AN ANALYSIS OF ALIASING AND IMAGE RESTORATION PERFORMANCE FOR DIGITAL IMAGING SYSTEMS Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Master of Science in Electrical Engineering By Iman Namroud UNIVERSITY OF DAYTON Dayton, Ohio May, 2014 AN ANALYSIS OF ALIASING AND IMAGE RESTORATION PERFORMANCE FOR DIGITAL IMAGING SYSTEMS Name: Namroud, Iman APPROVED BY: Russell C. Hardie, Ph.D. John S. Loomis, Ph.D. Advisor Committee Chairman Committee Member Professor, Department of Electrical Professor, Department of Electrical and Computer Engineering and Computer Engineering Eric J. Balster, Ph.D. Committee Member Assistant Professor, Department of Electrical and Computer Engineering John G. Weber, Ph.D. Tony E. Saliba, Ph.D. Associate Dean Dean, School of Engineering School of Engineering & Wilke Distinguished Professor ii c Copyright by Iman Namroud All rights reserved 2014 ABSTRACT AN ANALYSIS OF ALIASING AND IMAGE RESTORATION PERFORMANCE FOR DIGITAL IMAGING SYSTEMS Name: Namroud, Iman University of Dayton Advisor: Dr. Russell C. Hardie It is desirable to obtain a high image quality when designing an imaging system. The design depends on many factors such as the optics, the pitch, and the cost. The effort to enhance one aspect of the image may reduce the chances of enhancing another one, due to some tradeoffs. There is no imaging system capable of producing an ideal image, since that the system itself presents distortion in the image. When designing an imaging system, some tradeoffs favor aliasing, such as the desire for a wide field of view (FOV) and a high signal to noise ratio (SNR). The reason is that aliasing is less disturbing visually if compared against the noise and blur. Some previous research attempted to define the best combination of the optics and pitch that would result in the best image quality that can be achieved practically. However, those studies may have not considered that the post processing can be conducted inside the imaging system. In this work, we reinspect the optics of the imaging system by taking the post image processing into account. Among the optics, we are more concerned about the aspect of the f-number. Varying the f-number iii controls the aperture and the focal length, which affect the number of passing light photons, the width of FOV, and the speed of the shutter. Optimizing the f-number would impact the amount of noise, blur, and undersampling contained in an image. To simulate the post processing, various restoration methods are used. The restoration methods are the adaptive Wiener filter (AWF), Wiener filter, lanczos, and the bicubic interpolation. We mainly focus on the AWF and its performance, since it is a super resolution (SR) algorithm that is designed to restore images that are sampled below the Nyquist rate. Despite the fact that the AWF is a SR algorithm, it was built to expect multiple low resolution (LR) images as an input, and was never used to restore images from only one LR image. So, we employ the AWF as a single frame SR algorithm for the first time, and compare its performance against the other three methods, in order to achieve the best f-number that would introduce the best image quality available. iv To my parents, my husband, and my two children. You enrich my life. v ACKNOWLEDGMENTS I would like to thank all of the professors who taught me during my studies at the University of Dayton. It was pleasure to be in each of their classes and to learn from each of them. I want to thank Russel C. Hardie, Ph.D. for his inspiration, patience, and cooperation. Working under his supervision was very challenging, yet very beneficial for me and my career. I was able to present this work because of his instructions and guidance. Finally, I thank my parents, my husband, and my entire family for their ongoing sup- port. vi TABLE OF CONTENTS ABSTRACT . iii DEDICATION . .v ACKNOWLEDGMENTS . vi LIST OF FIGURES . ix LIST OF TABLES . xi I. INTRODUCTION . .1 II. OBSERVATION MODEL . .5 2.1 The Point Spread Function . .5 2.2 The Discrete Model . .9 2.3 The Impact of the F-Number on the Degradation Process . 10 2.4 The Ratio of λf=p ............................. 14 III. ADAPTIVE WIENER FILTER . 17 3.1 The AWF Algorithm . 17 IV. EXPERIMENTAL RESULTS . 22 4.1 Degradation Results . 24 4.2 Restoration Comparison . 24 4.3 Desired and Restored Images Comparison . 30 vii V. CONCLUSION . 43 BIBLIOGRAPHY . 45 viii LIST OF FIGURES 2.1 Observation model. .6 2.2 Uniform detector array. .7 2.3 Observation model. 10 2.4 A 3D plot of the impulse invariant PSF models for different f-numbers. (a) PSF for f/4 ; (b) PSF for f/8; (c) PSF for f/12; (d) PSF for f/16. 12 2.5 Undersampling Vs. f-number. 15 3.1 The block diagram of the original AWF algorithm . 18 4.1 Ideal test images. (a) Motocross bikes; (b) Chirp; (c) Aerial ; (d) River. 23 4.2 Degradation example. (a) Desired image ; (b) Image degraded at f/1; (c) Image degraded at f/4.5; (d) Image degraded at f/8; (e) Image degraded at f/11.5;(f) Image degraded at f/15. 25 4.3 The degradation and restoration of motocross bikes at f/2. (a) motocross bikes. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 27 4.4 The degradation and restoration of motocross bikes at f/6. (a) motocross bikes. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 28 4.5 The degradation and restoration of motocross bikes at f/10. (a) motocross bikes. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 29 ix 4.6 The degradation and restoration of the aerial image at f/1. (a) Aerial. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 31 4.7 The degradation and restoration of the aerial image at f/8. (a) Aerial. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 32 4.8 The degradation and restoration of the aerial image at f/15. (a) Aerial. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 33 4.9 The restored images of the motocross bikes image at f/4. (a) restored using bicubic filter; (b); restored using lanczos filter (c) restored using Wiener filter; (d) restored using AWF filter. 34 4.10 The degradation and restoration of the chirp image at f/4. (a) The chirp. (b) degraded ; (c) restored using bicubic filter; (d); restored using lanczos filter (e) restored using Wiener filter; (f) restored using AWF filter. 35 4.11 Error between the desired and the restored images of the aerial image. The four curves are for the filters, bicubic (green), lanczos (magenta), Wiener (red), and AWF (blue) (a) MAE. (b) MSE ; (c) SSIM. 37 4.12 Error between the desired and the restored images of river. The four curves are for the filters, bicubic (green), lanczos (magenta), Wiener (red), and AWF (blue) (a) MAE; (b) MSE ; (c) SSIM. 38 4.13 The mean square error MSE between the desired and the restored images of motocross bikes. The four curves are for the filters, bicubic (green), lanczos (magenta), Wiener (red), and AWF (blue). 40 4.14 The mean square error MSE between the desired and the restored images of the chirp. The four curves are for the filters, bicubic (green), lanczos (magenta), Wiener (red), and AWF (blue). 41 4.15 MSE Error between the desired and the restored images of river. (a) Using scaled noise; (b) Using a constant noise. 42 x LIST OF TABLES 4.1 The values of Q that introduce the minimum MSE for all test images using the bicubic interpolation, lanczos, Wiener, and the AWF methods. 40 xi CHAPTER I INTRODUCTION In imaging systems design, it is preferable to achieve a wide field of view (FOV), faster shutter speed, more depth of view, and high image resolution. Although it depends on the application, this is true in most cases. The optics of the camera and its pitch control such features. The attempt to design the optics to improve one of these qualities might reduce the chances to boost the other. For example, to get better resolution, the number of detectors of the focal plane array (FPA) needs to be increased, and the size of the detectors needs to be decreased, but small detectors would not be able to collect enough light photons. The attempt to optimize all the imager properties can only be physically limited and expensive. All images acquired from any imaging system are not ideal. The images are always susceptible to blur, noise, and undersampling caused by the camera. In order to overcome the distortion, the image can be processed later to restore the information that was lost during the process of image acquisition. There are many restoration methods that are used to process images, such as linear filters. One of the most commonly used linear filters is the Wiener filter. Wiener filter attempts to find an estimate image of a desired one by minimize the mean square error between them. The Wiener filter finds the estimated image by filtering an observed noisy image.