Diffuse Optical Tomography, Inverse Problems, and Optimization. Mary Katherine Huffman. Undergraduate Research Fall 2011 Spring 2012

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1 Diffuse Optical Tomography, Inverse Problems, and Optimization Mary Katherine Huffman Undergraduate Research Fall 11 Spring Introduction. This paper discusses research conducted in order to investigate improvements to an optimization problem regarding diffuse optical tomography, an imaging technique in which near-infrared light illuminates tissue. The scattering and absorption of the light is recorded and studied, and this light allows for a picture of the inner characteristics of the tissue to be formed [2]. Specifically, the optimization problem arises after a parametric level sets-based regularization method adequately models an anomaly and its background. This research examines possible improvements regarding decreasing the amount of parameters, generating a good initial guess as to the location of the anomaly, and introducing a second optimization phase after a first has been completed. 2. Background. Before describing the analysis and results of the research, this paper will discuss three background concepts. Section 2.1 will address diffuse optical tomography and its beginnings. Section 2.2 will then explain the nature of inverse problems and illustrate the parametric level sets method as a representation of an anomaly and a medium using a modest number of parameters. Finally, 2.3 will discuss the resulting optimization problem. 2.1 Diffuse Optical Tomography. As previously stated, diffuse optical tomography (DOT) is a technique used to illuminate tissue and develop a model to reconstruct a picture of the inner characteristics. From [2], sources and detectors are placed around the tissue, and this model is created based on the measurements that are calculated. Currently, diffuse optical tomography is being applied to the detection of tumors in the brain and the breast. Before DOT was used, photon migration imaging was explored, and to calculate photon measurements, either illumination by pulses of light, continuous wave illumination, or radio-frequency amplitude modulated illumination was employed. Other pre-dot applications include pulse oximetry and near-infrared spectroscopy. Different choices of layouts for the sources and detectors have been proposed. Light sources, times, and frequencies can deviate; also, the use of virtual sources can be considered. Sources and detectors may be located on opposite sides of the tissue, gathered on the same side of the tissue, or placed in rings around the tissue. The illumination is typically either continuous wave or radio-frequency amplitude modulated illumination. The concept of virtual sources addresses the idea of an optical source inside the tissue; however, since the implementation of sources inside human beings is not always practical, frequencies are used instead. Ultimately, to retrieve the wanted information from DOT, an inverse problem must be solved to recover the desired parameters. Recent research explores parametric level set methods for the solving of inverse problems. 2.2 Parametric Level Sets. Inverse problems are abundant in many fields of science and engineering, including DOT [1]. Generally, a measure is taken from some medium, and the inverse problem serves as a means of finding a physical model to explain the observations.

2 Many times in real-world applications, the inverse problem is ill-conditioned, meaning that it is sensitive to noise; thus, retrieving true, accurate information is often extremely difficult. To deal with the ill-conditioned inverse problem, regularization is used, which stabilizes the reconstruction. The goal of solving the inverse problem is to identify characteristics about a particular region in a medium. As a way of approximating the region of interest along with the adjacent regions in the medium, the parametric level sets (PaLS) method has been introduced. For inverse problems regarding the retrieving of the geometry of a region, the foreground and background are viewed as separate cases, and the boundary between the two is represented as the level set of a function. For images involving DOT, we are interested in the background tissue and the anomaly located inside of the region. The PaLS method assumes that the characteristics for the scattering and absorption of light are significantly different in the anomaly and the background. The level set function is defined as where c ϵ R and is a parameter vector. If the function value is greater than c, we consider that region inside the anomaly. If the function value is equal to c, the function is on the boundary between the anomaly and the background, and if the function value is less than c, the function is outside of the anomaly. We assume the unknown property p is defined as Above, denotes the property values inside of the domain, and denotes the values outside of the domain. H is the Heaviside function, defined as H is approximated by a smooth Heaviside function,. The PaLS method represents the level set function as a summation of radial basis functions, defined as.. where α is the parameter vector of expansion coefficients, β is the vector of dilation coefficients, χ denotes the vector of the centers of the radial basis functions, and ψ denotes a smooth radial basis function. The norm utilized is an approximated Euclidean norm, which we define as where v is a small number. This parameterization regularizes the problem, and now the reconstruction of the image is an optimization problem.

3 2.3 Optimization. Specifically, the solution to an inverse problem is based on a measurement u of the region of interest and is defined by a model M, a function of the property p of the region, and noise η [3]. We represent u as where u is the actual measurement, and p, we solve for the residual operator, defined as is the assumed model. In order to approximate For ease, we will refer to any function as. The function then to minimize is To solve this minimization problem, iterative methods such as the Gauss-Newton (GN) method, Levenberg-Marquardt (LM) method, or TREGS method can be used [3]. The GN method approximates the function as.. where is the current approximate solution and is the ill-conditioned Jacobian of the residual operator. The LM method replaces in the GN method with where λ > 0 and controls the step size. TREGS uses the ideas of singular value decomposition and a trust region approach to approximate more accurate updates. For further detail regarding these methods, refer to [3]. 3. Analysis Overview. Current MATLAB code utilizes the PaLS and TREGS methods to reconstruct an image for a DOT model. The region of interest is constructed in a grid with both the x and y axes ranging from 0 to. The anomaly is within this region. The code assumes distinct property values for both the anomaly and the background and begins the process using an initialized guess of the anomaly location based on the radial basis functions. Then, µ is updated until the norm of the function is below an inputted tolerance. My research aimed to improve the optimization method in several ways. Section 3.1 will discuss holding certain parameter values constant and thus inhibiting the optimization to change them. Section 3.2 examines the results of using only a single radial basis function to define the parametric level sets function to locate the anomaly. Once the method produced a result using this single function, based on the reconstruction I strategically placed radial basis functions around the specified region and began a second optimization. I refer to this process as reseeding. Finally, section 3.3 will discuss three other reseeding options using more than one radial basis function for the first phase of optimization. 3.1 Constant Parameters. Recall that the parameters of the parametric level sets function include an α vector, β vector, and coordinates of the centers of the radial basis functions. By removing one of these vectors from the function, the optimization updates µ with respect to the remaining parameters. Thus, instead of optimizing with minor updates to both the dilation and expansion coefficients, the method makes larger updates to only one vector. Running the

4 optimization method once with a constant parameter vector did not always produce a better reconstruction, as seen in Figure 1 below. Figure 1. The actual image (A), the reconstruction after the original optimization (B), the reconstruction after the optimization with a constant β vector A. B. C. 3.2 A Single Radial Basis Locating Approach. In order to cheaply locate the anomaly in the region, I experimented with reconstructing the image in a first optimization phase with only one radial basis function. The theory was that the optimization would update this single function until the general area of the anomaly was located; then, in a second optimization phase, I would reseed the region with more strategically placed radial basis functions based on the first optimization of the single function. I began the optimization with a single function in the middle of the region, giving the radial basis function a center of (2., 2.). Figure 2 provides an example. The single radial basis function was able to locate the anomaly when only one anomaly existed. After this first reconstruction, the code updated five different radial basis functions. These functions were given centers corresponding to the final update of the first reconstruction. The x and y coordinates of the centers were a combination of the x and y coordinates of this final update and those values plus or minus 0.. Let (x*, y*) denote the center of the final update of the first function. Then, the second phase included five functions with centers (x*, y*+0.), (x*, y* 0.), (x*,y*), (x*+0., y*), and (x* 0., y*). Figure 2. The actual image (A), the single radial basis function (B), the first reconstruction with the single function (C), the second reconstruction with five functions (D) A. B. C. D. While the single function was able to locate the anomaly and approximate its size, the second optimization was able to capture the shape to a greater extent. However, if more than one anomaly was present in the region, the single radial basis function was not able to find the distinct locations. Instead, the optimization updated the function to ultimately locate only one anomaly or match the background. Figures 3 and 4 demonstrate these findings respectively.

Figure 3. The actual image (A), the reconstructed image (B). 4 x -3 9 3 8. 1 2 1 8 7. 2 2 7 3 3 6. 6 4 4. 0 0 0.00 0 0 A. B. Figure 4. The actual image (A), the reconstructed image (B). x -3 4 8 1 3 2 1 7.

Additionally, I examined three options regarding reseeding the method with radial basis functions after a first optimization phase using more than one function.

5 Figure 3. The actual image (A), the reconstructed image (B). 4 x A. B. Figure 4. The actual image (A), the reconstructed image (B). x A. B. 3.3 Other Reseeding Methods. Additionally, I examined three options regarding reseeding the method with radial basis functions after a first optimization phase using more than one function. All examples begin with the same original image, and the first phase of the optimization utilizes fifteen radial basis functions distributed uniformly throughout the grid region. One option included using the centers of the final update of the radial basis functions after the first reconstruction as the centers of the functions for the second reconstruction. As shown in Figure, using radial basis functions already known to be close to the anomaly provided a more accurate reconstruction. The second reconstruction results in less background noise. Figure. The actual image (A), the reconstruction after a first optimization phase (B), the reconstruction after using the centers of the final update of the first phase functions as the centers for the radial basis functions of the second phase A. B. C.

Another option used the centers of the final update of the radial basis functions after the first reconstruction that also had positive expansion coefficients as the centers of the functions for the

6 Another option used the centers of the final update of the radial basis functions after the first reconstruction that also had positive expansion coefficients as the centers of the functions for the second reconstruction. These results also showed much improvement, as demonstrated in Figure 6. The higher accuracy could be due to the fact that the functions with positive expansion coefficients are the functions that are seen in the reconstructed image. Figure 6. The actual image (A), the reconstruction after a first optimization phase (B), the reconstruction after using the centers of the final update of the first phase functions with positive expansion coefficients as the centers for the radial basis functions of the second phase A. B. C. The third option used the centers of the final update of the radial basis functions after the first reconstruction with dilation coefficients greater than 0.9 as the centers of the functions for the second reconstruction. For the image used, this option appeared to be the most accurate. A possible explanation could be that since the functions with a dilation coefficient of a certain value or higher result from the optimization of the first reconstruction, these functions were clearly those most utilized in the first reimaging. Figure 7 provides the graphic results. Figure 7. The actual image (A), the reconstruction after a first optimization phase (B), the reconstruction after using the centers of the final update of the first phase functions with dilation coefficients greater than 0.9 as the centers for the radial basis functions of the second phase A. B. C. 4. Conclusions. After reviewing the results of the experiments, this research suggests that in general, a second optimization phase with reseeded radial basis functions produces a better reconstruction than a single phase of optimization. The three options for the reseeding procedure examined were (1) using the centers of the final updates of the functions from the first phase of optimization as the beginning centers of the functions for the second phase, (2) using the centers

7 of the final updates of the functions from the first phase of optimization with positive expansion coefficients as the beginning centers of the functions for the second phase, and (3) using the centers of the final updates of the functions from the first phase of optimization with dilation coefficients greater than 0.9 as the beginning centers of the functions for the second phase. In all three instances, a more accurate reconstruction does in fact appear to make sense. In (1), the first optimization phase produces an accurate reconstruction with radial basis functions spread uniformly throughout the region; therefore, the second phase should naturally produce accurate results when the radial basis functions are located more specifically around the anomaly. In (2), the reseeding process utilizes the updated functions of the first phase with positive expansion coefficients. This accurate image results from the fact that the functions with positive expansion coefficients are the functions that are above a specific level and thus contribute much to what is seen of the image. In (3), I postulate that using the functions with dilation coefficients greater than 0.9 created a more accurate image because this value would indicate that the corresponding function was utilized to a greater extent in the first optimization phase. The dilation coefficients are all initialized at 0.8. Therefore, with a final value of 0.9, these functions were beneficial to the reconstructed image and would be of greater use than the others. The method of using a single radial basis function is advantageous when only one anomaly exists in the region of interest. The single function accurately captures the location and relative size of the anomaly; however, when more than one anomaly exists, the single function is optimized to match the background or locates only one anomaly. I would not recommend this method of location in human applications unless it is known that only one anomaly is in existence. Regarding keeping a parameter vector constant, it appears that this method is not as accurate as utilizing all parameters. If only one parameter vector is available, this procedure can provide an approximate location for the anomalies. Unfortunately, the sizes and shapes of the anomalies are not as accurately displayed as with other methods. Finally, further research aims to investigate improving the optimization by reducing the Jacobian matrix to exclude certain columns. This method would eliminate the columns with norms below a tolerance determined by the matrix norm. The reduced matrix would in turn produce a more useful singular value decomposition for solving ill-conditioned matrix equations.

8 Works Cited [1] Aghasi, A., Kilmer, M., & Miller, E. L. (11). Parametric Level Set Methods for Inverse Problems. SIAM J. Imaging Sciences, [2] Boas, D. A., Brooks, D. H., Miller, E. L., DiMarzio, C. A., Kilmer, M., Gaudette, R. J., et al. (01). Imaging the Body with Diffuse Optical Tomography. IEEE Signal Processing Magazine, [3] de Sturler, E. & Kilmer, M. E. (11). A Regularized Gauss-Newton Trust Region Approach to Imaging in Diffuse Optical Tomography. SIAM J. Scientific Computing, 7-86.

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