DOI: http://dx.doi.org/10.15446/ing.investig.v36n2.56426

Probability of correct reconstruction in compressive spectral imaging

Probabilidad de reconstrucción exitosa en el sensado compreso de imágenes espectrales

Samuel E. Pinilla¹, Héctor M. Vargas², and H. Arguello³

¹ System Engineer, Masters in Math, Universidad Industrial de Santander, Colom­bia. Affiliation: Department of Computer Science, Universidad Industrial de Santander, Bucaramanga, Colombia. E-mail: samuel.pinilla@correo.uis.edu.co.
² Electronic Engineer, Masters in System Engineering, Universidad Industrial de Santander, Colombia. Affiliation: Department of Computer Science, Universidad Industrial de Santander, Bucaramanga, Colombia. E-mail: hector.vargas@correo.uis.edu.co.
³ Electrical Engineer, Masters in Electrical Power, Universidad Industrial de Santander. Ph.D in Electrical and Computer Engineering, University of Delaware, USA. Affiliation: Department of Computer Science, Universidad Industrial de Santander, Bucaramanga, Colombia. E-mail: henarfu@uis.edu.co.

How to cite: Pinilla, E., Vargas, M., & Arguello, H. (2016). Probability of correct reconstruction in compressive spectral imaging. Ingeniería e Investigación, 36(2), 68 - 77. DOI: 10.15446/ing.investig.v36n2.56426.

ABSTRACT

Keywords: Restricted Isometry Property, RIP, CASSI, compressive sensing, spectral imaging, coded aperture.

RESUMEN

El sistema de adquisición de imágenes espectrales de única captura basado en apertura codificada (CASSI), capta información tridimensional (3D) espacio-espectral de una escena, usando un conjunto de medidas bidimensionales (2D) proyectadas en un FPA (Focal Plane Array). Para recuperar el cubo de datos a partir de las proyecciones en el FPA, se usa un algoritmo de reconstrucción basado en la teoría de muestreo compresivo. En CASSI la calidad de la reconstrucción de imágenes espectrales depende exclusivamente de la matriz de sensado, que es determinada por la estructura estadística del código de apertura. La propiedad restringida isométrica (RIP) de la matriz de sensado CASSI es usada para determinar la probabilidad de una correcta reconstrucción de la imagen. Este artículo describe la arquitectura óptima CASSI y desarrolla la RIP para las matrices de muestreo, para la captura de la información del cubo de datos. En efecto, la RIP provee la guía para determinar el mínimo número de capturas FPA necesarias para la reconstrucción de una imagen. Más adelante, la RIP es usada para encontrar la estructura óptima de las proyecciones de los códigos de apertura de CASSI. Las simulaciones muestran alta calidad de la reconstrucción obtenida de las imágenes espectrales cuando se satisface la condición impuesta por la RIP. También muestran el más alto rendimiento obtenido de las estructuras óptimas de las proyecciones CASSI.

Palabras clave: Propiedad Isométrica Restrictiva, RIP, CASSI, muestreo compresivo, imágenes hiperespectrales, códigos de apertura.

Received: April 23rd 2016 Accepted: July 18th 2016

Introduction

Imaging spectroscopy requires sensing a large amount of spatial information across a multitude of wavelengths. The sensed signals are titled multispectral or hyperspectral images. Traditional imaging spectroscopy sensing techniques scan adjacent zones of the underlying spectral scene and merge the results to construct a spectral 3-Dimensional (3D) data cube. Push broom spectral imaging sensors, for instance, capture a spectral data cube with one FPA measurement per spatial line of the scene (Gehm et a/., 2008). Spectrometers based on optical band-pass filters spectrally scan the source data cube (Gupta, 2008). These sensing techniques obey the well-known Nyquist criterion, which imposes a severe limit on the number of samples needed. More specifically, these methods require scanning a number of zones linearly in proportion to the desired spatial or spectral resolution. As the desired resolution increases, the required number of samples grows considerably, such that the cost of sensing a hyperspectral image is extremely high. Recently, with a mathematical technique for reconstructing sparse signals, called Compressive Sensing (CS), one can use fewer samples much lesser then predicted by the Nyquist criterion. This new technique involves diverse mathematical areas such as numerical optimization, signal processing, random matrix analysis, and statistics. The enormous potential of CS has been recently applied in areas such as microscopy, holography, tomography and spectroscopy (Arce et a/., 2014; Brady et a/., 2009; Studer et al., 2012; Wagadarikar et a/., 2008; Yu and Wang, 2009). CS allows sensing a signal with a fewer number of samples than that required by the Nyquist criterion. Thus, CS has allowed overcoming a diverse number of physical sensing limits. This paper focuses on the application of CS in spectral imaging which is coined Compressive Spectral Imaging (CSI). CSI senses 2D coded random projections of the underlying scene, such that the number of required projections is far less than the linear scanning case. The Coded Aperture Snapshot Spectral Imaging (CASSI) system is an imaging architecture that effectively implements CSI. Thus, CASSI senses the 3D spectral information of a scene using 2D random projections. Figure 1 illustrates the CASSI optical architecture. The main components in CASSI are the coded aperture, the prism and the Focal Plane Array (FPA). The coded aperture patterns are the only varying elements in CASSI, the other optical elements remain fixed during the operation of the instrument.

Figure 2 shows the main components of the CASSI architecture. In CASSI, the coding is applied to the image source density f₀(x, y, λ,) by the coded aperture T(x, y), where (x, y) are the spatial coordinates. The resulting coded density f₁(x,y, λ,) = T(x,y) f₀(x,y, λ,) is spectrally dispersed by a dispersive element before it impinges in the focal plane array (FPA), resulting in the coded field

where T(x, y) is the transmission function representing the coded aperture, h(x - S(λ) - x, y' - y) is the optical impulse response of the system, and S(λ) is the dispersion induced by the prism. A linear dispersion by the prism is assumed. The compressive measurements across the FPA are obtained by the integration of the field f₂(x,y,λ) over the detector spectral range. The discretized FPA measurements can be thus modeled as

= floor is the intensi­ty measured at the (j,) position of the FPA with is the discretized data cube, is the discretized coded aperture with two-dimensional structure, and (ω_jt is the noise system.

The single-shot CASSI system has been extended to a multi-shot system architecture such that multi-frame coded measurements are acquired as separate FPA measurements, each with a distinct coded-aperture pattern (Kittle et a/., 2010).

Equation (2) can be written in matrix form as y = Hf+ω, where y is the measurements vector, H is the matrix that represents the coded apertures and prism effect and f is the vectorized spectral data cube. Given the compressive measurements y, the objective of CS is to recover the signal f. Compressive Sensing (CS) dictates that an n-long signal f can be recovered from m « n random projections y = H Ψ θ. Moreover, if it defines the m x n matrix A = Hψ as the sensing matrix, then the spectral data cube f = Ψθ is sparse in the base Ψ. The RIP in CASSI is thus fundamental to determine both the minimum number of shots needed for correct reconstruction and for the design and optimization of the coded aperture patterns (Wagadarikar, 2010). CS structured random matrices have been extensively analyzed for Toeplitz, random partial Fourier, and partial circulant matrices (Fornaiser, 2010). Recently a generalization of structurally random matrices has been developed (Do et a/., 2012). The RIP in CASSI was first considered in Wagadarikar (2010), where it is assumed that the RIP in CASSI is satisfied. The work in Wagadarikar (2010) then derives conditions on the coded apertures so that the RIP is better satisfied. These results, however, do not develop the explicit parameters for the bounds needed in the RIP, such as the probability of error, or the minimum number of FPA measurements. In this paper the RIP in CASSI is derived, the structure of the CASSI sensing matrix is formulated, and the RIP constants are expressed as a function of the structure of the random coded aperture patterns. A minimum number of random projections are needed so as to satisfy the RIP of the sensing matrix A (Candes and Tao, 2005). This paper establishes the theoretical probability of correct recovering in CSI. The results of this work can be used in the optimization of coded aperture patterns and to estimate a bound for the minimum number of shots needed for correct reconstruction.

CASSI Sensing Matrix

The goal in this section is to present the discrete model of the optical process in CASSI system in matrix form. Moreover, using this model allows carrying out the RIP analysis on the CASSI sensing matrix. The discretized output at the detector corresponding to the i^th coded aperture is given by

for i = 1,...,K, where is the intensity at the position at the detector, whose dimensions are N x N+L -1, F represents an N x N x L spectral data cube and ω_j is the white noise of the sensing system. In this work, the entries of the coded aperture are limited to . The varying terms in Equation (3), indexed by i are: first, the discrete coded apertures T¹ or its vector representation t ¹ whose entries are realizations of a Bernoulli random variable with parameter and second, the number of shots K. (See Appendix A).

where is a V-long vector representation of the measurement . in Equation (3), where represents the coded aperture and the dispersive element where n = N²L, is the vector representation of the data cube F, and ω is the noise of the system. More specifically, the entries of f_k can be expressed as

where . The vectorization of the code aperture T ⁱ is defined as

where . Similarly, the vectorization of the output Y ⁱ is defined as

where . Using the above matrix representation, the output y_t can be expressed as

The matrix H_i can be expressed as (Arguello and Arce, 2012)

where 0_R is a N² x N(L- 1) zero-valued matrix, the structure of the Vx V permutation matrix Θ_v is

The n x n matrix Θ_n has the same structure of Θ_v in Equation (10) and where I is an N² x N² identity matrix, are N- times the usual matricial product, and 0 is a N² x N²(L - 1) zero valued matrix.

The ensemble of CASSI outputs can be rewritten as

where the CASSI sensing matrix is A = HΨ, the matrix , and θ is a sparse representation of f in the base Ψ. Notice that where m = KV, and m ≤ n. Figure 3 and Figure 4 depict an example of the matrices H and A respectively for small values of N, L, and K. In practice, the dimensions of these matrices are much larger (Arguello and Arce, 2011). Figure 4 uses where Ψ_2D is the 2D Symmlet Wavelet Transform.

Given the measurements y, the signal f is estimated by solving

where τ is the regularization constant. Several algorithms have been developed to solve this l₂ - l₁} optimization. In this paper the Gradient Projection for Sparse Reconstruction (GPSR) algorithm is used (Figueiredo et al., 2007).

The RIP in CASSI

The sth restricted isometry property (RIP) δ_s =δ_s (A) of the matrix is defined as the smallest δ such that

for all s-sparse vectors Θ. Notice that equivalently Equation(13) is given by

where A_S^H H is the conjugate transpose of A_s and I_d ∈ℝ^SxS is the identity matrix. Since the matrix A_S^TA_S-I_d is Hermitian, we have from the Equation (15)

Thus, from the definition of the RIP in Equation (13) and Equation (16) we have that the constant δ_s is given by

Defining the matrix A_TT = A_S ^TA_S, Equation (17) can be rewritten as

where λ_max (·) denotes the largest eigenvalue (Fornaiser, 2010). A small value of δ_S in Equation (18) for a large value of S indicates that the RIP condition is satisfied. The RIP condition implies a stable recovery of the signal Θ from the projections AΘ using a l1 optimization algorithm (Fornaiser, 2010). Let the entries of Ψ be Ψ_j,k, then using the structure of the matrices Hi in Equation (9), the entries of A_S can be written as

for j=0, ..., S-1 and j ? k. When j=k then (A_TT)_jk = K. Without loss of generality, we analyze the case when Ψ = diag (Ψ_2D, ... , Ψ_2D) and Ψ 2D is the 2D Symmlet Wavelet Transform. Given that Ψ is a block diagonal matrix, the sums over u and r in Equation (20) have only one nonzero element, then Equation (20) can be expressed as

where are realizations of a Bernoulli random variable with parameter p^* = p² + (1- p)² . Notice that is a Binomial random variable. More specifically, (x)_ℓ -- B (K, p*). Notice also that with Ω_j and Ω_k varying randomly, it can be considered a random variable with E ((y)_ℓ) = 0. Then, Equation (21) can be rewritten as

and (A_TT )_jk = K for j = k. Because (A_TT)_jk for j ? k is the sum of i.i.d zero mean random variables, and applying the central limit theorem when V → ∞ , then (A_TT)_jk → N (0,σ²) where σ² = V Var ((x)_ℓ (y)_ℓ). A demonstration of the applicability of the central limit theorem in Equation (22) can be found in Do et al. (2012). Figure 5 depicts the matrix A_TT for k = 2, N = 8, and S = 180. It can be seen in this figure that the matrix A_TT is dense and its entries can be seen as a Gaussian random variable. Given that x and y in Equation (22) are independent random variables then V_xy = Var((x)_ℓ (y)_ℓ) can be estimated as

Since (x)_ℓ is a Binomial random variable, then Var((x)_ℓ)= 4Kp* (1- p*) and (E((x)))= K(2p* -1). Given the specific structure of Ψ_2D, the variance Var ((y)_ℓ)= 1/N⁴. The variance V_xy in Equation (23) can be rewritten as

Given that the entries of the matrices A_TT are Gaussian, then it is possible to apply the concentration inequalities developed by Ledoux to calculate the distribution of λ_max in Equation (17) (Candes and Tao, 2005; Ledoux, 2001). More specifically, for each fixed t > 0 the concentration inequality is given by

where σ_max indicates the maximum singular value and o(1) is a small term tending to zero as K → ∞ which can be estimated as Using Equation (25) and following a similar approach as that of Candes in (Candes and Tao, 2005), the RIP in CASSI can be written as

where and . Using Equation (26), the constant δ_s is found as .

Simulations

In order to verify the RIP in CASSI in Equation (26), an algorithm is proposed based on Monte Carlo Uniform Sampling. Also, two spectral data cubes are used to simulate the CASSI real system reconstructions. All experiments have been carried out in Matlab 2010b on a 3GHz Intel Core i7 and 32GB memory desktop computer.

To estimate numerically the restricted isometry constant δ_s of order S from matrix A, Monte Carlo simulations are computed by sampling the sub-matrices (A^T_S A_S- I_d) from Equation (17).

An important parameter of the CASSI system is the transmittance of the coded apertures. Algorithm 1 was developed in order to determine the relationship between the transmittance of the coded apertures and the RIP constant δ_s. The objective of this section is to determine the transmittance that provides the best value of the constant δ_s.

The steps 1-3 in Algorithm 1 are used to construct the CASSI sensing matrix with parameters p and K. The coded apertures Tⁱ for i ∈{0,...,K -1} are N x N symmetric matrices whose entries {1,-1} follow a Bernoulli distribution with probability p. The CASSI Sensing matrix A is formed by multiplying the CASSI matrix H with the sparsity basis Ψ. Subsequently, an iterative method is implemented to find the maximum eigenvalue of the possible sub-matrices (A_S^T A_s - I_d).

The number of iterations in Algorithm 1 is defined as maxlter=n!/(n-s) !s!. Since the number of all possible combinations of the sub-matrices is very large, the constant RIP δ_s can be determined using Monte Carlo simulations. Therefore the index subspace S is sampled by using a uniform distribution (step 5) with size |S| ≤ s. The eigenvalues in all experiments are stored in the vector C. Finally, the probability of correct reconstruction is calculated from probability distribution of C by determining the number of eigenvalues smaller than 1. In this work, the parameters used in Algorithm 1 were N = 16, L = 8 , maxlter = 1 000 000 and Ψ = Ψ_2D⊗ I_L, where Ψ_2D ∈ℝN² x N² is the 2D-Wavelet Symmlet 8 basis and ℝL² x L² is the identity matrix.

Figure 6 shows the variation of δ_s for different number of shots K, p, transmittances and S/n = 0,005. The constant δ_s is minimum for P {(t )_ℓ = l} = p = 0,5, or equivalently when the transmittance of the code apertures is 50 %. Notice that a value of δ_s = 1 indicates that reconstruction is not possible.

Similar performance is obtained for other values of S/n. The probability of reconstruction 1 -P_e (see Equation (26)) is also compared to the results provided by a set of Monte-Carlo simulations. The results are shown in Figure 7, where a high probability depends on small sparsity and a higher number of shots.

Experiments with real data

In this section, the performance of CASSI sensing matrix is evaluated in the reconstruction of three spectral images (Foster et al., 2006). An example of the first database of 533 nm is shown in Figure 8a. This is a natural scene with 256 x 256 pixels of spatial resolution and L = 33 spectral bands ranging from 400 nm to 720 nm. An example of the second spectral database at 533 nm is shown in Figure 9a. The second spectral image is a city scene with 256 x 256 pixels of spatial resolution and L = 33 spectral bands ranging from 400 nm to 720 nm. The third database is a color scene with 256 * 256 pixels of spatial resolution and L = 31 spectral bands ranging from 400 nm to 700 nm at 10 nm steps. In In Figure 10a, an example is shown where the spectral band selected is 500 nm. For all the databases, the reconstruction is compared for different sparsity values using Bernoulli distributed coded apertures in CASSI with transmittance 50 %.

The reconstruction is realized using the GPSR algorithm (Figueiredo et al., 2007). The base representation Ψ = diag(Ψ_2D,..., Ψ_2D) where Ψ_2D is the 2D-Wavelet Symmlet 8 basis. The data cube F was approximate for different values of the sparsity S/n by removing the n - S less significant coefficients in magnitude from the sparse representation Θ. The performance of matrices is analyzed under different sparsity levels. Figure 8d, Figure 9d and Figure10d indicate the performance (PSNR) analysis as a function of the sparsity and number of shots.

The results of 100 reconstructions experiments given K and S/n are condensed in Figure 8d and Figure 9d (with S/n = 0,065 in the first image and S/n = 0,125 in the second and third images), where the PSNR of the reconstruction images verifies that the performance (PSNR) increases as the sparsity and the number of shots K increases.

Minimum number of shots for reconstruction

While the Nyquist Shannon sampling theorem states that a certain minimum number of samples is required for sampling the spectral data cube, compressive sensing can reduce the number of measurements that need to be stored by the prior knowledge that spectral signal is sparse in a known basis and that sensing matrix satisfied the RIP. Figure 8d, Figure 9d and Figure 10d indicate the respective RIP constant 3 for correct reconstruction. The results indicate s clearly that there are a minimum number of shots for correct reconstruction as a function of the sparsity for a noiseless case. The reconstruction of the band depicted in Figure 8a is shown in Figure 8b and Figure 8c for 4 and 9 shots respectively. Figure 8b illustrates the effect of using a number of shots less than that established by the RIP where the PSNR was 25 dBs with sparsity S/n = 0,065. For Figure 8c the PSNR was 31 dBs with sparsity S/n = 0,065. Notice that the number of shots used in Figure 8c is higher than the minimum recommended in Figure 8d. The number of shots used in Figure 8d is similar than the minimum recommended in Figure10d. The artifacts in the reconstructed image in Figure 8b, Figure 9b and Figure10c, with low PSNR, are clearly noticeable. On the contrary, Figures 8c, 9c and 10b show high quality in the reconstructed image. For Figure 10b and 10c the reconstruction performance are 28 and 24 dBs respectively, for the given number of shots 12 and 8.

Conclusions

The structured CASSI sensing matrix has been formulated and its RIP has been demonstrated. Simulations verified the theoretical RIP CASSI bounds. For optimal reconstruction, the coded aperture patterns must be designed with p = 0,5 or equivalent with a transmittance of fifty percent. Given a sparsity condition for signal representation, the minimum number of shots for reconstruction is determined by the variance of the representation basis and the variance of the random structure of the code apertures. An algorithm based on Monte Carlo simulations was developed to determine the optimal transmittance of the coded apertures. The experiments indicate the influence of the sparsity level and the number of shots in the performance of the system.

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Appendix A

Let Γ be a coded aperture such that (F_jt) = 1 for all j,ℓ . Suppose that ^y_Γ is the FPA measurement when the aperture Γ is used, H' be the CASSI matrix associated to the coded Γ and thus y_r = H'f . Let y = be the CASSI output when the coded aperture Tⁱ (j,ℓ) ∈{0,1} is used. Define the FPA measurement y_i as where it can be noticed that . Thus, the binary coded apertures can be implemented by subtracting the reference signal ^y_Γ from boolean output 2. Notice that this type of coded aperture cannot be implemented directly. That means that the implementation of these coded apertures requires one additional FPA CASSI measurement and the noise in the emulated measurements y is 5 times higher than the noise in the original measurements 2.