Edge Detection and Enriched Subspaces Pubblico

Abstract

We describe an image reconstruction algorithm that reconstructs the image using approximate image basis obtained from the image segmentation algorithm. Unlike similar approaches which reconstruct the image directly, our algorithm highly improves the accuracy of reconstruction in most cases especially with only a limited range of X-ray projection angles. We first use the image segmentation method to detect the edge of the object, then use the segmentation result as the approximate image basis to reconstruct the image. Moreover, our algorithm can be applied to different kinds of objects, such as a single object, separated objects and overlay objects, and the accuracy of the reconstruction image is always improved.

Table of Contents

1 Introduction 1

2 Background 4

2.1 Computational Diculties . . . . . . . . . . . . . . . . . . . . 5

2.2 Regularization Methods . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Truncated SVD . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Tikhonov Regularization . . . . . . . . . . . . . . . . . 9

2.2.3 Modied Tikhonov Regularization . . . . . . . . . . . . 11

3 Image Reconstruction 13

3.1 Math Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Reconstruction Example . . . . . . . . . . . . . . . . . . . . . 16

4 Image Segmentation 19

4.1 Segmentation Example . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Segmentation Method . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Step 1: Forward Model . . . . . . . . . . . . . . . . . . 23

1

4.2.2 Step 2: Attenuation Update . . . . . . . . . . . . . . . 24

4.2.3 Step 3: Curve Update . . . . . . . . . . . . . . . . . . 25

4.3 Comparison of Reconstruction and Segmentation . . . . . . . 28

5 Enriched Krylov Subspace Methods 30

5.1 Krylov Subspace Method . . . . . . . . . . . . . . . . . . . . . 30

5.2 Enriched Subspace . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Alternative Approach - Modied CGLS . . . . . . . . . . . . . 35

6 Numerical Experiments 40

6.1 Segmentation Experiments . . . . . . . . . . . . . . . . . . . . 40

6.1.1 Single Object . . . . . . . . . . . . . . . . . . . . . . . 41

6.1.2 Separated Objects . . . . . . . . . . . . . . . . . . . . 47

6.2 Reconstruction Experiment . . . . . . . . . . . . . . . . . . . 52

6.2.1 Single Object . . . . . . . . . . . . . . . . . . . . . . . 52

6.2.2 Separated Object . . . . . . . . . . . . . . . . . . . . . 63

6.2.3 Overlay Object . . . . . . . . . . . . . . . . . . . . . . 72

7 Concluding Remarks 82

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Parola chiave	X-ray tomography image reconstruction regularization inverse problem iterative method image segmentation
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Victoria Powers, Emory University Bree Ettinger, Emory University Davide Fossati, Emory University Yuanzhe Xi, Emory University

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