# A Novel Dictionary-Based Image Reconstruction for Photoacoustic Computed Tomography
## 2 Methods
### The Proposed Method  
We model the procedure of PACT using \(b=Ax\), where \(A\) is the measurement matrix, and \(x\) is the assumed phantom in the forward problem and the reconstructed image in the backward problem. \(b\) is the detected signals by the ultrasound transducers. When the number of unknowns (number of pixels of the PA image) is greater than the number of equations (number of recorded samples), which usually happens in PACT, especially when a low number of view angles is used for decreasing the data acquisition time, we have an underdetermined system. In this case, there is no exact solution. One commonly used method to solve an underdetermined problem is the least square technique, giving \(x_{est}=(A^{T}A)^{(-1)}A^{T}b\). The least square method uses the error minimization to obtain a solution, but the answer is not the best and the most accurate one that can be obtained. The sparse component analysis and sparse representation of signals can be used to solve the underdetermined problem of the PA image reconstruction. Having a prior knowledge, we can promote sparsity using \(l_{0}\), \(l_{1}\), and \(l_{p}\) (\(0<p<1\)) norms to obtain a more accurate solution [30]. In other words, we can improve the image quality by assuming that the imaging target is sparse. Targets in PA imaging are composed of many singularities. A singularity is the point of an exceptional set where it fails to be well-behaved in differentiability. These singularities can be considered as non-zero values of \(x\). Such an assumption in the procedure of the backward problem leads to a more accurate solution in comparison with algorithms that use error minimization. The representation of the backward problem is given in Equation (4). \[\min_{x}J(x)\quad s.t.\quad b=Ax \tag{4}\]  
where \(J(x)\) is the prior knowledge function that is used to promote sparsity. The basis pursuit method uses the \(l_{1}\)-norm to sparsify the problem and transfers it into a linear or quadratic equation (see Equation (5)). \[\min_{x}\frac{1}{2}\|b-Ax\|_{2}^{2}+\lambda\|x\|_{1} \tag{5}\]  
where \(\lambda\) is the scalar regularization parameter, and \(\|.\|_{1}\) and \(\|.\|_{2}\) indicate the \(l_{1}\)-norm and \(l_{2}\)-norm, respectively. The first and second terms of Equation (5) represent the error of estimation and level of sparsity of estimated \(x\), respectively. An effective sparsity method should represent all of the most important features of the PA images. Considering the diagnostically relevant features in a PA image, significant features can be edges, singular points, and homogenous texture. A standard transform function can be used for signal sparsification. A single transformation, however, can well represent only one of the major features. In order to obtain an image having all these features, we propose combining some of the well-known standard transform functions. Although WT's magnitude will not oscillate around singularities, and it uses continuous transform to characterize the oscillations and discontinuities, the images generated by WT are blurred. DCT helps separate the image into spectral sub-bands of differing importance. In addition, it preserves homogeneous textures better than WT. However, it provides some blocking artifacts in the image. The blocking artifact is a distortion that appears as abnormally large pixel blocks. Therefore, WT and DCT cannot capture two-dimensional singularities, i.e., curves and edges in an image. On the other hand, the TV is an operator that works based on the local variations in a signal. It well preserves the edges in the image. However, the artifacts in the initial image introduced by limited view angles significantly affect the performance of TV [37]. As illustrated, all the three mentioned transformation functions have some benefits and disadvantages.