Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00003833/00001
## Material Information- Title:
- Implementation of a trous algorithm using lifting for the image processing
- Creator:
- Patil, Kunal B
- Publication Date:
- 2010
- Language:
- English
- Physical Description:
- vii, 94 leaves : illustrations ; 28 cm
## Subjects- Subjects / Keywords:
- Image processing -- Digital techniques ( lcsh )
Digital images ( lcsh ) Electronic noise ( lcsh ) Data compression (Computer science) ( lcsh ) Algorithms ( lcsh ) Wavelets (Mathematics) ( lcsh ) Algorithms ( fast ) Data compression (Computer science) ( fast ) Digital images ( fast ) Electronic noise ( fast ) Image processing -- Digital techniques ( fast ) Wavelets (Mathematics) ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 92-94).
- General Note:
- Department of Electrical Engineering
- Statement of Responsibility:
- by Kunal B. Patil.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- |Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 656396335 ( OCLC )
ocn656396335 - Classification:
- LD1193.E54 2010m P37 ( lcc )
## Auraria Membership |

Full Text |

IMPLEMENTATION OF A TROUS ALGORITHM USING
LIFTING FOR THE IMAGE PROCESSING by Kunal B. Patil A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2010 This thesis for the Master of Science Degree by Kunal B. Patil Dr. Hamid Fardi o4/'^ Date Patil, Kunal B. (M.S. Electrical Engineering) Implementation of A Trous Algorithm using Lifting for the Image Processing Under the guidance of Dr. Jan Bialasiewicz Abstract Wavelet transforms have become useful tools for various image processing applications. Primarily, in this study we are dealing with the image denoising and compression. In this study, lifting wavelet transform is used for the processing of real images. We have implemented lifting wavelet transform along with the two algorithms, called a trous algorithm, and conjugate gradient algorithm. In a trous algorithm, unlike other algorithms, instead of down sampling the filtered image during decomposition, dilation of the filter is done by creating holes (trous is French). Only horizontal and vertical details along with the corresponding low frequency approximations are generated from this algorithm. Further, the a trous algorithm is used, within an iterative conjugate gradient technique that uses multiscale edge detection to reconstruct an image. Signed ACKNOWLEDGEMENT I am heartily thankful to advisor, Dr.Jan Bialasiewicz, whose guidance from the initial to the final level enabled me to develop an understanding of the subject. Above all and the most needed, he provided me unflinching encouragement and support in various ways. I am thankful to him for introducing me to the world of wavelets and opening up new opportunities in my professional career. I would also like to thank my parents. They were always supporting me and encouraging me with their best wishes. Lastly, I offer my regards and blessings to all of those who supported me in any respect during the completion of the project. TABLE OF CONTENTS Figures........................................................vii Chapter 1. Wavelet Transforms............................................1 2. The Discrete Wavelet Transform................................4 3. Condition for Perfect Reconstruction..........................7 4. Lifting......................................................11 5. A trous Algorithm............................................41 6. Image Denoising..............................................50 7. Image Reconstruction using Modulus Maxima....................54 8. Conclusion...................................................61 Appendix A. Image Analysis using Lifting Wavelet Transform.............62 B. Implementation of A Trous Algorithm using Lifting...........65 C. Image Denoising.............................................70 D. Modulus Maxima Mask Generation using A Trous Algorithm......74 E. Reconstruction of the Image using the Conjugate Gradient Algorithm 80 V F. Modulus Maxima............................................82 G. Generation of the Initial Search Direction Used in the Conjugate Gradient Algorithm...........................................87 H. Conjugate Gradient Program................................89 I. Program For Decomposing the Search Direction Used in the Conjugate Gradient Algorithm.................................90 References...................................................92 VI LIST OF FIGURES Figure 2.1 Filter Bank Implementations of the Wavelet Transform.......11 Figure 2.2. Filter Bank Implementations of the Inverse Wavelet Transform..................................................12 Figure 3.1 Two channel Analysis and Synthesis..........................14 Figure 4.1. Lifting Steps, Split, Predict (P), and Update (U)..........19 Figure 4.2: DWT in the z-representation................................23 Figure 4.3. Level 1 Approximation and Details..........................39 Figure 4.4. Level 2 Approximation and Details..........................40 Figure 4.5. Level 3 Approximation and Details..........................40 Figure 5.1 .Decomposition Filter Bank for the A trous Algorithm.......43 Figure .5.2. Reconstruction Filter Bank for the A trous Algorithm......44 Figure 5.3. Scaling Function...........................................48 Figure 5.4.Wavelet Function............................................48 Figure 5.5.Three level Image Decomposition: Approximations, Horizontal, and Vertical Details .......................................49 Figure 6.1 .Scaling and Wavelet functions of the Symlet filters used in the Denoising operations........................................52 Figure 6.2. Image Denoising: Original Image, Noisy Image, Denoised Image (From Left to Right)..................................53 Figure 7.1: Three Level Modulus Maxima Matrix of Image.................58 Figure 7.2: Original Image.............................................60 Figure 7.3: Reconstructed Image........................................60 vii 1. Wavelet Transforms Signal analysis has traditionally been the domain of Fourier transforms based techniques. Fourier analysis suffers from the fundamental lack of time localization, and therefore is not suited to the analysis of signals containing localized features such as transients, and edges. The Fourier transform's utility lies in its ability to analyze a signal in the time domain for its frequency content. The transform works by first translating a function in the time domain into a function in the frequency domain. The signal can then be analyzed for its frequency content because the Fourier coefficients of the transformed function represent the contribution of each sine and cosine function at each frequency. An inverse Fourier transforms data from the frequency domain into the time domain. For image analysis, the Fourier transform identifies the spectral components present in an image but it does not provide information as to where certain components occur within the image. If we are only interested in stationary signals, the Fourier transform provides complete spectral analysis because all spectral components exist at all times. However, if we are interested in non-stationary signals, signals with transient phenomena, the Fourier transform would only give us the l spectral components within an image but not there location [3]. The wavelet transform solves this problem by using a variable length window called wavelet. The wavelet transform or wavelet analysis is probably the most recent solution to overcome the shortcomings of the Fourier transform. Wavelet analysis can provide both time and frequency localization, which led to efficient representations of the signal. Over the last decade, wavelets have found applications in numerous areas of mathematics, engineering, computer science, statistics, physics, etc. [1]. Wavelet transform provides high time resolution, and low frequency resolution for high frequencies (low scales); and low time resolution, and high frequency resolution for low frequencies (high scales). The wavelet transform itself offers great design flexibility. It uses two functions -scaling and wavelet functions for the transform. The wavelet function is equivalent to a high pass filter and produces the high frequency components (details) of the signal at its output. The scaling function is equivalent to a low pass filter and passes the low frequency components (approximations) of the signal. The wavelet coefficients are retained and represent the details in the signal. The scaling coefficients are decomposed further using another set of low 2 pass and high pass filters. Fast implementation of wavelet transforms using a filter-bank framework enables real time processing capability. 3 2. The Discrete Wavelet Transform The Discrete Wavelet Transform (DWT) provides sufficient information for both analysis and synthesis of the original signal, with a significant reduction in the computation time and resources. The DWT, where the signal is decomposed into so-called approximations & details, can be realized by the multi-rate filter bank. Approximations represent low frequency signal content, and details correspond to high frequency components of the processed signal. Each set of the components of a signal can decompose further into other levels of approximations & details. The DWT represents a signal in terms of shifts, and dilations of a low- pass scaling function,
is multiscale, in that it creates a set of coarse coefficients that |