Research On Construction Method Of Compressed Sensing Measurement Matrices

Benefiting from the development and transitions of the data era,new technologies such as cloud computing,5G,artificial intelligence,Internet of Things and edge computing have been applied to people’s daily life.With the arrival of the big data era,various types of information are growing at an increasingly rapid rate.During the process of data collection,transmission and storage,hardware devices are facing more and more severe pressure.Meanwhile,in the process of data processing,there are certain security risks to information because the channel is not necessarily secure.However,data leakage incidents caused by disclosure,theft and breach of confidentiality are now commonplace.The leakage of some sensitive information may have a negative impact on human property and life safety.Therefore,in the process of information processing,there is an urgent requirement for a new data processing method to ensure the security of data transmission and release the pressure of hardware devices such as memory and sensors.In recent years,a breakthrough theory for information acquisition has been proposed,namely,compressed sensing theory.No matter what type of signal,this theory indicates that there is always a sparse or compressible representation of the signal in the original domain or some transformation domains.In order to ensure accurate or high probability reconstruction of the signal,linear projection values that are far lower than traditional Nyquist sampling are used in the transmission process.This provides a new optimization method for signal recovery,thereby broadening the new perspective of mathematical theory in engineering applications.It serves as a connection to the measurement matrix in compressed sensing.In the accurate or high probability reconstruction of sparse signals,the measurement matrix is crucial,and the compressed signal must contain enough important information;Otherwise,the original signal can not be reconstructed.A measurement matrix with good performance can not only ensure the accuracy or high probability of signal reconstruction,but also improve the performance of reconstruction algorithms.In the encryption process,the measurement matrix is regarded as an encryption key,which not only achieves the encryption effect,but also reduces the need for sensitive information to obtain resources,thereby effectively protecting the security of information.Signal processing and image encryption expand the application field of compressed sensing measurement matrices,so it is of great theoretical and practical significance to study the construction method of measurement matrices with good performance and design a secure and efficient image encryption scheme.This dissertation focuses on the construction of measurement matrices based on square matrices and image security transmission problems,and conducts research from three aspects:the construction method of measurement matrices based on high coherence square matrices,the construction method of measurement matrices based on low coherence square matrices,and image encryption schemes based on compression sensing.The main achievements and innovations of this dissertation are presented as follows:(1)For the incidence square matrix with high coherence in combinatorial design,a method that combines vertical expansion,horizontal expansion and embedded operation is proposed to construct the measurement matrix of compressed sensing,so that the constructed matrix meets the requirements of the special properties and special dimensions of the measurement matrix.The specific methods are given as follows.Firstly,based on a finite set,we propose a construction method of symmetric balanced incomplete block design,whose incidence matrix is a square matrix and has high coherence.Secondly,based on the idea of removing four rings of low-density parity check codes,a vertical expansion method is proposed to ensure the low coherence of the constructed matrix by increasing the number of rows of the incidence matrix.Then,a horizontal expansion method is proposed based on the same principle.Under the condition of maintaining the coherence,the number of columns of the incidence matrix is increased to ensure that the constructed matrix has a large dimension.Further,by embedded operation,the number of columns of the matrix is multiplied to make the constructed matrix meet the requirements of special properties and special dimensions of the measurement matrix.Finally,simulation experiments show that the constructed matrices have better reconstruction performance under OMP algorithm and IST algorithm compared with several typical matrices.(2)For the incidence square matrix with low coherence in projective plane,an effective method is proposed to construct the structured random measurement matrix in semi-tensor product compressed sensing,which makes the constructed matrix not only satisfies the requirements of the special properties and special dimensions of the measurement matrix,but also reduces storage space and computational complexity of the measurement matrix.The specific methods are shown as follows.Firstly,taking the corresponding relationship between the finite order projective plane and the incidence matrix of Steiner system in combinatorial design as the starting point,the full rank binary square matrix with low coherence is studied.Secondly,selecting the random matrix with appropriate dimension to carry out Gram Schmidt orthogonalization and unitization,the resulting matrix is also a random matrix with the same dimension.Then,using the selected two matrices as seed matrices,a structured random matrix is constructed through embedded operation,in which the elements of the matrix are arranged in a structured manner according to some certain rules,thus a smaller dimensional random matrix is needed.Finally,we propose a semi-tensor product compressed sensing model,which not only reduces storage space,but also improves transmission efficiency.Especially,when the scale of the matrix is large,the advantage of structured random matrix is more obvious.Experimental results show that when reconstructing one-dimensional signals and two-dimensional images,the constructed matrix has better reconstruction performance than several typical matrices.(3)Aiming at the secure storage and transmission of images in mobile devices with limited resources,an image encryption scheme based on parallel compressed sensing and secret sharing is proposed to ensure the secure and efficient transmission of image information.The specific methods are shown as follows.Firstly,the original image is measured by parallel compressed sensing,which realizes compression and encryption at the same time.This method not only reduces the computational complexity and storage space,but also improves the transmission efficiency.Secondly,the measured image is scrambled by block Arnold transform,and the obtained image is called the secret image,so as to achieve the effect of scrambling and encryption again.Then,the secret image is divided into n shadow images by Shamir’s(t,n)-threshold secret sharing scheme,which greatly reduces the size of the encrypted image that needs to be stored.To increase the applicability of mobile devices with limited resources,different application scenarios can be realized by adjusting the size of shadow images.Further,to ensure the security of shadow images,Zigzag confusion is used to confuse n shadow images in order to achieve the encryption effect.The simulation experiments analyzed that the proposed encryption scheme has good performance in terms of encryption and decryption,security and comparison with existing schemes.It is shown that the proposed scheme achieves the security and availability of the network system under the condition of limited resources.