1057-7149 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2019.2959737, IEEE Transactions on Image Processing IEEE TRANSACTIONS ON IMAGE PROCESSING, 2018 1 Sensing Matrix Design for Compressive Spectral Imaging via Binary Principal Component Analysis Jonathan Monsalve, Hoover Rueda-Chacon, Member, IEEE, and Henry Arguello, Senior Member, IEEE Abstract—Compressive spectral imaging (CSI) is a framework that captures coded-and-multiplexed low-dimensional projections of spectral data-cubes. In general, the sensing process in many CSI architectures is described using binary matrices, so-called sensing/projection matrices, whose elements can be either ran- dom or designed. However, some characteristics of the spectral data, such as the ℓ2-norm or the second moment statistics, can be lost when this dimensionality reduction is performed. Similarly, principal component analysis (PCA) is a data dimensionality reduction technique that minimizes the least-squared error be- tween the spectral data and its low-dimensional projection, but preserving its structure or variance. Thus, PCA can be used to guide the CSI acquisition process by designing the binary sensing matrix. Nonetheless, PCA requires to know the spectral image a-priori, and also, its associated projection matrix is not binary, as required by CSI optical architectures. Therefore, in this paper, an algorithm to design CSI sensing matrices by exploiting the structure-preserving property of the PCA projection is proposed. First, a set of compressive measurements obtained with random sensing matrices is used to rapidly estimate the covariance matrix associated with the spectral data. Then, a new sensing matrix is designed by solving a non-convex optimization problem that finds a set of binary vectors that approximate the principal components of the covariance matrix, thus maximizing the explanation of the data variance. Experimental results show an improvement of up to 3 dB in image reconstruction quality, in terms of the peak signal to noise ratio (PSNR), when the binary PCA- based sensing matrices are used and compared with conventional random sensing matrices and state-of-art designed matrices based on PCA. Index Terms—Compressive spectral imaging, binary principal component analysis, sensing matrix design. I. I NTRODUCTION C OMPRESSIVE spectral imaging (CSI) is a framework to acquire and compress spectral images by means of coded bi-dimensional projections, such that, the number of required measurements for reconstruction are fewer than those needed by traditional techniques based on the Shannon- Nyquist sampling theorem [1], [2], [3], [4]. CSI exploits the fact that natural scenes can be accurately represented in a lower dimensional subspace. This concept is known as sparsity or low rank behavior [3], [5], [6], [7]. Further, the linear projector, so-called sensing matrix, used to capture the compressed Jonathan Monsalve is with Department of Electrical Engineering, Hoover Rueda-Chacon and Henry Arguello are with the Department of Computer Sci- ence, Universidad Industrial de Santander, Bucaramanga, Colombia, 680002 e-mail: henarfu@uis.edu.co This paper has supplementary downloadable material available at http://ieeexplore.ieee.org provided by the author. The material includes proofs and additional figures. MATLAB code can be downloaded from https://codeocean.com/capsule/8658864/. Contact henarfu@uis.edu.co for fur- ther questions about this work. version of the spectral image has to be incoherent with the representation basis, where the data becomes sparse. This, in turn, guarantees with high probability an accurate reconstruc- tion, since an incoherent matrix has a dense representation in the basis domain, and so, no assumption on the behavior of the data is required. Principal component analysis (PCA) is a technique used to reduce the dimensionality of a signal by projecting it into a lower dimension, such that, most of its variance is explained [7], [8], [9]. In particular for spectral images, PCA projects the spectral data using the eigenvectors associated with the m greatest eigenvalues of the covariance matrix Σ, resulting of the signal F =[f 1 , ··· , f n ] ∈ R l×n , where l is the number of spectral bands, n the number of spatial pixels, and f i ∈ R l is a pixel, for i =1, ··· ,n. Thus, a matrix W m ∈ R l×m , with the m eigenvectors as columns, is used to project the data as ˜ F = W T m F, with ˜ F ∈ R m×n and m<l. This formulation has shown to achieve a small error in the Euclidean sense, described by ||F − W m ˜ F||, while preserving the structure of the data in the low-dimensional space, and thus the direction of greatest variability [10]. In a similar way, the noiseless CSI sensing procedure can be expressed as Y = Q T F, where Q ∈ R l×m is the sensing matrix and F is the input image [7], [11]. CSI can be categorized as a dimensionality reduction technique, since it projects the spectral signal in a low-dimensional subspace spanned by the rows of the sensing matrix Q. Note however that, Q is either randomly generated or designed based on, the restricted isometry property (RIP) or its incoherence with a representation basis [12], [13], [14], [15]. In other words, Q does not rely, conventionally, on the input signal. Therefore, much effort has been done in the signal processing community to design Q such that, the structure of the data is preserved in the low-dimensional subspace. Figure 1 shows an example of how PCA better preserves the direction of greatest variability of the data compared to random matrices. In this figure a dataset in R 3 is projected onto a R 2 subspace using the eigenvectors associated with the covariance matrix of the data and compared against a conventional random matrix. Note that the PCA projection better preserves the data separability in the R 2 subspace whereas the random projection mixes them all. Remark that, if the CSI sensing matrix Q is equal to the PCA matrix W m , the compression or low-dimensional projec- tion, can be considered optimal in the least squares sense [16]. However, PCA is data-dependent, which requires to know the spectral image to be compressed beforehand, thus prohibiting its usage in CSI, where the target data are unknown a priori. Nevertheless, important information about the spectral data This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIP.2019.2959737 Copyright (c) 2019 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.