A Novel Image Compressive Sensing Method Based on Complex Measurements Nandini Ramesh Kumar and Wei Xiang Faculty of Engineering & Surveying University of Southern Queensland Toowoomba, QLD 4350, AUSTRALIA Email: nandini.rameshkumar@usq.edu.au Jeffrey Soar School of Information Systems Faculty of Business & Law University of Southern Queensland Toowoomba, QLD 4350, AUSTRALIA Abstract—Compressive sensing (CS) has emerged as an effi- cient signal compression and recovery technique, that exploits the sparsity of a signal in a transform domain to perform sampling and stable recovery. The existing image compression methods have complex coding techniques involved and are also vulnerable to errors. In this paper, we propose a novel image compression and recovery scheme based on compressive sensing principles. This is an alternative paradigm to conventional image coding and is robust in nature. To obtain a sparse representation of the input, discrete wavelet transform is used and random complex Hadamard transform is used for obtaining CS measurements. At the decoder, sparse reconstruction is carried out using compressive sampling matching pursuit (CoSaMP) algorithm. We show that, the proposed CS method for image sampling and reconstruction is efficient in terms of complexity, quality and is comparable with some of the existing CS techniques. We also demonstrate that our method uses considerably less number of random measurements. Index Terms—Compressive sensing, image representation, CS reconstruction, CoSaMP, complex Hadamard transform. I. I NTRODUCTION The popularly known Shannon sampling theory states, the sampling rate must be twice the signal bandwidth in order to avoid the loss of information while capturing a signal. The traditional approach of data acquisition is to sample at Nyquist rate followed by coding methods for compression. It is also true that any compressible signal can be well approximated using sparse representation and hence could be exploited for reduction in complexity of encoding. Compressive sensing [1] [2] (also called compressive sampling) is an emerging theory based on sparse representations. It provides a dramatic reduction in sampling rates and computation complexity in data compression. It aims to measure sparse and compress- ible signals close to their intrinsic information rate rather than their Nyquist rate [1]. It addresses the shortcomings of the traditional transform-based methods by directly acquiring compressed samples. It uses the concept that a small group of non-adaptive linear projections of a sparse signal contain enough information to reconstruct the complete data and also preserve the originality of the signal. An appropriate way to obtain linear measurements is by using incoherent sampling in a transform domain that is equipped with fast transform algorithms [3]. Hence, the CS theory has been rapidly gaining more attention in image/video processing due to the requirement for processing large data. The key elements of image CS are measurement matrix and reconstruction algorithm. The measurement matrix is selected based on a sufficient condition that it satisfies the restricted isometric property. Several matrices have been proposed in literature for video CS, like independent identically distributed Gaussian matrix [4], Bernoulli matrices as in [5] [6]. Their main advantage is that they are universally incoherent with any sparse signal and thus, the number of compressed mea- surements required for exact reconstruction is almost minimal. However, they inherently have two major drawbacks in practi- cal applications: huge memory buffering for storage of matrix elements and high computational complexity due to their completely unstructured nature [2]. Another group of matrices based on Fourier and Hadamard were also proposed, as in [7] where it is called the scrambled block Hadamard ensemble. Partial Fourier exploits the fast computational property of Fast Fourier Transform (FFT) and thus, significantly reduces the complexity of a sampling system. However, partial Fourier matrix is only incoherent with signals which are sparse in the time domain, severely narrowing its scope of applications. A few reconstruction algorithms have been in popular use for image processing in CS. To name a few, orthogonal matching pursuit (OMP) [8], a modified version of gradient projection for sparse reconstruction (GPSR) [9]. Though these algorithms are fast, they require a large number of samples which could be tedious to acquire. Also, algorithms like GPSR and its many versions are computationally burdensome. There are few more reconstruction algorithms like compressive sampling matching pursuit (CoSaMP) [10] that have been suggested for image/video processing theoretically but have not been practically explored. This algorithm considers the shortcom- ings of other existing reconstruction algorithms and guarantees speed and are effective. More theoretical approaches for exact reconstruction have been proposed, but only a couple of them work practically specially in case of image processing. Considering the literature available for CS based system for image compression and reconstruction, it is evident that an efficient method is necessary which addresses the drawbacks of the existing methods. Most of the literature concentrate on using random measurement ensembles, but none to the best of 2011 International Conference on Digital Image Computing: Techniques and Applications 978-0-7695-4588-2/11 $26.00 © 2011 IEEE DOI 10.1109/DICTA.2011.36 173 2011 International Conference on Digital Image Computing: Techniques and Applications 978-0-7695-4588-2/11 $26.00 © 2011 IEEE DOI 10.1109/DICTA.2011.36 175