A quantitative and qualitative performance analysis of compressive spectral imagers

Context: Spectral images (SI) contain spatial-spectral information about a scene arranging in a data cube, which often comprises a significant amount of data. However, traditional (SI) systems acquire data ignoring the high correlation between the measurements and the samples are redundant. Compressive spectral imaging systems compress spectral data in the acquisition step, so it allows reducing redundancy and the data amount. Recently, several spectral imaging systems have become available, providing new functionality for users and opening up the field to a wide array of new applications. For instance, the CASSI, SCSI, SSCS, and HYCA systems are four of the most outstanding systems. Methods: Some review works have provided comprehensive surveys of the available technologies and have shown how the new capabilities of spectral imaging approaches can be utilized. However, selecting a specific architecture requires a quantitative and qualitative comparison of these systems in the same scenarios. Results: This paper analyzes the qualitative and quantitative performance of these four compressive spectral imaging systems to evaluate them in the same scenarios. For that, the architectures are modeled as a system of linear equations; then, image reconstructions are accomplished through the same optimization approach, transmittance, coded aperture, and shot numbers. Conclusion: Results show that the performance of the SSCSI system attains better quality reconstruction in terms of PSNR.

Spectral remote sensing is a task that contains a large volume of data, regularly redundant.This volume of information demands memory, transmission bandwidth, and information recovery.Recently, compressive sampling (CS) theory has been developed to make efficient signal sampling and reduce redundancy by taking advantage of the correlation between measurements (Baraniuk, 2007).
Sin embargo, para la selección de una arquitectura específica se requiere una comparación cuantitativa y cualitativa de estos sistemas en los mismos escenarios.Resultados: En este trabajo se analiza el rendimiento cualitativo y cuantitativo de estos cuatro sistemas de compresión de imágenes espectrales para evaluarlos en los mismos escenarios.Para ello, cada arquitectura se modela como un sistema de ecuaciones lineales y el proceso de reconstrucción de las imágenes se logra con el mismo enfoque de optimización transmitancia, código de apertura y número de proyecciones.Conclusión: Se muestra que el sistema SSCSI alcanza el mejor rendimiento en la reconstrucción con el valor más alto PSNR.Palabras clave: matriz de muestreo, arquitecturas de muestreo compresivo, imágenes espectrales.represent the spatial dimension and L the spectral dimension), allows to recover the signal without distortion (Baraniuk, 2007).The vector f can also be expressed by f �Ψ s s where contains only non-zero elements and is a sparseness operator.The measurements vector is obtained as g=Hf, where is a sensing matrix.By choosing q projections of H and assuming sparseness of f in the ψ domain, the signal f can be recovered from model g ��H � s s (Donoho, 2006).
In recent years, CS was adopted as an SI sampling protocol.The following architectures were selected because they are widely used in CS studies such as Coded Aperture Snapshot Spectral Imaging system (CASSI) (Arguello & Arce, 2010).In this case, the main elements are coded aperture, a dispersive element, and the sensor responsible for capturing the energy of the encoded scene, the coded apertures are matricial arrays composed by translucent optical elements that block or unblock the path of light through the system.Another well-know system is the Spatial Spectral encoded Compressive hyper-Spectral Imaging system (SSC-SI) (Lin, Liu, Wu, & Dai, 2014).In this system the spatial-spectral sampling encoded scheme modulates the RGB bands achieving a spatially changing spectral coding that provides a higher degree of randomness in the measured projections, and a coding mask that encodes each spectral channel independently.On the other hand, a recent architecture is the Snapshot Colored Compressive Spectral Imager system (SCCSI) (Correa, Arguello, & Arce, 2014).In this architecture, an array of optical filters encodes the input source, and then a dispersive element disperses it; both processes are integrated along the spectral range sensitivity of the detector.The last architecture presented in this paper is the Hyper-spectral Coded Aperture system (HYCA) (Martin, Bioucas-Dias, & Plaza, 2015), which is a theoretical architecture based on compression blocks of pixels in the spectral bands according to the compression window size, which allows simultaneous sampling and compression of SI.
This paper analyzes qualitative and quantitative performance of these four compressive spectral imaging systems in the same conditions.For that, the architecture are modeled as a system of linear equations, and then image reconstructions are accomplished with the same optimization approach.This paper is organized as follows: section II shows the formulation of the sampling model for each system, section III presents the description of the reconstruction algorithm; simulation results are presented in section IV.Finally, conclusions are described in section V.

COMPRESSIVE SAMPLING ARCHITECTURES
Compressive sampling architectures can be modeled by the system as g = hf, where H is the sampling matrix that characterizes the system.The parameters of the sampling matrices depend on the specific architecture (CASSI, SSCSI, SCCSI, or HYCA).In this section, the formulation of the sampling model of each system is shown by characterizing sampling matrices, correlation between the spectral and spatial dimensions of the data cube components, coded aperture pattern, and dispersive elements.

Source: own work
Where T(x' ,y') represents the transfer function of the coded aperture, is the impulse response of the optical system, and s 1 λ is the dispersion induced by the dispersive element, assumed as linear dispersion.Compressed measurements are acquired when the field is integrated into the detector (Arguello & Arce, 2010).
Equation (1) can be expressed as a system of linear equation g = H CASSI f where H CASSI represents the CASSI sensing matrix.The matrix H CASSI accounts for the effects of the coded aperture and the dispersive element.Figure 2 shows the CASSI sampling matrix , where q is the number of coded projections (shots) for N = 5, L = 3 and q = 3.The number of measurements is . The diagonal patterns that show in the horizontal direction are the coded aperture pattern, each time with an N unit downward shift, as many times as the quantity of spectral bands.

Spatial-spectral encoded compressive HS imager, SSCSI
The Spatial-Spectral encoded Compressive HS imager (SSCSI) is illustrated in figure 3.This system allows the reconstruction of a high-resolution spectral image from a single sensor, due to the design of the optical camera being based on the sampling of spatial-spectral encoded image, which is given by for three-dimensional hyper-spectral images (Lin et al., 2014).The image is formed by the projection of the hyper-spectral image through the spectral dimensions Ω λ ; it can be expressed as equation ( 2). (2) The coded sensor image can be expressed as equation ( 3). (3) Figure 3 shows the calibration parameter for converting spectral coordinate to spatial coordinate; and b is the distance relation between mask to sensor and between spectral planes to sensor.The equation (2) can be modeled as the matrix system SSCSI g = H SSCSI f where .The number represents the SSCSI sensing matrix.The matrix accounts for the effects of the coding mask that encodes each spectral channel independently.
Figure 4 shows the SSCSI sampling matrix The number of measurements is .The diagonal patterns repeated in the horizontal direction correspond to each coded spectral channel pattern independently, as many times as the quantity of spectral bands.(FPA) by sensors with patterns of optical filters.Furthermore, a dispersive element allows capturing both spatial and spectral information from the spectral image in only one shot.This architecture has an arrangement of optical filters in the detector, with different spectral response, which allows obtaining specific spectral components in each pixel.In this way, the coding scheme is enriched, allowing the reconstruction problem to be better conditioned.The small number of optical elements makes a compact system that generates a considerable cost reduction and helps its implementation (Correa et al., 2014).

Snapshot colored compressive spectral imager, SCCSI
In the SCCSI system, a dispersive element spectrally disperses the multispectral image is spectrally dispersed by, generating the field given by equation ( 4).(4) Where is the dispersion and h is the impulse response of the system, f 1 is encoded by the optical filters array located in the FPA detector, to obtain a coded and dispersed version of the original signal, expressed as equation ( 5). (5) A quantitative and qualitative performance analysis of compressive spectral imagers [ 60 ] Where the system output � � ��� �� is obtained by integrating product of f 1 .The equation ( 5) can be expressed as g = H SCCSI f where H SCCSI represents the SCCSI sensing matrix.The matrix H SCCSI is determined by the spectral response of the optical filter matrix structure and represents the proper operation of dispersive elements.
Figure 6 shows the sensing matrix of the SCC-SI system , for N = 5, L = 3 and q = 3.The number of measurements is The diagonal patterns that repeat in the horizontal direction correspond to spectral response of the optical filter matrix structure pattern every time with a unit N downward shift, as many times as the quantity of spectral bands.

Hyper-spectral coded aperture, HYCA
The methodology based on Hyper-spectral Coded Aperture (HYCA) relies on two characteristics of hyper-spectral images: first, the low dimensional space where the hyper-spectral vector belongs; second, the high correlation between the spectral and spatial dimensions of the data cube components.The vectored data cube is defined as , and the measurement vector is modeled as equation ( 5). ( 6) Where is an operator that calculates M inner products between the hyper-spectral vector and f, and w represents the noise level.As H is a linear operator, then H(f) can be rewritten as H HYCA f where is the matrix modeling the linear operator H.
The matrix H is constructed as equation ( 7). ( Where is the block diagonals, and is the matrix that measures each pixel along the spectral dimension (Martin et al., 2015).In contrast to other architectures, HYCA measurement process selects sub-compression matrices using Gaussian or Bernoulli i.e. random projections as illustrated in figure 7, where the data cube is split into a non-overlapping square window ws x ws, and ws is the window size.Figure 8 shows the sensing matrix of the HYCA system , for N = 5, L = 3 and q = 3.The number of measurements is .The diagonal patterns that repeat correspond to equation ( 7).

Coded aperture and transmittance
The coded aperture and transmittance define the information that the architecture has on each scene.Each one of the architectures has their self-definition of these elements.The coded aperture is an element that encodes the light before it is integrated with the detector.The amount of light that passes is given by the transmittance, as illustrated in figure 9 with different transmittance percentages.Transmittance can be written as equation ( 8).

Summary of architectures
Table 1 shows the characteristics of the different architectures, such as the compression, matrix H and the way each element is coded.Furthermore, the architectures with the best compression rate are SSCSI and, HYCA.However, HYCA is not a real architecture, but a theoretical method.

Image reconstruction
The reconstruction is based on responding a linear undetermined system of equations from estimating f with a minimization function, which is expressed as equation ( 9).
Where the parameter τ is regularization constant and H can be the mentioned sensing matrices in table 1.The reconstruction requires using algorithms to converge quickly such problems.For example the GPSR algorithm (Figueiredo, Nowak, & Wright, 2007); the TWIST algorithm, which uses an implicit dispersed dictionary as total variation to solve the optimization problem (Bioucas-Dias & Figueiredo, 2007); the SPARSA algorithm, designed to solve optimization problems on a large scale, and it involves the sum of a term of soft error and a regularizing term correlated to the GPSR that is not limited to the regularization l 1 (Wright, Nowak, & Figueiredo, 2009).

METHODOLOGY
The simulations for this paper were developed on two test datasets and several experiments with measurements obtained from datasets with different compression rates from 12,50% to 52,73%, in order to analyze the reconstruction performance of each architecture.We used the Indian Pines and Salinas datasets from AVIRIS sensor for the simulations (Indiana's, 1992).Indian Pines dataset had 145 x 145 pixels, 220 spectral bands ranging from 0.2 to 2.4 mm with a spatial resolution of 20 m.Salinas dataset had 512 x 217 pixels, 224 spectral bands, and spectral resolution of 3.7 m. (Grupo de Inteligencia Computacional de la Universidad del País Vasco (UPV/EHU), 2014).We used 8 spectral bands with 128x128 pixels from each dataset.The reconstruction algorithm used was GPSR (Wright et al., 2009) with up to 200 iterations; the transmittance in the random matrices was 50%, and shots up to q = 4.

RESULTS
For simulations, sensing matrices depicted in ).Additionally, in all architectures the higher the number of shots, the higher the PSNR of reconstructed images.Figure 10 and figure 11 show the Indian Pines and Salinas data cube reconstructions for CASSI, SSCSI, SCCSI and HYCA architectures, with their respective PSNR.In addition, the best reconstruction was 35,06 [dB], performed with a 25% compression of the SSCSI data measurements.ends at 10% below; HYCA spectral signature has intensity value at 80% above and finishes at 10% below.This can be generalized for both pixels in both images.

CONCLUSIONS
In this work, we presented a coded aperture pattern, data compression, peak signal-to-noise ratio (PSNR) and dispersive elements analysis of CASSI, [ 66 ] SSCSI, SCCSI, and HYCA compressive sampling systems.In order to compare qualitatively, the sampling matrices of each architectures were depicted.The main difference between the architectures is the way to compress the image; CASSI spatially encodes each spectral band equally, and the measurements are the integration on the detector of a coded and shifted version of the spectral bands.On the contrary, SCCSI codes in a space-spectral way, so coding each band is different.The SSCSI measurements are multiplexed without a shift; and HYCA is a theoretical compression method that poses the spectral attenuation of pixels multiplexed without a shift in the spectral bands.
The construction of the measurements affects the quality of the reconstruction in each architecture, and this is proved in the results of the simulations, in which the same parameters are set up while comparing the different architectures.SSCSI reaches the highest PSNR value (35.06 [dB]) with the 25 % of data compression.

FerleyFigure 1 .
Figure 1.Optics CASSI architecture show scene capture until coding element aperture to create a block or unblock each coded spectral channel.

Figure 2 .Figure 3 .
Figure 2. Sampling matrix H CASSI , N = 5, L = 3 and q = 3 and coded aperture pattern with downward shift for each number of spectral bands Source: own work

Figure 5 Figure 4 .
Figure 5 Optics SCCSI architecture shows that the Snapshot Colored Compressive Spectral Imager (SCCSI) replaces the traditional focal plane array

Figure 5 .
Figure 5. Optics SCCSI architecture show scene capture until coding element aperture to create each coded spectral channel by optical filters array.Source: own work.

Figure 6 .
Figure 6.Sampling matrix H SCCSI , N = 5, L = 3, q = 3 and coded aperture pattern with downward shift for each number of spectral bands.Source: own work.

Figure 7 .Figure 8 .
Figure 7. Strategy to take measures hyper-spectral data cube, (a) is the way as the data cube is Split into a nonoverlapping square windows ws x ws, (b) code aperture the matrix that measures each pixel along the spectral dimension.Source: own work

Table 1 .
Features of the compressive sampling architectures Source: own work.

Table 2 .
Mean reconstruction PSNR in [dB] compressive sampling architectures