Color image denoising method of special affine matrix wavelet based on quaternion representation

Abstract

The invention relates to the technical field of digital image processing, in particular to a quaternion representation-based color image denoising method for special affine matrix wavelets, which comprises the following steps of: for a to-be-denoised color image, firstly, enabling R, G and B components of the color image to correspond to three imaginary parts of a quaternion so as to represent the three imaginary parts as quaternion signals; secondly, mapping the quaternion signal into a second-order complex matrix value signal through an equidistant operator, decomposing the matrix value signal by using multi-resolution analysis related to special affine convolution in a matrix value function space, performing threshold processing on detail coefficients, reconstructing the processed coefficients by using a fast algorithm, and finally obtaining a reconstructed quaternion signal; and finally, restoring the matrix value signal into a quaternion signal through inverse mapping of an equidistant operator, reading component values of each channel, and outputting a denoised color image. According to the method, the color image is processed as a whole, the complexity of quaternion convolution is avoided, and the defects in the prior art are overcome.

Description

Technical Field

[0001] This invention relates to the field of digital image processing technology, and specifically to a color image denoising method based on a special affine matrix wavelet with quaternion representation. Background Technology

[0002] Color images, as an important carrier of modern information, have always been a hot research topic in processing techniques. Unlike binary and grayscale images, color images more closely resemble the colors of the real world, and there is a close relationship between the R, G, and B channels of a color image. Traditional methods process the R, G, and B channels of a color image independently. This method ignores the spectral relationships between the channels and the correlation between the primary color components, often leading to color distortion in the image.

[0003] As research has deepened, quaternions have been introduced into color image processing. Each channel component of a color image corresponds to one of the three imaginary parts of a quaternion, thus representing a color image as a pure quaternion signal. This allows the color image to be treated as a whole during processing, effectively preserving the correlation of primary color components and the spectral relationships between channels. However, the non-commutative nature of quaternions makes it very difficult to establish a wavelet analysis framework based on structurally simple convolutions in the quaternion domain. On the other hand, special affine Fourier transforms, by introducing additional free parameters, provide greater flexibility and freedom in signal analysis, but they still lack the ability to perform joint time-frequency localization analysis of signals. Summary of the Invention

[0004] The purpose of this invention is to provide a color image denoising method based on a special affine matrix wavelet with quaternion representation, which aims to solve the problems of complex convolution and low efficiency in wavelet transform in the quaternion domain, and the inability of traditional methods to balance color fidelity and processing flexibility in color image processing.

[0005] To achieve the above objectives, this invention provides a color image denoising method based on a special affine matrix wavelet with quaternion representation, comprising the following steps:

[0006] Step 1: Input a noisy color image and convert it into a quaternion signal representation;

[0007] Step 2: Use equidistant mapping to map the quaternion signal into a second-order complex matrix value signal;

[0008] Step 3: Based on the definition of affine transformation in the function space of a second-order complex matrix, construct a special affine multiresolution analysis framework to decompose the matrix-valued signal and perform thresholding on the detail coefficients;

[0009] Step 4: Use a special affine multiresolution analysis framework to quickly reconstruct the processed detail coefficients to obtain the denoised matrix value signal;

[0010] Step 5: Restore the matrix value signal to a quaternion signal, and combine them to output the denoised color image.

[0011] Optionally, in step 1, the values ​​of the R, G, B components of the color image are read. It is represented as a pure quaternion signal ,in Represents the three distinct imaginary parts of a quaternion.

[0012] Optionally, in step 2, the equidistant operator is used. Map each quaternion to a second-order complex matrix, let , The mapped matrix is

[0013]

[0014] in, Indicates conjugate;

[0015] The original quaternion signal is mapped to a second-order complex matrix-valued function space. The signal in the diagram, where R and C represent the set of real numbers and the set of complex numbers, respectively. The specific form is

[0016]

[0017] L 2 (R) denotes all square-integrable functions.

[0018] Optionally, in step 3, in the second-order complex matrix valued function space Define affine transformation

[0019]

[0020]

[0021] Kernel function

[0022]

[0023] It is a parameter matrix and satisfy diag represents a diagonal matrix; This represents a special affine Fourier transform of a square-integrable function f. (The parameters A, B, C, D, p, and q have no practical meaning; they are simply adjustable parameters derived from the definition of the special affine Fourier transform.)

[0024] For any Special affine convolution Defined as

[0025]

[0026] in , , Represents a diagonal matrix. This represents a type of convolution between matrix-valued functions.

[0027]

[0028] * indicates classical convolution. .

[0029] Optionally, the rapid refactoring process in step 4 includes the following steps:

[0030] Step 4.1: Select appropriate basis functions and decomposition levels to obtain approximation coefficients at different scales. and detail coefficient (The formulas and symbols here have been modified.)

[0031]

[0032]

[0033] in, These represent a matrix-valued special affine low-pass filter and a matrix-valued special affine high-pass filter, respectively. This represents the conjugate transpose of a matrix. Indicates the downsampling operator. Indicates a mirror filter;

[0034] Step 4.2: For all detail coefficients Perform threshold denoising;

[0035] Step 4.3: Reconstruct the processed coefficients using a fast reconstruction algorithm to obtain the denoised matrix value signal. The fast reconstruction formula is as follows:

[0036]

[0037] in, This represents the upsampling operator.

[0038] Optionally, in step 5, the inverse operator of the equidistant operator is used. Reconstruct the matrix-valued signal into a quaternion signal. It reads the R, G, and B component values ​​of the quaternion signal, reassembles them into a denoised color image, and outputs it.

[0039] This invention provides a color image denoising method based on a special affine matrix wavelet representation using quaternions. For the color image to be denoised, the R, G, and B components are first represented as the three imaginary parts of a quaternion, thus representing it as a quaternion signal. Next, an isometric operator is used to map the quaternion signal into a second-order complex matrix-valued signal. Then, multi-resolution analysis related to special affine convolution in the matrix-valued function space is used to decompose the matrix-valued signal, and thresholding is applied to the detail coefficients. A fast algorithm is then used to reconstruct the processed coefficients, obtaining the denoised matrix-valued signal. Finally, the matrix-valued signal is restored to a quaternion signal through the inverse mapping of the isometric operator, and the values ​​of each channel component are read to output the denoised color image. This invention treats the color image as a whole, avoiding the complexity of quaternion convolution and overcoming the shortcomings of existing technologies. Attached Figure Description

[0040] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0041] Figure 1 This is a schematic diagram illustrating the specific process of a color image denoising method based on a special affine matrix wavelet with quaternion representation according to the present invention.

[0042] Figure 2 This is a schematic diagram illustrating the implementation process of isometric mapping in the method of this invention.

[0043] Figure 3 This is a schematic diagram of the fast decomposition algorithm based on a novel special affine convolution of the present invention.

[0044] Figure 4 This is a schematic diagram of the fast reconstruction algorithm based on a novel special affine convolution of the present invention. Detailed Implementation

[0045] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0046] This invention provides a color image denoising method based on a special affine matrix wavelet represented by quaternions, comprising the following steps:

[0047] Step 1: Input a noisy color image and convert it into a quaternion signal representation;

[0048] Step 2: Use equidistant mapping to map the quaternion signal into a second-order complex matrix value signal;

[0049] Step 3: Based on the definition of affine transformation in the function space of a second-order complex matrix, construct a special affine multiresolution analysis framework to decompose the matrix-valued signal and perform thresholding on the detail coefficients;

[0050] Step 4: Use a special affine multiresolution analysis framework to quickly reconstruct the processed detail coefficients to obtain the denoised matrix value signal;

[0051] Step 5: Restore the matrix value signal to a quaternion signal, and combine them to output the denoised color image.

[0052] Specific details are as follows Figure 1 As shown, firstly, for noisy images, the R, G, and B channel component values ​​of each pixel are read and stored as quaternions. Secondly, the quaternions are processed into a second-order complex matrix signal through an isometric operator. Then, an appropriate basis function and decomposition level are selected, and a fast decomposition algorithm corresponding to the proposed special affine matrix wavelet transform is used to decompose the image to obtain approximate coefficients and detail coefficients. The detail coefficients are then processed by selecting an appropriate threshold. Furthermore, a fast reconstruction algorithm is used to reconstruct the processed coefficients to obtain the denoised matrix value signal. Finally, the inverse operator of the isometric operator is used to convert the matrix value signal into a quaternion signal, and the R, G, and B component values ​​are extracted to output the denoised image.

[0053] The following provides further explanation in conjunction with specific implementation steps (please refer to [link]). Figures 2 to 4 ): .

[0054] Step 1: Image Input and Quaternion Representation

[0055] Input a noisy image to be processed, and read the values ​​of the R, G, and B components of the color image. It is represented as a pure quaternion signal Therefore, a color image is represented as a two-dimensional quaternion signal. .

[0056] The isometric mapping in step 2 to the matrix-valued function space;

[0057] Using the equidistant operator Each quaternion is mapped to a second-order complex matrix. The specific implementation is as follows: Figure 2 As shown: Let , The mapped matrix is

[0058] .

[0059] Through this operation, the original quaternion signal is mapped to a second-order complex matrix-valued function space. The signal in the equation, where R and C represent the set of real numbers and the set of complex numbers, respectively. The specific form is

[0060] ,

[0061] Let denote the set of all square-integrable functions.

[0062] Figure 2 The specific details of the action of the equidistant operator are shown. For example... Figure 2 As shown, the R, G, and B components of a pixel are extracted and combined into a pure quaternion. Then extract and Construct a second-order complex matrix.

[0063] In step 3, affine transformation and a novel special affine convolution:

[0064] In the space of second-order complex matrix-valued functions Define affine transformation

[0065]

[0066] ,

[0067] Kernel function

[0068] ,

[0069] It is a parameter matrix and satisfy diag represents a diagonal matrix; This represents the special affine Fourier transform of a square-integrable function f. (The parameters A, B, C, D, p, and q have no practical meaning; they are simply adjustable parameters derived from the definition of the special affine Fourier transform.)

[0070] Subsequently, a new special affine convolution is defined: for any Special affine convolution Defined as

[0071] ,

[0072] in , , Represents a diagonal matrix. A type of convolution between matrix-valued functions.

[0073] ,

[0074] * indicates classical convolution. .

[0075] Furthermore, a scale space is constructed based on special affine convolution.

[0076] ,

[0077] Where j represents the number of decomposition levels, M represents a fixed parameter matrix, and c represents a matrix space with a special form of square summability. sequence in Let Z denote a special affine scaling function, and Z denote the set of integers. Based on the monotonicity of space, the scaling equation is:

[0078] ,

[0079] in, This represents a matrix-valued special affine low-pass filter. Indicates the constitutive space The basis functions.

[0080] Then, find a class of functions This makes it a wavelet space The orthonormal basis, and has

[0081]

[0082] in, and Let represent orthogonal and wavelet functions, respectively.

[0083] Finally, based on the spatial decomposition, wavelet equations are constructed.

[0084]

[0085] in This represents a special affine high-pass filter with matrix values.

[0086] Figure 3 The specific process of the fast decomposition algorithm is demonstrated. First, the approximation coefficients of the previous layer are... and filter , Perform a special affine convolution; then, combine the convolution result with the diagonal matrix. Multiply the results and downsample the result; finally, combine the results with... Multiply to output the approximation coefficients and detail coefficients for the next layer. and This completes one decomposition.

[0087] Step 4: Denoising in the wavelet transform domain using a special affine matrix.

[0088] In the second-order complex matrix valued function space, the special affine multiresolution analysis framework constructed in this invention is applied, and the specific steps are as follows:

[0089] (1) Decomposition: A fast decomposition and reconstruction algorithm based on a novel special affine convolution is used, such as... Figure 3 As shown, by selecting appropriate basis functions and the number of decomposition layers, approximation coefficients at different scales can be obtained. and detail coefficient (The formulas and symbols here have been modified.)

[0090]

[0091]

[0092] in These represent a matrix-valued special affine low-pass filter and a matrix-valued special affine high-pass filter, respectively. This represents the conjugate transpose of a matrix. Indicates the downsampling operator. This represents a mirror filter.

[0093] (2) Threshold denoising: Soft threshold denoising is performed on all detail coefficients.

[0094] (3) Signal Reconstruction: Using a fast reconstruction algorithm, the processed coefficients are reconstructed to obtain the denoised matrix signal. The fast reconstruction formula is as follows:

[0095] ,

[0096] in This represents the upsampling operator.

[0097] Figure 4 The specific process of the fast reconstruction algorithm is demonstrated. First, the approximation coefficients of the previous layer are... Upsampling; secondly, the upsampling result is compared with the filter. , Perform a special affine convolution; sum the final results and output the approximation coefficients for the next layer. This completes a refactoring process.

[0098] Step 5 includes the inverse mapping and output process:

[0099] (1) Inverse mapping: through the inverse operator of the isometric operator Reconstruct the matrix-valued signal into a quaternion signal. .

[0100] (2) Output: Read the R, G, B component values ​​of the quaternion signal, recombine them into a denoised color image and output it.

[0101] In summary, the present invention has the following beneficial effects:

[0102] This invention represents a color image as a complete signal for processing, fundamentally ensuring color fidelity and achieving holistic processing of color images. By using an equidistant operator to map quaternion signals into second-order complex matrix value signals, it cleverly avoids the problems of complex and inefficient convolution calculations when generating wavelet transforms in the quaternion domain. It integrates a special affine Fourier transform with multiple free parameters into the wavelet transform framework, introducing a high-degree-of-freedom analysis tool and enhancing the flexibility and adaptability of the wavelet method. It gradually establishes a complete theoretical framework, including the convolution theorem, multi-resolution analysis, and fast algorithms, constructing a solid and complete theoretical system to ensure the rigor and scalability of the method. It provides a fast decomposition and reconstruction algorithm corresponding to the theory, and its effectiveness in color image denoising has been verified, completing the transition from theory to practice and possessing strong practical value.

[0103] The above description discloses only one preferred embodiment of the present invention, and should not be construed as limiting the scope of the present invention. Those skilled in the art will understand that all or part of the processes of the above embodiments can be implemented, and equivalent changes made in accordance with the claims of the present invention are still within the scope of the invention.

Claims

1. A color image denoising method based on a special affine matrix wavelet with quaternion representation, characterized in that, Includes the following steps: Step 1: Input a noisy color image and convert it into a quaternion signal representation; Step 2: Use equidistant mapping to map the quaternion signal into a second-order complex matrix value signal; Step 3: Based on the definition of affine transformation in the function space of a second-order complex matrix, construct a special affine multiresolution analysis framework to decompose the matrix-valued signal and perform thresholding on the detail coefficients; Step 4: Use a special affine multiresolution analysis framework to quickly reconstruct the processed detail coefficients to obtain the denoised matrix value signal; Step 5: Restore the matrix value signal to a quaternion signal, and combine them to output the denoised color image.

2. The color image denoising method based on special affine matrix wavelets with quaternion representation as described in claim 1, characterized in that, In step 1, the values ​​of the R, G, and B components of the color image are read. It is represented as a pure quaternion signal ,in Represents the three distinct imaginary parts of a quaternion.

3. The color image denoising method based on special affine matrix wavelets with quaternion representation as described in claim 2, characterized in that, In step 2, the equidistant operator is used. Map each quaternion to a second-order complex matrix, let , The mapped matrix is ； in, Indicates conjugate; The original quaternion signal is mapped to a second-order complex matrix-valued function space. The signal in the diagram, where R and C represent the set of real numbers and the set of complex numbers, respectively. The specific form is ； L 2 (R) denotes all square-integrable functions.

4. The color image denoising method based on special affine matrix wavelets with quaternion representation as described in claim 3, characterized in that, In step 3, in the second-order complex matrix valued function space Define affine transformation ； ； Kernel function ； It is a parameter matrix and satisfy diag represents a diagonal matrix; The special affine Fourier transform representing a square-integrable function f; For any Special affine convolution Defined as ； in , , Represents a diagonal matrix. This represents a type of convolution between matrix-valued functions. ； * indicates classical convolution. .

5. The color image denoising method based on special affine matrix wavelets with quaternion representation as described in claim 4, characterized in that, The rapid refactoring process in step 4 includes the following steps: Step 4.1: Select appropriate basis functions and decomposition levels to obtain approximation coefficients at different scales. and detail coefficient : ； ； in, These represent a matrix-valued special affine low-pass filter and a matrix-valued special affine high-pass filter, respectively. This represents the conjugate transpose of a matrix. Indicates the downsampling operator. Indicates a mirror filter; Step 4.2: For all detail coefficients Perform threshold denoising; Step 4.3: Reconstruct the processed coefficients using a fast reconstruction algorithm to obtain the denoised matrix value signal. The fast reconstruction formula is as follows: ； in, This represents the upsampling operator.

6. The color image denoising method based on special affine matrix wavelets with quaternion representation as described in claim 5, characterized in that, In step 5, the inverse operator of the equidistant operator is used. Reconstruct the matrix-valued signal into a quaternion signal. It reads the R, G, and B component values ​​of the quaternion signal, reassembles them into a denoised color image, and outputs it.

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