# Spatial Domain Terahertz Image Reconstruction Based on Dual Sparsity Constraints

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Spatial Domain Signal Model of THz-TDS System

**x**has m × n pixels. Let N = m × n, and randomly select the M positions from the N pixels for terahertz detection; then, a complete terahertz time domain waveform data will be obtained at each detection position in THz-TDS system. Select the peak value of each time domain waveform as the pixel value of the corresponding detection position; then, the sparse terahertz imaging system model in the spatial domain can be expressed as

**y**is the M dimensional terahertz measured data.

**x**is the sample image, and its m × n elements are arranged in an N dimensional column vector.

**R**is an M × N measurement matrix with only one element equal to 1 and the others are equal to 0 in each row, and the positions of the elements with the value of 1 are determined by the detection positions.

**x**from the sparse measured data

**y**. As the spectral density of an ordinary terahertz image is usually distributed in a low-frequency band, representing strong sparsity, the terahertz imaging can be transformed into the problem of sparse signal reconstruction, and the CS-based method can be used for terahertz image reconstruction [23]. Among the various frequency domain transform methods, wavelet transform has good spatial and frequency characteristics, and it is often used as the sparse transform for image reconstruction. By utilizing the sparsity of the terahertz image in the wavelet transform domain, the terahertz image can be reconstructed by

## 3. The Proposed DSC-THz Imaging Method

#### 3.1. Proposed DSC-THz Model

**x**) denotes the shift invariant wavelet transform of the image

**x**, $\mathsf{\Psi}$ denotes the shift invariant wavelet transform matrix, and

**r**is the shift invariant wavelet coefficient.

**x**, $\mathsf{\Psi}$ denotes the shift invariant wavelet transform matrix, and the wavelet coefficients are normalized here. a is a constant greater than 1, which is set to 10 in this paper. Figure 1 gives a simple example to compare the wavelet coefficients obtained by traditional wavelet transform and the proposed method. Figure 1a is a one-dimensional signal taken from a line of the two-dimensional terahertz image. Figure 1b,c are the wavelet coefficients obtained by traditional wavelet transform and the proposed method, respectively. From Figure 1, it is clear that the proposed method could further enhance the sparsity of the wavelet coefficients.

**x**represents gradient operation on the image

**x**. ${T}_{x}$ and ${T}_{y}$ denote the gradient operators on the horizontal and vertical directions, respectively.

**x**, it is very difficult to solve the optimization problem (11) involving multiple l

_{1}-norm terms.

#### 3.2. The Proposed Algorithm

**x**subproblem

**r**

_{e},

**d**

_{x},

**d**

_{y},

**b**

_{r},

**b**

_{x}and

**b**

_{y}, the optimization function of the

**x**is derived by splitting (12)

**x**from the l

_{1}portion of the optimization problem (12), the subproblem (16) that we must solve for

**x**is now differentiable, and optimality conditions for

**x**are easily derived. By differentiating with respect to

**x**and setting the result equal to zero, we get the update rule

**r**

_{e}subproblem

**x**,

**d**

_{x},

**d**

_{y},

**b**

_{r},

**b**

_{x}and

**b**

_{y}, the optimization function of the

**r**

_{e}is derived by splitting (12)

**d**

_{x}subproblem

**x**,

**r**

_{e},

**d**

_{y},

**b**

_{r},

**b**

_{x}and

**b**

_{y}, the optimization function of the

**d**

_{x}is derived by splitting (12)

**d**

_{y}subproblem

**x**,

**r**

_{e},

**d**

_{x},

**b**

_{r},

**b**

_{x}and

**b**

_{y}, the optimization function of the

**d**

_{y}is derived by splitting (12)

Algorithm 1. Proposed DSC-THz Imaging Algorithm |

Input:measurement y, measurement matrix R, exponential shift invariant wavelet basis W_{e}, horizontal gradient operator ${T}_{x}$, vertical gradient operator ${T}_{y}$. Initialization:${\mathit{x}}^{0}={\mathit{R}}^{-1}\mathit{y}$,${\mathit{r}}_{e}^{0}={\mathit{d}}_{x}^{0}={\mathit{d}}_{y}^{0}={\mathit{b}}_{r}^{0}={\mathit{b}}_{x}^{0}={\mathit{b}}_{y}^{0}=0$,$\mu $,$\lambda $,$\gamma $ Loop: set $i=0$ and repeat until ($||{\mathit{x}}^{i+1}-{\mathit{x}}^{i}|{|}_{2}<\delta $) ${\mathit{x}}^{i+1}={\left(\mu {\mathbf{R}}^{\mathrm{T}}\mathbf{R}+\gamma {T}_{x}^{T}{T}_{x}+\gamma {T}_{y}^{T}{T}_{y}+\lambda \mathbf{I}\right)}^{-1}{\mathit{z}}^{i}$ ${\mathit{r}}_{e}^{i+1}=\mathrm{shrink}\left({W}_{e}{\mathit{x}}^{i+1}+{\mathit{b}}_{r}^{i},1/\lambda \right)$ ${\mathit{d}}_{x}^{i+1}=\mathrm{shrink}\left({T}_{x}{\mathit{x}}^{i+1}+{\mathit{b}}_{x}^{i},1/\gamma \right)$ ${\mathit{d}}_{y}^{i+1}=\mathrm{shrink}\left({T}_{y}{\mathit{x}}^{i+1}+{\mathit{b}}_{y}^{i},1/\gamma \right)$ ${\mathit{b}}_{r}^{i+1}={\mathit{b}}_{r}^{i}+({W}_{e}{\mathit{x}}^{i+1}-{\mathit{r}}_{e}^{i+1})$ ${\mathit{b}}_{x}^{i+1}={\mathit{b}}_{x}^{i}+({T}_{x}{\mathit{x}}^{i+1}-{\mathit{d}}_{x}^{i+1})$ ${\mathit{b}}_{y}^{i+1}={\mathit{b}}_{y}^{i}+({T}_{y}{\mathit{x}}^{i+1}-{\mathit{d}}_{y}^{i+1})$ $i=i+1$ End loop Output: reconstructed terahertz image x. |

#### 3.3. Convergence Analysis

**Theorem**

**1.**

## 4. Experiments and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem**

**1.**

_{2}-norm on both sides of (A8), we obtain

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**Figure 1.**Comparison of the wavelet coefficients obtained by traditional wavelet transform and proposed method. (

**a**) Original one-dimensional signal; (

**b**) wavelet coefficients obtained by traditional wavelet transform; (

**c**) wavelet coefficients obtained by proposed method.

**Figure 4.**The full scan terahertz images of the two samples. (

**a**) Circular solids made from wheat flour; (

**b**) wheat seed.

**Figure 5.**Reconstructed results comparation of DSC-THz and SSC-THz with different sampling rates for the circular solids. (

**a**) DSC-THz with sampling rate 10%; (

**b**) DSC-THz with sampling rate 30%; (

**c**) SSC-THz with sampling rate 10%; (

**d**) SSC-THz with sampling rate 30%.

**Figure 7.**Reconstructed results comparation of DSC-THz and SSC-THz with different sampling rates for the wheat seed. (

**a**) DSC-THz with sampling rate 20%; (

**b**) DSC-THz with sampling rate 30%; (

**c**) DSC-THz with sampling rate 40%; (

**d**) SSC-THz with sampling rate 20%; (

**e**) SSC-THz with sampling rate 30%; (

**f**) SSC-THz with sampling rate 40%.

**Figure 8.**Reconstructed results comparation of DSC-THz and SSC-THz with different sampling rates at the selected regions. (

**a**) DSC-THz with sampling rate 20%; (

**b**) DSC-THz with sampling rate 30%; (

**c**) DSC-THz with sampling rate 40%; (

**d**) SSC-THz with sampling rate 20%; (

**e**) SSC-THz with sampling rate 30%; (

**f**) SSC-THz with sampling rate 40%.

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Ren, X.; Jiang, Y.
Spatial Domain Terahertz Image Reconstruction Based on Dual Sparsity Constraints. *Sensors* **2021**, *21*, 4116.
https://doi.org/10.3390/s21124116

**AMA Style**

Ren X, Jiang Y.
Spatial Domain Terahertz Image Reconstruction Based on Dual Sparsity Constraints. *Sensors*. 2021; 21(12):4116.
https://doi.org/10.3390/s21124116

**Chicago/Turabian Style**

Ren, Xiaozhen, and Yuying Jiang.
2021. "Spatial Domain Terahertz Image Reconstruction Based on Dual Sparsity Constraints" *Sensors* 21, no. 12: 4116.
https://doi.org/10.3390/s21124116