# Compressive Sensing Hyperspectral Imaging by Spectral Multiplexing with Liquid Crystal

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spectral Modulation for CS with LC

## 3. Reconstruction Process

#### Dictionary for Sparse Representation

**S**:

**f**to its sparse representation, $f={\mathsf{\Psi}}_{d}\mathsf{\alpha}$. Each column of ${\mathsf{\Psi}}_{d}$ is referred to as an atom of the dictionary. Therefore, the spectrum

**f**can be viewed as a linear combination of atoms in ${\mathsf{\Psi}}_{d}$ according to weights in $\mathsf{\alpha}$. Based on Equation (7), a corresponding system dictionary ${\mathsf{\Omega}}_{d}\in {\Re}^{M\times {N}_{d}}$ is created by the inner products of the spectral dictionary with the CS-MUSI sensing matrix, $\mathsf{\Phi}$:

## 4. Compressive Hyperspectral and Ultra-Spectral Imaging

#### 4.1. Camera Calibration

#### 4.2. Staring Mode

#### 4.3. Scanning Mode

## 5. 4D Imaging

## 6. Target Detection

**x**is the pixel signature,

**t**is the target spectral signature and

**m**is the estimated background. $\Gamma $ is the covariance matrix, which holds the statistics of the background and can be approximated using:

**x**’ is a pixel that contains the target and p is the ratio of the target present in the pixel.

## 7. Discussion

## 8. Patents

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spectral multiplexing. The figure represents three different examples of spectral multiplexing. Each sub-figure illustrates multiplexing of a few spectral bands onto a FPA.

**Figure 2.**LC cell phase retarder. The LC phase retarder is made of a Nematic LC layer (blue arrow) sandwiched between two glass plates and two linear polarizers (green layers). The glass plates are coated with Indium Tin Oxide (ITO, pink layers) and a polymer alignment layer (purple layers).

**Figure 3.**Measured spectral responses (intensity transmission vs. wavelength in nm) of the fabricated LC phase retarder. Each graph represents the spectral modulation with a different voltage applied on the LC cell (15 different voltages).

**Figure 4.**(

**a**) CS-MUSI acquisition process. (

**b**) CS-MUSI optical scheme diagram. The HS object $F(x,y,\lambda )$ is modulated according to ${\varphi}_{\mathrm{LC}}\left(\lambda ,{V}_{i}\right)$, yielding the multiplexed measurement ${G}_{i}(x,y)$.

**Figure 6.**CS-MUSI spectral response map for voltages from 0 V to 10 V (

**left map**) and a zoom in on the area where the voltages are from 1.3 V to 3.5 V (

**right map**).

**Figure 7.**Staring mode reconstruction result of three LED arrays. (

**a**) RGB color image of three LED arrays that were used as objects to be imaged with CS-MUSI. (

**b**–

**e**) Representative single exposure images for LC cell voltage of 0 V, 5.8373 V, 7.6301 V and 8.6552 V, respectively. (

**f**) RGB representation of the reconstructed HS image (700 × 700 pixels× 391 bands). (

**g**–

**i**) Reconstructed images at 460 nm, 520 nm and 650 nm, respectively. (

**k**–

**m**) Spectrum reconstruction for three points in the HS datacube and comparison to the measured spectra of the three respective LEDs with a commercial grating-based spectrometer.

**Figure 8.**Staring mode reconstruction result of six different markers. (

**a**) RGB representation of the reconstructed HS image (800 × 900 pixels × 1171 bands). (

**b**) Four reconstructed images at four different wavelengths (470 nm, 530 nm, 580 nm, and 630 nm).

**Figure 9.**Staring mode reconstruction results with the dictionary of (

**a**) outdoor and (

**b**) indoor HS images taken with CS-MUSI camera. The figures show RGB representation of the reconstructed HS datacube.

**Figure 10.**CS-MUSI camera along-track scanning. Each shot of the CS-MUSI camera, ${G}_{i}$, captures a shifted scene with a different LC spectral transmission (which depends on the voltage ${v}_{i}$).

**Figure 11.**Scanning mode (Figure 10) reconstruction result. (

**a**) RGB color image of three LED arrays. (

**b**–

**e**) representative single exposure images (frame #30, #90, #150 and #300, respectively) and (

**f**–

**i**) the RGB representation of the reconstructed HS image up to the appropriate column.

**Figure 13.**(

**a**) 4D Spectro-Volumetric imaging. (

**b**) Grayscale representation of HS images at three different depths (225 cm, 254 cm and 270 cm).

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## Share and Cite

**MDPI and ACS Style**

Oiknine, Y.; August, I.; Farber, V.; Gedalin, D.; Stern, A.
Compressive Sensing Hyperspectral Imaging by Spectral Multiplexing with Liquid Crystal. *J. Imaging* **2019**, *5*, 3.
https://doi.org/10.3390/jimaging5010003

**AMA Style**

Oiknine Y, August I, Farber V, Gedalin D, Stern A.
Compressive Sensing Hyperspectral Imaging by Spectral Multiplexing with Liquid Crystal. *Journal of Imaging*. 2019; 5(1):3.
https://doi.org/10.3390/jimaging5010003

**Chicago/Turabian Style**

Oiknine, Yaniv, Isaac August, Vladimir Farber, Daniel Gedalin, and Adrian Stern.
2019. "Compressive Sensing Hyperspectral Imaging by Spectral Multiplexing with Liquid Crystal" *Journal of Imaging* 5, no. 1: 3.
https://doi.org/10.3390/jimaging5010003