Research On Sparse Representation Method Based On Geometric Algebra

Signal sparse representation is a novel method of signal analysis and synthesis.Its purpose is to represent signals with as few atoms as possible in an over-complete dictionary.As an effective image representation model,sparse representation uses an over-complete dictionary containing a certain number of prototype atoms as elements.Images can be represented as sparse linear combinations of these atoms,which has become a hot research focus in recent years.Sparse representation has achieved a relatively advanced level in gray image processing.However,for color image,the traditional method is to treat three color channels of color image as three independent gray images for monochrome processing,which completely ignores the relationship among channels,thus possibly causing hue distortion in reconstruction results.The improved method is to connect RGB channels to form a common channel to alleviate hue distortion and train a learning dictionary to represent this common channel.However,due to the lack of explicit constraints on the relationship among the three channels,unsatisfying results are still obtained.Therefore,based on the compatibility between quaternion matrix and color image,an effective color sparse representation model has emerged to avoid the loss of correlation among three color channels.However,for higher dimensional signals such as multispectral images,quaternions are powerless.Fortunately,geometric algebra(GA)provides an effective tool for processing multi-channel signals.Based on GA,a novel sparse representation model for color images and multispectral images is proposed.Since the multiplication of GA is non-commutative,a reduced geometric algebraic(RGA)basis is proposed with commutative multiplication rules.A new sparse representation model based on reduced geometric algebra is presented,which greatly reduces computational complexity.This paper firstly analyzes the structures of existing sparse representation models and the framework of geometric algebraic theory.Based on GA,a new color image sparse representation model and a multispectral image sparse representation model are proposed.Sparse representation is applied to higher dimensional data processing.Then,aiming at the high computational complexity caused by the non-commutative multiplication of GA,a reduced geometric algebraic(RGA)basis with commutative multiplication rules,is proposed.Based on RGA,a novel sparse representation model for color images is proposed,aiming at improving the performance and reducing the computational complexity.In addition,based on GA,an algorithm for minimizing the L1-norm of multi-channel signals is proposed.The specific contents of this paper are listed as follows:(1)The framework of geometric algebra(GA)and reduced geometric algebra(RGA)are studied.GA provides a powerful image processing framework and an efficient representation method for multi-channel signals,that is,the dimensions of multi-channel signals are encoded into each component of GA for holistic processing.However,the multiplication of GA is non-commutative,which has the disadvantage of high computational complexity.To solve this problem,a reduced geometric algebraic(RGA)basis with commutative multiplication rules is proposed,which can greatly reduce the computational complexity while removing data redundancy.(2)A sparse representation model for color images based on GA is proposed.In this model,the color image is represented as the form of GA multivector,and a dictionary learning algorithm based on GA is proposed.In the GA space,the sparse basis is chosen to convert the channel image to the orthogonal color space.In the process of vector reconstruction,the inherent color structure of the color image can be completely preserved.In addition,due to the low redundancy between different channels,the sparse model is more effective in image restoration tasks compared with the existing sparse models.The experimental results show that the proposed sparse representation model for color images based on GA successfully avoids the problem of hue deviation,and demonstrates its potential as a versatile and powerful tool in the field of color image analysis and processing.(3)A sparse representation model for multispectral images based on GA is proposed.GA has been applied widely in image processing.It provides a powerful method for the representation of multispectral images.Based on GA,a multivector sparse representation model for multispectral images is proposed.The model takes full account of spatial information and spectral information,and represents multispectral images as GA multivectors.A dictionary learning algorithm based on GA is proposed.Taking into consideration the correlation between spectral channels in multispectral images,the artifacts and blurring effects are successfully avoided.The experimental results show that the proposed sparse model is more practical and effective than the existing multispectral image reconstruction and denoising methods.(4)A sparse representation model for color images based on reduced geometric algebra(RGA)is proposed.First,a new RGA theory is introduced,which includes commutative sparse basis and their geometric operations.Secondly,based on the theory of RGA,the three-channel color image is represented as an RGA multivector,and the space information and spectral information are preserved in the RGA space.A new dictionary learning algorithm based on RGA is proposed.Experimental results show that the proposed sparse representation model for color images based on RGA can remove data redundancy,reduce the computational complexity,and effectively retain the original color structure.(5)A L1-norm minimization for multi-channel signals based on GA is proposed.The L1-norm minimization problem,which plays an important role in compressed sensing theory,is discussed.The existing methods are designed for low dimensional signals.In order to solve the L1-norm minimization problem for multi-channel signals,a GA-based optimization algorithm is proposed.By transforming it into two order cone programming,the L1-norm minimization problem of noisy and noiseless multichannel signals can be solved.The multi-channel signal is represented as a form of GA multivector,and it is processed integrally without losing the correlation among different channels of multi-channel signals.Numerical experiments show that the algorithm is effective and robust to noise.