**2. Inverse model solution**

In most cases, the measurement data in DOT reconstruction is derived from the numerical solution of the forward problem. Regularization techniques are used to eliminate the obtrusive inverse problem variables. The measurement technology of optical devices is so limited that their existence cannot be accurately determined from all angles. Instead, it gets the average contact angle data for the phantom. The purpose is to reconstruct the image from known scattering and absorption coefficients, which are assumed to be known. The reconstructed result is obtained by comparing the true value to the measured value. A variety of practical reconstruction algorithms have been developed in tomography to implement the process of reconstructing a 3D object from a projection [16]. These algorithms are mainly based on the mathematics of statistical knowledge of the data acquisition process and the geometry of the data imaging system [17].

### **2.1 Levenburg-Marquardt method**

The inverse problem is used to reconstruct an image in the following ways by estimating the scattering coefficient, absorption coefficient, and optical flux [18, 19]. The noise level is present in the actual measurement; both actual measurement data and actual data are shown here [9]. The following nonlinear equation is used to solve the inverse problem of DOT for *i* = 1,…..,s. Assume you know the total attenuation coefficient.

$$F\_i(\mu\_t, \mu\_s) = \mathcal{M}\_i^\delta, (\mu\_t, \mu\_s) \in \mathcal{D} \tag{1}$$

The inverse problem of DOT is inappropriate and uses regularization techniques to reconstruct the image [8], i.e. the Tikhonov functional feature that is minimized for the coefficient. R (*μS*) is a penalty function for regularization [4]. By analyzing the minimization problem,

$$J(\mu\_s) = \frac{1}{2} \sum\_{i=1}^s \left\| F\_i(\mu\_s) - M\_i \right\|\_{L^2(d\mathcal{K})}^2 + a\mathcal{R}(\mu\_s) \tag{2}$$

*Soft Tissue Image Reconstruction Using Diffuse Optical Tomography DOI: http://dx.doi.org/10.5772/intechopen.102463*

Over the set,

$$Q\_{ad} = \{ \mu\_s \in L^\infty(X) \}\tag{3}$$

$$\mu\_{\mu\_s} \in Q\_{\text{ad}}^{\text{inf}} \; f(\mu\_s) \tag{4}$$

The standard reconstruction method is considered using Eqs. (3) and (4).

#### *2.1.1 Standard reconstruction*

Traditional norm-squared penalties are believed to reduce the following functions,

$$R(\mu\_s) = \frac{1}{2} \left\| \nabla \mu\_s \right\|\_{L^2(\mathcal{X})}^2 \tag{5}$$

$$J(\mu\_s) = \frac{1}{2} \sum\_{i=1}^s \left\| F\_i(\mu\_s) - \mathsf{M}\_i^\delta \right\|\_{L^2(d\mathbf{x})}^2 + \frac{\alpha}{2} \left\| \mu\_s - \mu\_s^\* \right\|\_{L^2(X)}^2 \tag{6}$$

In the inverse problem of DOT, the Levenberg–Marquardt regularization method [20] is used. The forward operator is linearized around the initial estimation for each;

$$F\_i(\mu\_s) = F\_i(\mu\_s^0) \left(\mu\_s - \mu\_s^0\right) + \mathcal{R}(\mu\_s^0; i) \tag{7}$$

where Eq. (7) denotes the Taylor remainder for the linearization around and the Frechet derivative is obtained by substituting the above equation and ignoring the higher-order remainder [13].

$$\inf\_{\mu\_t \in D} \frac{1}{2} \sum\_{i=1}^s \left\| F\_i \left( \mu\_s^0 \right) + F\_i' \left( \mu\_s^0 \right) \left( \mu\_s - \mu\_s^0 \right) - M\_i^\delta \right\|\_{L^2(\text{div})}^2 \tag{8}$$

The Euler equation for discrete problems is

$$\sum\_{i=1}^{s} F\_i' \left(\mu\_s^0\right)^\* \left(F\_i \left(\mu\_s^0\right) + F\_i' \left(\mu\_s - \mu\_s^0\right) - \mathcal{M}\_i^\delta\right) + a \left(\mu\_s - \mu\_s^0\right) = \mathbf{0} \tag{9}$$

Solving this (9) yields the final Equation [16]. That is,

$$\left(\sum\_{i=1}^{s} F\_i'(\mu\_s^0) \,^\*F\_i'(\mu\_s^0) + aI\right) \left(\mu\_s - \mu\_s^0\right) = -\sum\_{i=1}^{s} F\_i'(\mu\_s^0) \,^\*\left(F\_i(\mu\_s^0) - M\_i^6\right) \tag{10}$$

where *I* is the identity matrix to solve the new estimate of *μ<sup>s</sup>* based on the initial guess *μ*<sup>0</sup> *s* .

#### **2.2 Sparsity reconstruction**

Sparsity reconstruction function can be minimized as

$$J(\mu\_s) = \frac{1}{2} \sum\_{i=1}^s \left\| F\_i(\mu\_s) - M\_i^\delta \right\|\_{L^2(\text{div})}^2 + \frac{\alpha}{2} \left\| \mu\_s \right\|\_{l\_1} \tag{11}$$

$$\begin{aligned} \left(\mu\_s^k, d^k\right) &= \operatorname\*{argmin} \frac{1}{2} \sum\_{i=1}^s \left\| \left| F\_i(\mu\_s) - M\_i^k \right| \right\|\_{L^2(\text{div})}^2 + a \|d\|\_{l\_1} + \frac{\beta}{2} \left\| d - \mu\_s - b\_d^{k-1} \right\|\_2^2, \\\ b\_d^k &= b\_d^{k-1} + \mu\_s^k - d^k \end{aligned} \tag{12}$$

$$\mu\_s^k = \operatorname\*{argmin} \frac{1}{2} \sum\_{i=1}^s \left\| F\_i(\mu\_s) - \mathsf{M}\_i^\delta \right\|\_{L^2(\text{div})}^2 + \frac{\beta}{2} \left\| d^{k-1} - \mu\_s - b\_d^{k-1} \right\|\_2^2 \tag{13}$$

$$d^k = \underset{d}{\text{argmin}} \|d\|\_{l\_1} + \frac{\beta}{2} \left\|d - \mu\_s^k - b\_d^{k-1}\right\|\_2^2 \tag{14}$$

$$b\_d^k = b\_d^{k-1} + \mu\_s^k - d^k \tag{15}$$

$$\mu\_s^k = \operatorname\*{argmin}\_{\mu\_s} \frac{1}{2} \sum\_{i=1}^s \left\| \left| F\_i(\mu\_s^{k-1}) + F\_i'(\mu\_s^{k-1}) \left( \mu\_s - \mu\_s^{k-1} \right) - M\_i^\delta \right| \right\|\_2^2 \\ + \frac{\beta}{2} \left\| d^{k-1} - \mu\_s - b\_d^{k-1} \right\|\_2^2. \tag{16}$$

$$\begin{aligned} & \left( \sum\_{i=1}^{s} F\_i' (\mu\_s^{k-1}) \, ^\*F\_i' (\mu\_s^{k-1}) + \beta I \right) (\mu\_s - \mu\_s^{k-1}) \\ &= \beta \left( d^{k-1} - \mu\_s^{k-1} - b\_d^{k-1} \right) + \sum\_{i=1}^{s} F\_i' (\mu\_s^{k-1}) \, ^\* \left( F\_i (\mu\_s^{k-1}) - \mathcal{M}\_i^{\delta} \right) \end{aligned} \tag{17}$$

$$d^k = 
sh{
min}
\left(
\mu\_s^k + b\_d^{k-1}, \frac{a}{\beta}
\right) \tag{18}$$

$$Shrink(\mathbf{x}, t) = \text{sign}(\mathbf{x}) \max \left( |\mathbf{x}| - \mathbf{t}, \mathbf{0} \right) = \begin{cases} \mathbf{x} - t, \mathbf{x} \ge t, \\ \mathbf{0}, |\mathbf{x}| < t, \\ \mathbf{x} + t, \mathbf{x} \le -t \end{cases} \tag{19}$$

The rate of spilt Bregman is highly dependent on the rate of dissolution *F*<sup>0</sup> *<sup>i</sup> μ<sup>s</sup>* ð Þ.

## **3. Levenberg: Marquardt algorithm**

Because the DOT image has poor spatial resolution due to severe ill-posedness, the regularization technique is used in conjunction with reconstruction algorithms to reconstruct the images.

#### **Algorithm**

**Input:** Set the initial estimation *μ*<sup>0</sup> *<sup>s</sup>* ; The regularization parameter α,β. **Output:** Approximate minimizer *μs*. for *k* = 1, ……… .*k* do. for *i* = 1, ……… .*s* do. i. Compute the Frechet derivative *F*<sup>0</sup> *i* (*μ<sup>k</sup> <sup>s</sup>* ), and *F*<sup>0</sup> *i* (*μ<sup>k</sup> s* ) ∗ .

end for.


$$\left(\sum\_{i=1}^{\prime} F\_i^{\prime}(\mu\_s^k)^\* F\_i^{\prime}(\mu\_s^k) + \mathfrak{a}\mathbf{I}\right) \left(\mu\_s - \mu\_s^k\right) = -\sum\_{i=1}^{\prime} F\_i^{\prime}(\mu\_s^k)^\* \left(F\_i^{\prime}(\mu\_s^k) - \mathcal{M}\_i^{\delta}\right)$$

iii. Check the stopping criterion. end for.

### **4. Spilt Bregman algorithm**

**Input:** set the initial guess *μ<sup>s</sup>* **0**;

Regularization parameters *<sup>α</sup> <sup>&</sup>gt;* **<sup>0</sup>**, *<sup>β</sup> <sup>&</sup>gt;* **<sup>0</sup>**; *<sup>d</sup>***<sup>0</sup>** <sup>¼</sup> *<sup>b</sup>***<sup>0</sup>** *<sup>d</sup>* ¼ **0** and margins of error *ε*. **Output:** Outputs an approximate value *<sup>μ</sup><sup>s</sup>* <sup>¼</sup> *<sup>μ</sup><sup>k</sup> s* . While *μ<sup>k</sup> <sup>s</sup>* � *<sup>μ</sup><sup>k</sup>*�**<sup>1</sup>** *<sup>s</sup>* � � � � *> ε* do

i. For each **1** *i s*, calculate (17) to be acquired *μ<sup>k</sup> s* ;

$$\text{iii.} \, d^k = \text{shrink}\left(\mu\_s^k + b\_d^{k-1}, \alpha/\beta\right);$$

iii. Calculate *bk <sup>d</sup>* <sup>¼</sup> *<sup>b</sup><sup>k</sup>*�<sup>1</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>μ</sup><sup>k</sup> <sup>s</sup>* � *dk*

iv. *<sup>μ</sup><sup>s</sup>* <sup>¼</sup> *<sup>μ</sup><sup>k</sup> s*

end while.

The split Bregman method entails locating the Fréchet derivative, which is nothing more than the first order derivative function. The algorithm is built with regularization parameters in mind.
