Phase Retrieval of Sparse Signals from Fourier Transform Magnitude using Non-Negative Matrix Factorization Mohammad Shukri Salman, Alaa Eleyan Electrical and Electronic Engineering Department Mevlana University Konya, Turkey {mssalman, aeleyan}@mevlana.edu.tr Zeynel Deprem , A. Enis Cetin Electrical and Electronic Engineering Department Bilkent University Ankara, Turkey cetin@bilkent.edu.tr, zdeprem@ee.bilkent.edu.tr Abstract—Signal and image reconstruction from Fourier Transform magnitude is a difficult inverse problem. Fourier transform magnitude can be measured in many practical applications, but the phase may not be measured. Since the autocorrelation of an image or a signal can be expressed as convolution of ࢞ሺ࢔ሻ with ࢞ሺെ࢔ሻ, it is possible to formulate the inverse problem as a non-negative matrix factorization problem. In this paper, we propose a new algorithm based on the sparse non-negative matrix factorization (NNMF) to estimate the phase of a signal or an image in an iterative manner. Experimental reconstruction results are presented. I. INTRODUCTION Magnitude of the Fourier transform of a desired signal or an image can be measured or calculated in many practical problems ranging from astronomical imaging to crystallography. However, it may not be possible to measure the phase information. This inverse signal reconstruction problem is a difficult one and a wide range of algorithms have been proposed in the literature [1]-[12]. The inverse problem is especially difficult when the magnitude values are corrupted by noise. It is well known that when the desired signal ݔሾሿ is one dimensional and it consists of  samples the phase retrieval problem has 2 to the  solutions. When the signal has two or more dimensions the solution is unique in almost all cases [12]. The main reason for this difference between the 1-D and higher dimensional problems are that the fundamental theorem of algebra cannot be extended to higher dimensions. In this article we assume that the desired signal is a discrete signal and its Fourier Transform is available for all frequency values. We also assume that: (i) the signal is non-negative, (ii) sparse in time or spatial domain, and (iii) finite-extent, i.e., it consists of N samples in 1-D, and it consists of N×N samples. Sparsity is also assumed by Eldar et al. [15] Non-negativity of the signal samples were used by Fienup and his co-workers. In almost all image processing problems pixel values are positive therefore the non-negativity assumption is not a restrictive assumption. In this article, the nonnegative matrix factorization (NNMF) algorithm and its sparsified version [13], [14] are used to solve the phase retrieval problem. Since the inverse Fourier Transform of the squared magnitude is the autocorrelation sequence of the desired signal it is possible to formulate the phase retrieval problem as a non-negative matrix factorization problem. In Section 2, the NNMF based algorithm is described. In Section 3, simulation examples are presented. II. PHASE RETRIEVAL PROBLEM Let ݔሾሿ,  ൌ 0, 1, 2, . . . ,  െ 1be the signal to estimated. The Discrete Fourier Transform (DFT) of the signal is given by ሺሻ ൌ ∑ ݔሾሿ ௝ఠ௡ ேଵ ௡ୀ଴ . (1) The Fourier Transform magnitude |ሺሻ| information is equivalent to the autocorrelation sequence ݎሾሿ of the signal. This is because ݎሾሿ ൌ ݔሾሿ ݔ כሾെሿ ൌ ܨ ଵ ሼ|ሺሻ| ଶ ሽ, (2) where * is the convolution operator and ܨ ଵ ሼ. ሽdenotes the inverse Fourier Transform. This also can be represented using matrix multiplication as ൦ ݎሾ0ሿ ݎሾ1ሿ ڭ ݎሾ െ 1ሿ ൪ൌ൦ ݔሾ0ሿ ݔሾ1ሿ ڮݔሾ െ 2ሿ ݔሾ െ 1ሿ 0 ݔሾ0ሿ ڮݔሾ െ 3ሿ ݔሾ െ 2ሿ ڭ ڭ ڰ ڭ ڭ 0 0 ڮ0 ݔሾ0ሿ ൪൦ ݔሾ0ሿ ݔሾ1ሿ ڭ ݔሾ െ 1ሿ ൪, (3) where ࢘ൌሾݎሾ0ሿ ݎሾ1ሿ ڮݎሾ െ 1ሿሿ  is the autocorrelation sequence and ࢞ൌሾݔሾ0ሿ ݔሾ1ሿ ڮݔሾ െ 1ሿሿ  is the input signal. As a result the autocorrelation sequence can be represented as: ࢘ ൌ ࢄ࢞, (4) where ࢄ and ࢞ are defined in Eq. (3), respectively. In phase retrieval problems, it is assumed that the autocorrelation vector r is known but ࢄ and ࢞ are unknowns. Therefore it is possible to apply the NNMF algorithms to estimate ࢄ and ࢞. 1113 978-1-4799-0248-4/13/$31.00 ©2013 IEEE GlobalSIP 2013