X\left|\right|}_{*}=\underset{i=1}{\overset{\min \{m,n\}}{\mathrm{\Σ}}}{\mathrm{\σ}}_i\left(X\right)=\mathrm{trace}\left(\sqrt{X^{H}X}\right)---\left(8\right)$

σ _iThe singular value of () representing matrix.The mark of trace () representing matrix, i.e. diagonal of a matrix element sum.Subscript H represents conjugate transpose.

2) the Frobenius norm is defined as:

${\left|\right|X\left|\right|}_F=\sqrt{\underset{i=1}{\overset{\min \{m,n\}}{\mathrm{\Σ}}}{\mathrm{\σ}}_i^2\left(X\right)}=\sqrt{\mathrm{trace}\left(X^{H}X\right)}---\left(9\right)$

Attention: Quaternion Matrix X and its real number are expressed X ^RNorm equate, such as ‖ X ‖ _*=‖ X ^R‖ _*

Matrix fill-in (Matrix Completion:MC) is by Fazel, what the people such as Recht and Candes proposed minimizes by nuclear norm the theory that the Partial Elements that lacks in the low-rank matrix is filled, its theoretical frame as shown in Figure 1, " s.t. " expression in figure " Subject to (satisfying) ";

The expression complex field.

Specifically comprise the following steps:

Steps A, input size are the image array X of m * n;

Step B, at transmitting terminal, image array X is measured, namely utilize even stochastic sampling matrix that original size is sampled for the image array of m * n, and it is mapped as the vectorial b of p * 1, be b=Φ (X), wherein, X is that original size is the image array of m * n, Φ be from

The space is arrived

The linear mapping in space (that is: Φ:

).

Step C, the vectorial b that will measure store or transmit.

Step D, at receiving end based on the complex matrix filling theory, according to known linear mapping Φ: With the vectorial b(that measures be the parts of images matrix value), recover original entire image matrix, it is realized by the minimization problem of finding the solution following rank of matrix:

Wherein " min " expression " minimize(minimizes) ".

Existing image processing techniques relates to every aspect, but recovering this field with regard to image is generally to complete according to the nyquist sampling law reconstruction of thinking, this method is sampled with the twice greater than signal frequency, make the semaphore of sampling large, very high to memory requirement, in addition, existing technology is seldom the processing for coloured image.

Summary of the invention

Goal of the invention: the technical problem to be solved in the present invention is to solve than present complex field image recovers problem quaternion field color image restoration problem more widely, and a kind of quaternion field color image restoration method under low sampling rate is provided.

Technical scheme: a kind of quaternion field color image restoration method under low sampling rate, be included in transmitting terminal original quaternion field coloured image is carried out even stochastic sampling, and it is mapped to the less quaternionic vector of Spatial Dimension, thereby reach the purpose of low sampling rate.Specifically comprise the following steps:

steps A, hypercomplex number to coloured image is expressed, triple channel data value (R component with coloured image, the G component, the value of B component) be placed on respectively three imaginary parts the inside of each element of the Quaternion Matrix of a two dimension, the real part of each element of Quaternion Matrix is set to 0, namely a width size is the coloured image of m * n, its Quaternion Matrix size is m * n, expression-form is: q (x, y)=0+R (x, y) i+G (x, y) j+B (x, y) k, 0≤x≤m-1, 0≤y≤n-1, R (x wherein, y), G (x, y), B (x, y) represent respectively coloured image matrix (x, y) the R component of individual position, the G component, the value of B component, i, j, k are three imaginary units of hypercomplex number.

For example, coloured image partial pixel matrix is:

$\left[\begin{array}{ccc}\left(\mathrm{53,51,32}\right)& \left(\mathrm{53,51,10}\right)& \left(\mathrm{25,25,25}\right)\\ \left(\mathrm{25,25,25}\right)& \left(\mathrm{25,2525}\right)& \left(\mathrm{25,25,25}\right)\\ \left(\mathrm{25,25,25}\right)& \left(\mathrm{25,25,25}\right)& \left(\mathrm{25,25,25}\right)\end{array}\right]$

Corresponding hypercomplex number is expressed as:

$\left[\begin{array}{ccc}(0+53i+51j+32k)& (0+53i+51j+10k)& (0+25i+25j+25k)\\ (0+25i+25j+25k)& (0+25i+25j+25k)& (0+25i+25j+25k)\\ (0+25i+25j+25k)& (0+25i+25j+25k)& (0+25i+25j+25k)\end{array}\right]$

Therefore, can obtain size is 3 * 3 Quaternion Matrix.

Step B, with Quaternion Matrix X column vector, namely

Described above 3 * 3 Quaternion Matrix column vector turns to:

$\left[\begin{array}{c}(0+53i+51j+32k)\\ (0+25i+25j+25k)\\ (0+25i+25j+25k)\\ (0+53i+51j+10k)\\ (0+25i+25j+25k)\\ (0+25i+25j+25k)\\ (0+25i+25j+25k)\\ (0+25i+25j+25k)\\ (0+25i+25j+25k)\end{array}\right]$

Then utilize the gaussian random sampling matrix A that generates, size is p * mn, and the Quaternion Matrix after column vector is sampled, and obtains sampled data b, namely

The control of sampling rate is that the big or small p by the line number of sampling matrix realizes, for example, the column vector signal vec (X) of 9 * 1 is 3 * 9 random Gaussian sampling matrix A sampling with size, and its sampling rate is Obtain 3 sampled data (y ₁, y ₂, y ₃)

${\left[\begin{array}{cccc}{A}_{11}& A_{12}& \·\·\·& A_{19}\\ A_{21}& A_{22}& \·\·\·& A_{29}\\ A_{31}& A_{32}& \·\·\·& A_{39}\end{array}\right]}_{3\×9}{\left[\begin{array}{c}{x}_1\\ x_2\\ \·\\ \·\\ \·\\ x_9\end{array}\right]}_{9\×1}={\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]}_{3\×1}$

In addition, A can be real number Gauss sampling matrix, plural Gauss's sampling matrix or hypercomplex number Gauss sampling matrix, and is as follows:

(1) random Gaussian real number sampling matrix, namely the element in sampling matrix is real number, for example, the real number stochastic sampling matrix form of 3 * 3 is as follows:

$\left[\begin{array}{ccc}0.5377& 0.8622& -0.4336\\ 1.8339& 0.3188& 0.3426\\ -2.2588& -1.3077& 3.5784\end{array}\right]$

(2) random Gaussian complex sampling matrix, namely the element in sampling matrix is plural number, for example, the plural stochastic sampling matrix form of 3 * 3 is as follows:

$\left[\begin{array}{ccc}2.769+1.409i& 0.7254-1.207i& -0.205+0.4889i\\ -1.35+1.417i& -0.06305+0.7172i& -0.1241+1.035i\\ 3.035+0.6715i& 0.7147+1.63i& 1.49+0.7269i\end{array}\right]$

(3) random Gaussian hypercomplex number sampling matrix, namely the element in sampling matrix is hypercomplex number, for example, the plural stochastic sampling matrix form of 3 * 3 is as follows:

$\left[\begin{array}{cc}-0.1517+0.1626i-0.01503j-0.003425k& 0.4442-0.8558i+0.5466j+0.1857k\\ 0.1469-0.3775i-0.08244j+0.7663k& -0.5735-0.05112i+0.5546j-0.1128k\\ -0.3936+0.6851i+0.3139j-0.3848k& -0.5344-0.1207i-0.4318j+0.5587k\end{array}\right.$

$\left.\begin{array}{c}-0.4047+0.1596i+0.03868j-0.5545k\\ -1.472+0.1564i-0.6017j+0.01628k\\ 0.7192-0.4324i-0.5568j+0.2763k\end{array}\right]$

Step C, utilize matrix fill-in theoretical according to known A and b, recover original view picture coloured image matrix X, namely need to find the solution the minimization problem of the order of following matrix X:

min?rank(X)s.t.b=A·vec(X),

It is a NP-Hard problem that following formula is found the solution.Because the optimization of the nuclear norm of the optimization of the order of matrix X and matrix X has equivalence, following formula can be converted into the optimization problem under the nuclear norm of X so:

Further, for Quaternion Matrix vec (X), A and b, use the real number matrix vec (X) that has meaning of equal value with it ^R, A ^RAnd b ^RRepresentation, as follows:

Therefore formula (12) can be converted into:

min‖X ^R‖ _*?s.t.b ^R=A ^R·vec(X) ^R,

Formula (13) is about matrix X ^ROptimization problem can be further converted to about the optimization problem of matrix Y and Z and solve:

$\underset{Y,Z}{\min }\mathrm{trace}\left[\begin{array}{cc}Y& \\ & Z\end{array}\right]$

$s.t.\left[\begin{array}{cc}Y& X^R\\ {\left(X^R\right)}^T& Z\end{array}\right]\≥0,b^R=A^R\·\mathrm{vec}{\left(X\right)}^R---\left(14\right)$

Wherein, 〉=0 represents that matrix is positive semidefinite matrix,

With

So the optimum positive semidefinite square that will obtain is: $\left[\begin{array}{cc}Y& X^R\\ {\left(X^R\right)}^T& Z\end{array}\right],$ X ^RBe the optimum solution that will obtain.

Problem (14) solution form further clearly is expressed as follows: obtained a matrix X that size is m * n if coloured image carries out the hypercomplex number expression _{M * n}, the X column vector is obtained vec (X), namely

Then use size as the sampling matrix A(A of p * mn can be: real number, plural number, hypercomplex number).At first with sampling matrix, vec (X) is sampled and obtain sampled data

Then sampling matrix A and matrix vec (X) and the sampled data b that obtains are converted into corresponding real number matrix and express shape, namely by formula (7): A ^R* (vec (X)) ^R=b ^RIn view of SeDuMi is with the processing of objective matrix M column vector, i.e. processing array

Wherein,

So, $\mathrm{trace}\left[\begin{array}{cc}Y& \\ & Z\end{array}\right]$ Minimize and be:

$\min c\tilde{M}$

Formula: b ^R=A ^RVec (X) ^R

The basis on expression to utilize vec (X) ^R

The regularity of middle position, and then find corresponding

Realize So problem is expressed as:

$\min c\tilde{M}$

$\mathrm{mat}\left(\tilde{M}\right)\≥0$

Wherein,

Vec (b ^R) inverse transformation, namely column vector is become matrix.Matrix

Wherein,

$A_{i,j}^R=\left[\begin{array}{cccc}R\left(A_{i,j}\right)& -I\left(A_{i,j}\right)& -J\left(A_{i,j}\right)& -K\left(A_{i,j}\right)\\ I\left(A_{i,j}\right)& R\left(A_{i,j}\right)& -K\left(A_{i,j}\right)& J\left(A_{i,j}\right)\\ J\left(A_{i,j}\right)& K\left(A_{i,j}\right)& R\left(A_{i,j}\right)& -I\left(A_{i,j}\right)\\ K\left(A_{i,j}\right)& -J\left(A_{i,j}\right)& I\left(A_{i,j}\right)& R\left(A_{i,j}\right)\end{array}\right]$

Obtain

After, just can extract objective matrix X ^R, obtain the hypercomplex number of coloured image, and then just obtain the three-channel value of coloured image.

By realizing the method with SeDuMi software package and hypercomplex number QTFM software package.

Beneficial effect: compared with prior art, the invention has the advantages that the characteristics of utilizing the hypercomplex number algorithm restrictive strong, improved the precision of color image restoration under the condition of low sampling rate, complexity and memory data output when having simplified simultaneously sampling.

Description of drawings

Fig. 1 is the theoretical frame figure of matrix fill-in in prior art;

Fig. 2 is method flow diagram of the present invention;

Fig. 3 is the ten width coloured image samples that are used for testing restoration methods in the embodiment of the present invention, and size is 128 * 128;

Fig. 4 is that (sampling rate is followed successively by the coloured image effect of recovering with the real number matrix sampling in the embodiment of the present invention: 10%, 30%, 50%, 70%, 90%);

Fig. 5 is that (sampling rate is followed successively by the coloured image effect of recovering with the complex matrix sampling in the embodiment of the present invention: 10%, 30%, 50%, 70%, 90%);

Fig. 6 is that (sampling rate is followed successively by the coloured image effect of recovering with the Quaternion Matrix sampling in the embodiment of the present invention: 10%, 30%, 50%, 70%, 90%).

Embodiment

Below in conjunction with specific embodiment, further illustrate the present invention, should understand these embodiment only is used for explanation the present invention and is not used in and limits the scope of the invention, after having read the present invention, those skilled in the art all fall within the application's claims limited range to the modification of the various equivalent form of values of the present invention.

As shown in Figure 2, quaternion field color image restoration method under low sampling rate, below our the first width in Fig. 3 how to implement as example illustrates, because the accessible matrix size of internal memory restriction is limited, processes respectively and comprise the steps: so we are divided into some with a width coloured picture

Step 1, utilize quaternion field to express coloured image, 128 * 128 coloured images can be stored its triple channel data value with the Quaternion Matrix of 128 * 128, this matrix is divided into 256, and every block size is 8 * 8, and wherein first Quaternion Matrix is as follows:

Step 2, at first with the Quaternion Matrix column vector that obtains in the first step, the matrix of the random Gaussian matrix that then utilize to generate after to column vector sampled, and obtains sampled data.

Matrix after column vector is as follows:

${\left[\begin{array}{c}53I+51J+32K\\ 26I+26J+26K\\ \·\\ \·\\ \·\\ 26I+27J+27K\\ \·\\ \·\\ \·\\ \·\\ \·\\ \·\\ 22I+24J+21K\end{array}\right]}_{64\×1}$

Test figure shows that recovery effects can be better, namely selects random Gaussian complex sampling matrix better than the recovery effects of selecting random Gaussian real number sampling matrix when adopting restrictive strong sampling matrix that data are sampled; It is better than selecting random Gaussian complex sampling matrix recovery effects to select random Gaussian hypercomplex number sampling matrix.(seeing in detail test findings)

The control of sampling rate is that the size by the line number of sampling matrix realizes.

Here enumerating sampling rate is 10%, and namely the sampling matrix size is 6 * 64 real number matrix, complex matrix, Quaternion Matrix.

Real number matrix:

Complex matrix:

Quaternion Matrix:

Step 3, according to the step C in summary of the invention, transfer stochastic sampling matrix and the data that obtain of sampling to corresponding parameter matrix, use SeDuMi software package and hypercomplex number QTFM software package to obtain the optimum solution of every, then just obtain the data of view picture coloured picture.

Parameter matrix:

In order to verify the effect of the inventive method, carried out following experiment:

1, experiment condition:

Carry out confirmatory experiment on the computing machine of a 32-bit operating system, this allocation of computer is Intel (R) Core (TM) i5 four core processors (2.80G hertz) and 2G random-access memory (ram), and what programming language was used is the Matlab(2012a version).

2, experimental technique:

Choose ten width coloured images in Columbia University's image data base, size is 128 * 128, the three-channel data value of its RGB is mapped as hypercomplex number, obtain a size and be 128 * 128 Quaternion Matrix Q, then use Gauss's sampling matrix (real number matrix, complex matrix, Quaternion Matrix) A that this is sampled, obtain sampled data b, then according to Structural matrix Objective matrix is found the solution in the order of calling at last in SeDuMi.

3, the evaluation index of experimental result:

The hypercomplex number expression of test findings employing Recovery image is estimated the situation that image recovers with the size of the ratio of the F norm of the difference of former figure hypercomplex number expression and former figure hypercomplex number.

Error formula is:

$\mathrm{err}=\frac{{\left|\right|\mathrm{xrec}-\mathrm{xor}i\left|\right|}_F}{{\left|\right|\mathrm{xor}i\left|\right|}_F}=\frac{\mathrm{norm}(\mathrm{xrec}-\mathrm{xor}i)}{\mathrm{norm}\left(\mathrm{xor}i\right)}$

Wherein, xrec represents the hypercomplex number expression of Recovery image, and the former figure hypercomplex number of xori is expressed.

4, test findings (as shown in Fig. 3-6)

Table 1 is under the particular sample rate, when using respectively three kinds of sampling matrixs, and the effect of the ten width coloured picture Recovery images of selecting (mean value of err is less, and recovery effects is unreasonable to be thought).

Claims (3)

1. the method for quaternion field color image restoration under a low sampling rate, is characterized in that, comprises the steps:

At first coloured image is carried out hypercomplex number and express, namely a width size is the coloured image of m * n, and its hypercomplex number expression-form is: q (x, y)=0+R (x, y) i+G (x, y) j+B (x, y) k, 0≤x≤m-1,0≤y≤n-1, R (x, y) wherein, G (x, y), B (x, y) represents respectively coloured image matrix (x, y) the R component of individual position, G component, the value of B component; I, j, k are three imaginary units of hypercomplex number;

Then, the Quaternion Matrix X column vector that is m * n with original size turns to the big or small mn * 1 Quaternion Matrix vec (X) that is, utilize the size that generates to sample for the even stochastic sampling matrix A of p * mn, vec (X) is mapped as the quaternionic vector b of p * 1, be b=Φ (X)=Avec (X), Φ be from

The space is arrived

The linear mapping in space;

According to known sampling matrix A and the quaternionic vector b that sampling obtains, recover original view picture coloured image matrix X, specifically comprise the following steps:

min?rank(X)s.t.b=A·vec(X),

Rank (X) represents the order of matrix X, and finding the solution due to formula (1.1) is a NP-Hard problem, and the optimization of the nuclear norm of the optimization of the order of matrix X and matrix X has equivalence, and following formula can be converted into the optimization problem under the nuclear norm of X so:

Further, for Quaternion Matrix vec (X), A and b, use the real number matrix vec (X) that has meaning of equal value with it ^R, A ^RAnd b ^RRepresentation, as follows:

Therefore formula (1.2) can be converted into:

min‖X ^R‖ _*s.t.b ^R=A ^R·vec(X) ^R,

Formula (1.3) is about matrix X ^ROptimization problem be further converted to about the optimization problem of matrix Y and Z and solve:

$\underset{Y,Z}{\min }\mathrm{trace}\left[\begin{array}{cc}Y& \\ & Z\end{array}\right]$

$s.t.\left[\begin{array}{cc}Y& X^R\\ {\left(X^R\right)}^T& Z\end{array}\right]\≥0,b^R=A^R\·\mathrm{vec}{\left(X\right)}^R---\left(1.4\right)$

Wherein, 〉=0 represents that matrix is positive semidefinite matrix,

With

The optimum positive semidefinite square that obtains is: $\left[\begin{array}{cc}Y& X^R\\ {\left(X^R\right)}^T& Z\end{array}\right],$ X ^RBe the optimum solution that will obtain; In view of the SeDuMi software package is with the processing of objective matrix M column vector, i.e. processing array

Wherein,

So, $\mathrm{trace}\left[\begin{array}{cc}Y& \\ & Z\end{array}\right]$ Minimize and be:

$\min c\tilde{M}$

Formula: b ^R=A ^RVec (X) ^R

The basis on expression to utilize vec (X) ^R

The regularity of middle position, and then find corresponding

Realize So problem is expressed as:

$\min c\tilde{M}$

$\mathrm{mat}\left(\tilde{M}\right)\≥0$

Wherein,

Vec (b ^R) inverse transformation, namely column vector is become matrix; Matrix

Wherein,

$A_{i,j}^R=\left[\begin{array}{cccc}R\left(A_{i,j}\right)& -I\left(A_{i,j}\right)& -J\left(A_{i,j}\right)& -K\left(A_{i,j}\right)\\ I\left(A_{i,j}\right)& R\left(A_{i,j}\right)& -K\left(A_{i,j}\right)& J\left(A_{i,j}\right)\\ J\left(A_{i,j}\right)& K\left(A_{i,j}\right)& R\left(A_{i,j}\right)& -I\left(A_{i,j}\right)\\ K\left(A_{i,j}\right)& -J\left(A_{i,j}\right)& I\left(A_{i,j}\right)& R\left(A_{i,j}\right)\end{array}\right]$

Obtain After, extract objective matrix X ^R, obtain the hypercomplex number of coloured image, and then just obtain the three-channel value of coloured image.

2. the method for quaternion field color image restoration under low sampling rate as claimed in claim 1, it is characterized in that: A is real number stochastic sampling matrix, plural stochastic sampling matrix or hypercomplex number stochastic sampling matrix; P＜mn, the control of sampling rate size is that the big or small p by the line number of sampling matrix realizes.

3. the method for quaternion field color image restoration under low sampling rate as claimed in claim 1 or 2, is characterized in that: the triple channel data value that obtains coloured image by SeDuMi software package and hypercomplex number QTFM software package.

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