CN-122023565-A - Multi-dimensional color image filling method and system based on quaternion tensor

CN122023565ACN 122023565 ACN122023565 ACN 122023565ACN-122023565-A

Abstract

The invention relates to the technical field of computer vision and signal processing, in particular to a multi-dimensional color image filling method and system based on quaternion tensors. According to the method, RGB values of an image are encoded into pure quaternions through quaternion tensor representation, the pure quaternion tensor is organized into a quaternion tensor, inherent complex nonlinear characteristics in visual data are captured through nonlinear transformation in a quaternion domain, then two novel regularization terms are introduced through construction of a unified optimization model, global low rank performance and local smoothness are jointly encoded, finally, the model is solved through a nonlinear alternating direction multiplier method, and therefore quaternion representation, nonlinear transformation, low rank performance and smoothness prior effective fusion is achieved, and the model has higher expression capacity and robustness than that of a traditional linear method, and therefore multi-dimensional color image filling with higher quality and robustness is achieved.

Inventors

YANG LIQIAO
JIANG TAIXIANG
LIU GUISONG
MA AO
HU YEXUN
TANG ZIXIN

Assignees

喀什地区电子信息产业技术研究院
西南财经大学

Dates

Publication Date: 20260512
Application Date: 20251229

Claims (7)

1. A multi-dimensional color image filling method based on quaternion tensors, which is characterized by comprising the following steps: S1, acquiring multidimensional color image data to be filled, constructing quaternion tensors to be filled containing image missing information, and defining an observation set and a corresponding projection operator based on the image missing information; s2, respectively constructing a component nonlinear activation function according to the quaternion tensors to be filled, and defining a nonlinear transformation quaternion tensor total variation regularization term; S3, based on a component nonlinear activation function and a nonlinear transformation quaternion total variation regularization term, a multidimensional color image filling model is established by combining an observation set and a corresponding projection operator, an auxiliary variable and a Lagrangian multiplier are introduced, an augmented Lagrangian function is established, and sub-problem decomposition is carried out on the multidimensional color image filling model, so that a sub-problem set comprising an auxiliary variable updating sub-problem, a main variable updating sub-problem and a multiplier updating sub-problem is obtained; S4, according to the sub-problem set, iterative optimization is carried out on the quaternion observation tensor based on a nonlinear alternating direction multiplier method frame until all the sub-problems are converged, and the converged quaternion tensor is reconstructed into the filled multidimensional color image data.
2. The method of claim 1, wherein the constructing the quaternion observation tensor including the image missing information in step S1 includes extracting red, green and blue color channel values of each pixel point in the multidimensional color image data to be filled, encoding each color channel value into an imaginary part of a corresponding quaternion pixel, mapping the imaginary part to a quaternion algebraic domain, and obtaining the quaternion tensor to be filled, wherein the quaternion observation tensor is expressed as: ; Wherein, the Representing the quaternion tensor to be filled, The imaginary parts of the red, green and blue color channel values are represented respectively, Representing the real component of the signal, Coefficients representing the imaginary parts of the red, green, and blue color channel values, respectively; Defining the observation set and the corresponding projection operator based on the image missing information comprises extracting element values with observation values based on the quaternion tensor to be filled and establishing the observation set, defining the projection operator to keep the element values in the observation set unchanged, and setting the element values except the observation set to be zero.
3. The method of claim 2, wherein the component nonlinear activation function in step S2 is configured to act on each algebraic component of the quaternion tensor independently, and is defined as: ; Wherein, the Representing a component-type nonlinear activation function, Representing a nonlinear function from the real number domain to the real number domain.
4. A multi-dimensional color image filling method based on quaternion tensors according to claim 3, wherein the non-linear transformation quaternion tensor total variation regularization term in step S2 comprises a regularization term based on summation calculation And a canonical term based on product calculation Expressed as: ; ; Wherein, the Representing a set of modes that have been set in advance, The pattern set cardinality is represented, The current mode is indicated and the current mode is indicated, Representing the total number of data dimensions, Representing the current dimension of the data, Representing the product of the quaternion tensor modes, A row-wise cyclic matrix is represented, Representing a quaternion tensor kernel norm; the regular term based on summation calculation is used for carrying out differential operation on the quaternion tensors after nonlinear transformation along each mode direction, calculating the quaternion tensor kernel norms of each differential result, and carrying out weighted summation on the kernel norms of all the mode directions; the regular term based on product calculation is used for sequentially carrying out sequential differential operation on the quaternion tensor after nonlinear transformation along all mode directions to obtain a comprehensive differential tensor, and calculating the quaternion tensor kernel norm of the comprehensive differential tensor.
5. The method of claim 4, wherein when the non-linear transformation quaternion tensor total variation regularization term is a regularization term based on a summation calculation And introducing a first auxiliary variable to constrain the multi-dimensional color image filling model, and constructing a first augmented Lagrangian function expressed as: ; Wherein, the 、、 Representing the auxiliary variables introduced when computing the regularization term based on summation, 、 Representing the lagrangian multiplier and, Representing a positive penalty parameter; representing an indicative function when When the value is 0, otherwise, the value is , Representing the operation of taking the real part.
6. The method of claim 4, wherein when the non-linear transformation quaternion tensor total variation regularization term is a product-based regularization term And introducing a second auxiliary variable to constrain the multi-dimensional color image filling model, and constructing a second augmented Lagrangian function expressed as: ; Wherein, the 、 Representing the auxiliary variables introduced when employing product-based calculation of the regularization term, 、 Representing the lagrangian multiplier and, Representing penalty parameters; representing an indicative function.
7. A quaternion tensor-based multi-dimensional color image filling system implemented based on the quaternion tensor-based multi-dimensional color image filling method according to any one of claims 1-6, comprising: The input module is used for acquiring multidimensional color image data to be filled, constructing quaternion tensors to be filled containing image missing information, and defining an observation set and a corresponding projection operator based on the image missing information; The model parameter generation and storage module is used for respectively constructing component type nonlinear activation functions according to the quaternion tensors to be filled, defining nonlinear transformation quaternion tensor total variation regularization items and storing the nonlinear transformation quaternion tensors; the model construction and subproblem decomposition module is used for constructing a multi-dimensional color image filling model based on component nonlinear activation functions and nonlinear transformation quaternion total variation regularization terms, combining an observation set and corresponding projection operators, introducing auxiliary variables and Lagrangian multipliers, constructing an augmented Lagrangian function, and carrying out subproblem decomposition on the multi-dimensional color image filling model to obtain a subproblem set comprising auxiliary variable update subproblems, main variable update subproblems and multiplier update subproblems; The optimization processing module is used for calling the sub-problem set, carrying out iterative optimization on the quaternion observation tensor based on a nonlinear alternating direction multiplier method frame until each sub-problem is converged, and reconstructing the converged quaternion tensor into filled multidimensional color image data; The processing module comprises a regularization operator solving sub-module and a nonlinear approximation sub-module, wherein the regularization operator solving sub-module is used for iteratively solving non-smooth items in the model, and the nonlinear approximation sub-module is used for solving non-convex items in the model.

Description

Multi-dimensional color image filling method and system based on quaternion tensor Technical Field The invention relates to the technical field of computer vision and signal processing, in particular to a multi-dimensional color image filling method and system based on quaternion tensors. Background Compared with the traditional two-dimensional image, the color imaging data can provide richer visual information, such as distinguishing the inherent properties (such as color, material and texture) of an object, providing chromaticity clues of illumination and shadow, and improving the accuracy of target detection and scene understanding, so the color imaging data are widely applied to high and new technical fields such as virtual reality, remote sensing, medical imaging and the like. However, in actual engineering processes such as data acquisition, imaging device manufacturing defects, or limited wireless transmission conditions, pixel missing (i.e., tensor complete problem) is inevitably present in high-dimensional image data. In view of the great influence of image information loss on subsequent applications (such as object tracking and three-dimensional reconstruction), how to recover color image information from incomplete observation data with high precision has become one of the core research hotspots in the fields of computer vision and application mathematics. In order to effectively preserve the inherent correlation between the three color channels of red (R), green (G) and blue (B), quaternion algebra has become a natural and powerful representation. In this paradigm, the pixel values of three color channels of a pixel are encoded into three imaginary parts of a pure quaternion. This overall processing of color has proven to be superior to the independent processing channels or the method of converting it to gray scale and has been successfully applied to color image denoising, restoration, and color video restoration. Expanding this paradigm to multi-dimensional color data, a quaternion tensor (quaternion tensor, QT) model is generated. The QT model naturally represents a multi-dimensional color image by stacking quaternion matrices along a third or higher order pattern, e.g., stacking color image frames to yield QT representation of color video. Although the relationship between RGB channels is captured within the quaternion-based model, their application in multi-dimensional color image modeling is relatively underexplored. The existing QT completion method is mainly focused on a low-rank linear model that expands real-valued tensors. This facilitates the definition of various QT ranks, including ranks based on a turner, tensor train (tensor train), and tensor loop (tensor ring) decomposition. And a quaternion tensor singular value decomposition (TQt-SVD) and a corresponding TQt-rank based on linear transformation under the QT product framework. Despite the above advances made by existing color multidimensional image filling methods, there are two fundamental challenges, one, the breaking of color structures in conjunction with priors. Conventional real-valued tensor models, while widely used and capable of efficient recovery by hybrid regularization (combining global low rank and local smoothness or sparsity), typically simply stack color channels, destroying the inherent color structure of the data. The efficient hybrid regularization idea is adapted to the quaternion domain to form a structured collaborative priori, and the method still needs to be perfected. Secondly, the capture of the nonlinear structure is insufficient. Real world visual data typically exhibits complex nonlinear structures resulting from motion, illumination changes, object interactions, etc., which cannot be adequately captured by the linear quaternion tensor model described above. While some recent work has explored the application of nonlinear transformations in real-valued tensor recovery, integrating them into the quaternion tensor framework, and corresponding optimization methods and convergence analysis, remains a challenge to be solved. Therefore, how to construct a high-precision tensor filling model which can preserve the color relation integrally, effectively capture nonlinear characteristics and cooperatively utilize the structure prior is a technical problem to be solved currently. Disclosure of Invention In order to solve the technical problems, the invention provides a multi-dimensional color image filling method based on quaternion tensors, which is characterized by comprising the following steps: S1, acquiring multidimensional color image data to be filled, constructing quaternion tensors to be filled containing image missing information, and defining an observation set and a corresponding projection operator based on the image missing information; s2, respectively constructing a component nonlinear activation function according to the quaternion tensors to be filled, and defining a nonlinear transformati