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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
This book dissects a very common but important image processing application: digital image denoising. The dissection begins with the most fundamental of digital images, noise sources, and the effect of noise on different image features. A variety of denoising techniques are reviewed, analyzed, and compared in this book to provide the readers with a complete picture of digital image denoising. The book has placed special emphasis on the theory behind each algorithm and how it can be for real world application. * Excellent explanatory introduction to image denoising methods; * thorough treatment of the theoretical foundations, with comprehensive analytical analysis of each presented algorithm; * discussion in detail - including the assumptions and limitations - of each presented algorithm; * detail devoted to MATLAB® implementation of the presented algorithm; * coverage of classical denoising methods, and advanced model based denoising methods; * simulation results are presented for comparative analysis; * summary of important results obtained in each chapter given at its end; and * theoretical, analytical, and computational exercises are provided at the end of each chapter. MATLAB code, solution manual, and PPT for lecturing will be provided.
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Seitenzahl: 309
Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
Nomenclature
About the Companion Website
Note
1 Digital Image
1.1 Color Image
1.2 Alternate Domain Image Representation
1.3 Digital Imaging in MATLAB
1.4 Current Pixel and Neighboring Pixels
1.5 Digital Image Noise
1.6 Mixed Noise
1.7 Performance Evaluation
1.8 Image Quality Measure
1.9 Structural Similarity
1.10 Brightness Normalization
1.11 Summary
Exercises
Notes
2 Filtering
2.1 Mean Filter
2.2 Wiener Filter
2.3 Transform Thresholding
2.4 Median Filter
2.5 Summary
Exercises
3 Wavelet
3.1 2D Wavelet Transform
3.2 Noise Estimation
3.3 Wavelet Denoise
3.4 Thresholding
3.5 Threshold Value
3.6 Wavelet Wiener
3.7 Cycle Spinning
3.8 Fusion
3.9 Which Wavelets to Use
3.10 Summary
Exercises
4 Rank Minimization
4.1 Singular Value Decomposition (SVD)
4.2 Threshold Denoising Through AWGN Analysis
4.3 Blocked SVD
4.4 The Randomized Algorithm
4.5 Summary
Exercises
5 Variational Method
5.1 Total Variation
5.2 Gradient Descent ROF TV Algorithm
5.3 Staircase Noise Artifacts
5.4 Summary
Exercises
6 NonLocal Means
6.1 NonLocal Means
6.2 Adaptive Window Size
6.3 Summary
Exercises
7 Random Sampling
7.1 Averaging Multiple Copies of Noisy Images
7.2 Missing Pixels and Inpainting
7.3 Singular Value Thresholding Inpainting
7.4 Wavelet Image Fusion
7.5 Summary
Exercises
Appendix A: MATLAB Functions List
References
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Statistical properties of the subband signals of AWGN corrupted
S
...
Table 3.2 Wavelet threshold values variations obtained from universal thres...
List of Illustrations
Chapter 1
Figure 1.1 Illustration of capturing an image by digital camera.
Figure 1.2 Representation of the digital image
Sculpture
: (a) a grayscale p...
Figure 1.3 Three separate RED, GREEN, and BLUE channels are combined to crea...
Figure 1.4 RED, GREEN, and BLUE samples obtained from
Bayer pattern
color fi...
Figure 1.5 Current pixels and its neighborhood.
Figure 1.6 Boundary extension: (a) symmetric extension on image
Sculpture
, (...
Figure 1.7 (a) A total dark image taken by a DSLR and (b) the same photo wit...
Figure 1.8 Additive Gaussian noise‐corrupted (a)
Sculpture
image with zero m...
Figure 1.9 The histogram of the noise variance computed from masks of (a) ...
Figure 1.10 Salt and pepper noise with total noise density being 0.05 over (...
Figure 1.11
Sculpture
image corrupted with (a) AWGN with and then SAP with...
Figure 1.12 Image denoising quality computation.
Figure 1.13 Different regions of interest in
Sculpture
image.
Figure 1.14 The
Sculpture
image corrupted by AWGN: (a) uniformly across the ...
Figure 1.15 Image feature corruption under different noise effects.
Chapter 2
Figure 2.1 (a) Noise‐free step edge image with pixel intensities at 128 and ...
Figure 2.2 The average spectral power at radian frequency of the
Sculpture
Figure 2.3 Ideal lowpass filter: (a) the filter in 2D view in frequency do...
Figure 2.4 Mean filter: (a) plot of a mean filter in frequency domain, (b) t...
Figure 2.5 Effect of mean filter kernel size versus different sources of noi...
Figure 2.6 (a) Spatial frequency response of the Gaussian filter along ‐axi...
Figure 2.7 Effect of the size of a Gaussian filter with identical passband r...
Figure 2.8 Wiener filtering denoised
Sculpture
images of (a) AWGN corrupted ...
Figure 2.9 Denoised images from a AWGN corrupted
Sculpture
image with =50 (...
Figure 2.10 Different shapes of the median filter windows (a) square, (b)
Figure 2.11 Median filter denoised images of (a) SAP corrupted
Sculpture
ima...
Figure 2.12 Median filter with adaptive window size ranging from to deno...
Figure 2.13 Median filter with mask
medneigh
denoised image: (a) SAP corrupt...
Figure 2.14 Median of median filter denoised image: (a) SAP corrupted
Sculpt
...
Chapter 3
Figure 3.1 Illustration of 2D wavelet decomposition of the
Sculpture
image. ...
Figure 3.2 The histogram of subband.
Figure 3.3 Column 400 of the
Sculpture
image: (a) from the noise‐free image,...
Figure 3.4 The three steps involved in the wavelet denoising process.
Figure 3.5 Threshold functions .
Figure 3.6 The MSE of the wavelet thresholding denoised AWGN corrupted
Sculp
...
Figure 3.7 Wavelet thresholding denoised
Sculpture
images corrupted by AWGN ...
Figure 3.8 Wavelet hard threshold denoising result with adaptive threshold f...
Figure 3.9 Denoising result on AWGN corrupted
Sculpture
image with =25: (a)...
Figure 3.10 Boundary extension in the function
cyclespin
: (a) The formation ...
Figure 3.11 The influence of the number of translations (
spinsize
) on the wa...
Figure 3.12 Wavelet hard threshold denoising with scale shrink threshold and...
Figure 3.13 DWT fusion: wavelet based image combining algorithm.
Figure 3.14 Wavelet fusion: (a) one of the input image () applied in wavele...
Chapter 4
Figure 4.1 Squares of the singular values of (a) AWGN corrupted
Sculpture
im...
Figure 4.2 Denoising result obtained by SVD singular value optimal threshold...
Figure 4.3 Denoising result obtained by block SVD by hard thresholding and e...
Figure 4.4 Denoised result obtained by SVD hard thresholding on mixed AWGN w...
Figure 4.5 Denoised image obtained from RSVD (Listing 4.4.2), where the sour...
Figure 4.6 Denoised image obtained from iterative RSVD (Listing 4.4.3), wher...
Figure 4.7 (a) AWGN corrupted noisy image ; (b) symmetrical extension patch...
Chapter 5
Figure 5.1 ROF TV denoised image using MATLAB Listing 5.2.2 with various , ...
Figure 5.2 Staircase noise artifact: (a) noise‐free 1D ramp signal; (b) nois...
Chapter 6
Figure 6.1 Distance function in (a) Gaussian filtering with distance betwe...
Figure 6.2 Pixel averaging operations: (a) examples of pixels and patches; (...
Figure 6.3 NLM filter image denoising results for (a) AWGN corrupted
Sculptu
...
Figure 6.4 Neighboring pixels considered in Gaussian filter and NLM filter: ...
Figure 6.5 Hard thresholding NLM filter image denoising results for (a) AWGN...
Figure 6.6 The PSNR performance of NLM filtering on AWGN corrupted
Sculpture
Figure 6.7 NLM filtering with adaptive patch size image denoising results fo...
Figure 6.8 NLM filtering with adaptive search window size image denoising re...
Chapter 7
Figure 7.1 Noisy
Sculpture
becomes less noisy by down‐ and up‐sampling, wher...
Figure 7.2 (a) One of the 10 AWGN corrupted
Sculpture
images with . (b) The...
Figure 7.3 Singular value thresholding inpainting: (a) AWGN corrupted
Sculpt
...
Figure 7.4 Averaging of multiple denoised image: (a) by averaging images of ...
Figure 7.5 Wavelet image fusion denoised image: (a) by fusing average image ...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
Nomenclature
About the Companion Website
Begin Reading
Appendix A: MATLAB Functions List
References
Index
End User License Agreement
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Digital Image Denoising in MATLAB
Chi-Wah Kok and Wing-Shan TamCanaan Semiconductor Pty LtdAdelaide, Australia
MATLAB® is a trademark of the MathWorks, Inc.. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
This edition first published 2024.© 2024 John Wiley & Sons Ltd
All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Chi‐Wah Kok and Wing‐Shan Tam to be identified as the authors of this work has been asserted in accordance with law.
Registered OfficesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
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In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging‐in‐Publication Data:
Names: Kok, Chi‐Wah, author. | Tam, Wing‐Shan, author.Title: Digital image denoising in MATLAB / Chi‐Wah Kok, Wing‐Shan Tam.Description: First edition. | Hoboken, NJ : Wiley, 2024. | Includes bibliographical references and index.Identifiers: LCCN 2023055005 (print) | LCCN 2023055006 (ebook) | ISBN 9781119617693 (cloth) | ISBN 9781119617754 (adobe pdf) | ISBN 9781119617730 (epub)Subjects: LCSH: MATLAB. | Image processing–Digital techniques.Classification: LCC TA345.5.M42 K65 2024 (print) | LCC TA345.5.M42 (ebook) | DDC 006.6–dc23/eng/20240527LC record available at https://lccn.loc.gov/2023055005LC ebook record available at https://lccn.loc.gov/2023055006
Cover Design: WileyCover Image: © Yuen Wai Lan
To my love, Annie, from Ted, for putting up over and over again
To my grandmother, Shui King, from Shan
About the Authors
Chi‐Wah Kok was born in Hong Kong. He was granted with a Ph.D. degree from the University of Wisconsin Madison. Since 1992, he has been working with various semiconductor companies, research institutions, and universities, which include AT&T Labs Research, Holmdel, SONY U.S. Research Labs, Stanford University, Hong Kong University of Science and Technology, Hong Kong Polytechnic University, City University of Hong Kong, Lattice Semiconductor, etc. He founded Canaan Semiconductor Pty Ltd. in Adelaide, South Australia, a fabless IC company with products in mixed‐signal IC, high‐performance audio amplifier, high‐power MOSFETs, and IGBTs. Dr. Kok embraces new technologies to meet the fast‐changing market requirements. He has extensively applied signal processing techniques to improve the circuit topologies, designs, and fabrication technologies within Canaan. This includes the application of semidefinite programming to circuit design optimization, abstract algebra in switched capacitor circuit topologies, nonlinear optimization method to optimize high‐voltage MOSFET layout and fabrication. He was MPEG (MPEG 4) and JPEG (JPEG 2000) standards committee member. He is an Associate Editor of Digital Signal Processing, Elsevier since 2018, and is the founding Editor‐in‐Chief of the journal Solid State Electronics Letters since 2017. He also is the author of four books by Prentice Hall and Wiley‐IEEE, and has written numerous papers on digital signal processing, multimedia signal processing, and CMOS circuits, devices, fabrication process, and reliability.
Wing‐Shan Tam was born in Hong Kong. She received her Ph.D. degree in Electronic Engineering from the City University of Hong Kong. She has been working in different telecommunication and semiconductor companies since 2004 and is currently the Engineering Manager and co‐founder of Canaan Semiconductor Pty Ltd. in Adelaide, South Australia, where she works on both advance CMOS sensor design, and high‐power device structure and process development. Dr. Tam has participated in professional services actively, in which she has been researcher in different universities since 2007. She has been an invited speaker for different talks and seminars in numerous international conferences and renowned universities. She has served as Guest Editor in several journals published by IEEE and Elsevier. She has been the founding editor of the journal Solid State Electronics Letters since 2017. She is a co‐author of Wiley‐IEEE textbooks, and research papers with award quality. Her research interests include image interpolation algorithm, color enhancement algorithm, mixed‐signal integrated circuit design for data conversion and power management, device fabrication process, and new device structure development.
Preface
This is the second book of an ambitious project on digital image processing using MATLAB. The first book “Digital Image Interpolation” was published in February 2019. Then it is this book, “Digital Image Denoising.” The last one of this project is “Hollywood Image Processing,” which we are looking forward to finish writing it before the end of 2025.
The purpose of this book is to take a step further in digital image processing. Instead of considering the spatial and spectral domain to process an image, as that in the first book, this book discusses functional representation, and optimization in image processing through a practical example, “Image Denoising.” Instead of suppressing undesirable signals created by processing the image artificially, the problem considered in this book is to suppress noise in image captured by modern electronic devices. The analytical tools developed in this book will be applicable to other image processing problems, as well as signal processing problems, such as pattern recognition and communication.
The book starts with discussions on noise removal through filtering, which include frequency domain‐based filtering, such as mean filter and Wiener filter; and spatial domain‐based rank order filtering, such as median filtering. The importance of filtering threshold selection is explained with the application of the general orthogonal transform (which include Fourier transform and Wiener filter, and hence spectral filtering as special cases). Discussions on adaptive window/block size are presented for both spectral and spatial filtering‐based denoising techniques.
The moving window filtering‐based denoising techniques are extended to block transform‐based denoising, and further extended to time‐frequency packet‐based wavelet transform in the wavelet chapter. Cycle spinning technique is applied to improve the robustness of the image denoising algorithm. Important mathematical relation between wavelet space representation and functional representation in Fourier space will be discussed.
The point‐based denoising algorithm is extended to functional optimization problem in the subsequent chapters. The first set of techniques considered in this book are the low‐rank matrix completion, approximation, and optimization methods. The image denoise problem is formulated as a functional optimization. However, when seeking solution to this optimization problem, it has been shown that the optimal solution is given by simple hard thresholding of the singular value. The first functional optimization‐based image denoising method presented in this book is the set of variational image denoising methods. Image denoising can be achieved by variational minimization that mixes a fit to the data and the prior. In this chapter, we shall discuss the Rudin–Osher–Fatemi (ROF) total variation image denoising method, and construct the MATLAB implementation to seek the solution to this problem.
Intuitively, the image processing problem can be formulated as “deterministic” or “probabilistic” problems. Techniques presented in initial chapter have treated the image denoising problem using deterministic signal model. The last two chapters of the book will discuss techniques that make use of the probabilistic signal model. We start with the patch‐based image denoising, the NonLocal mean image denoising method. With limited space and the objective of presenting algorithms that are generally useful across various signal processing field, we have made our choice to discuss patch‐based image denoising with self‐similarity that does not need an added dictionary, nor a prior training.
The last chapter will discuss the application of random sampling to mix and match various denoising algorithms together to achieve better result. It also demonstrates how to mix and match signal processing techniques developed with different models and in different signal spaces to achieve a better denoising result.
All discussions are accompanied with a thorough discussion on MATLAB implementation, where source codes are provided and embedded into the text as part of the discussions, and explanation on the difficult mathematics. A unified set of test images is applied throughout the whole book to allow reader to easily appreciate, compare, and observe the pros and cons of various discussed algorithms. The noise model being considered are additive white Gaussian, and Salt and Pepper (Poisson) noise, which are noises commonly found in modern digital camera photos. Their characteristics, generation, and visual appearance will be presented in the first chapter, where the first chapter will also help to warm up the reader by introducing notations being applied in the book, together with some basic MATLAB programming techniques for image processing. It will also present image quality metric and their development in MATLAB, and other mathematical and MATLAB tools that are required in the latter chapters. Only one image will be used throughout the book to provide consistency and ease of comparison. The chosen image is the “Sculpture” image on the front cover of the book. This is a very well‐crafted sculpture located at the Hong Kong Museum of Art and was captured by Miss. W. L. Yuen who provided us permission to use it in this book. Besides being an excellent photo, this image contains important features that have made explaining image denoising in this book easier. At the same time, we encourage the readers to experience the performance of algorithm presented in this book and also algorithm developed by themselves after reading each chapter with other images of their choice.
Chi‐Wah Kok, Wing‐Shan Tam
January 2024Adelaide, Australia
Acknowledgments
I would like to express my profound gratitude to my wife, Annie. After a long day of work, I find my reward is to have your head lying on my shoulder with the satisfaction of feeling comfortable and safe. I thank my beautiful and intelligent wife, with whom I can share this and so many other things, whose love and support through the years have had an immeasurable impact on my life. I sincerely believe that she deserves much more than what I can express with my words. She is always the beautiful music in my heart.
I am fortunate to see the transformation of Dr. Tam after a very long apprenticeship instead of a clone of myself. Dr. Tam has a transformed mind, a transformed way of thinking, talking, performing, arguing, writing, and even a transformed way of walking. The idea of apprenticeship is to share work, discover new theories together, but with the immense benefit of the master's experience being challenged by the apprentice's fearless questioning. Dr. Tam's sharp and judicious remarks greatly helped me to better describe many of the ideas found in this book. Dr. Tam is always the voice of challenge in my curiosity cabinet of creativity.
Take away from Me the noise of your songs; I will not even listen to the sound of your harps.
– Amos 5:23 (NIV)
Chi‐Wah Kok
The “transformation” is a process to reshape an object or a collection of objects in between different planes or coordinate systems. There are different types of “transformation,” e.g. reflection, rotation, shearing, etc., which are common operations in image processing. Different types of transformation can be applied independently or multiple of them can be invoked simultaneously to achieve the desired effect. No matter what type of transformation that we are talking about, we can formulate a well‐defined mathematical function to describe and to direct such transformation. There would be no ambiguous and unexpected results in the course of the transformation in the mathematical world. However, this might not be the case in the human world.
In human perspective, a transformation is sometimes unexpected, and the outcome is almost not predictable. The transformation requires courage brought by encouragement, persistence fostered by immaculate caring and understanding, and insight ignited by unreserved guidance. I am fortunate enough to experience an amazing transformation and gaining much power and strength from it. I would like to express my gratitude to everyone who nurtured my transformation. I would especially like to thank my parents for raising me up with full of encouragement, caring, and love. I would also like to thank my master, the co‐author of this book, Dr. Kok, for his inspiring and endless guidance. All the tangible and intangible support from the people surrounding me have paved the way for my transformation and it will continue.
I do not take the opportunity for writing my third book for granted. I cherish the opportunity and trust this is part of the plan from God, reminding me to be humble to learn, to be rigorous to write, to be grateful for all opinions, and to be joyful to share the Good News.
Praise the LORD, for the LORD is good; sing praise to his name, for that is pleasant.
– Psalm 135:3 (NIV)
Wing‐Shan Tam
Nomenclature
1D
one‐dimensional
2D
two‐dimensional
ADC
analog‐to‐digital converter (A/D)
AWGN
additive white Gaussian noise
CCD
charge‐coupled device
CMOS
complementary metal‐oxide semiconductor
CFA
color filter array
dB
decibel
DCT
discrete cosine transform
DFT
discrete Fourier transform
DSLR
digital single lens reflex
DTFT
discrete time Fourier transform
DWT
discrete wavelet transform
FFT
fast Fourier transform
FIR
finite impulse response
FRIQ
full reference image quality index
HR
high‐resolution
HVS
human visual system, describing how humans perceive and interpret visual images
IID
independent and identically distributed
IDCT
inverse discrete cosine transform
IDFT
inverse discrete Fourier transform
IDWT
inverse discrete wavelet transform
IIR
infinite impulse response
JPEG
joint photographic experts group
LPF
lowpass filter
LR
low‐resolution
MATLAB
high‐level technical computing language by MathWorks Inc.
MAD
median absolute deviation
MED
median
MOS
mean opinion score
MSE
mean squares error
MSSIM
mean structural similarity
NRIQ
no reference image quality index
probability density function
PSNR
peak signal‐to‐noise ratio
RGB
red, green, and blue color space
RMSE
root mean squares error
SAP
salt and pepper noise
SNR
signal‐to‐noise ratio
SSIM
structural similarity
SVD
singular value decomposition
WSS
wide‐sense stationary
WT
wavelet transform
YCbCr
luminance, blue chrominance, red chrominance color space
Ceiling operator that returns the largest integer lesser than or equal to
the set of integer
the set of positive integer (greater than 0)
the set of real number
the set of complex number
vector defined in continuous domain
vector defined in discrete domain
a 2D function defined on continuous Cartesian domain
a 2D function defined on discrete Cartesian domain
arbitrary matrix of size constructed by matrix entrance with
identity matrix of size
the space of all squares summable discrete functions/sequences
the space of all Lesbeque squares integrable functions
teal part of a number, matrix, or function.
imaginary part of a number, matrix, or function.
Sinc function
Kronecker delta, or Dirac‐Delta function, or unit impulse with infinite size
‐th root of unity and equals to
discrete Fourier transform operator
inverse discrete Fourier transform operator
discrete Fourier transform matrix of size ; . The Fourier matrix is of arbitrary size when is missing
a window that specifies a collection of pixel locations around the .
convolution operator
interval in domain ; the interval domain is arbitrary when is missing
A word on notations
(
Indices
): We denote continuous variable and discrete variable indexed signals as and , respectively.
(
Vector‐matrix
): The blackboard bold is used to represent matrix‐valued signal and function, and is used to represent the vector‐valued signal and function. The normal characters are used to represent signal in scalar form.
(
Rows versus columns
): For vector‐matrix multiplication written as , we may take vector as a row vector.
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/kokDeNoise
This website includes:
MATLAB codes
PowerPoint files
1
Solutions Manuals
1
Note
1
PowerPoint files and Solution manuals are available upon registration for Professors/lecturers who intend to use this book in their courses.
1Digital Image
An image is a two‐dimensional (2D) light intensity function , where is a coordinate system of interest. Without loss of generality, and to simplify our discussions, the rest of the book will concentrate on the case of 2D Cartesian coordinate system. The value of at the coordinates is proportional to the brightness of the image at that point. While digital images can be generated/acquired by a number of methods, primarily, the image is converted to a digital image through cameras using a 2D image sensor array. These sensors are typically constructed with charge‐coupled devices (CCD) and complementary metal oxide semiconductor (CMOS) technologies. Camera constructed with CCD or CMOS works in a similar fashion, where the light reflected from an object will impinge onto the face of the sensor array, such that each sensor element in the array will generate an electrical signal (for which the coordinate can be considered to be the digitized coordinate of ). Figure 1.1 illustrates the construction of a color digital camera which is used to capture the Sculpture image. The light bounced off the Sculpture will be focused onto the sensor array through the lens. Consider a sensor array with ‐rows and ‐columns, the output of the sensor array will be a matrix with , and . As a result, the arrangement of the image sensor array is also known as the sampling grid, where the intersection of a row and a column will be assigned with an integer coordinate in the discrete Cartesian coordinate system. The output of each sensor element represents the number of photons that react with the sensor at location . The output of the sensor array is not a digital image yet. The subsequent analog‐to‐digital converter (A/D converter) accomplishes the quantization processes of the light intensity at all locations to generate the digital image. The sampled image obtained from the sampling and quantization process, as shown in Figure 1.2(a), is the discrete image which forms a matrix . Each entry in this array, , records the number of photons sensed by the corresponding sensor in the arrays and is termed a pixel. Thus, a digital image obtained by a digital camera will look like
Figure 1.1 Illustration of capturing an image by digital camera.
The values assigned to every pixel are the brightness recorded by the image sensor, which is also interpreted as the pixel intensity (also known as the gray‐level or grayscale).1 To store, transmit, and visualize the discrete image, the pixel intensity of the discrete image will be rounded to the nearest integer value within different gray levels through the quantization process performed within the A/D converter. This process will produce the digital image, which can be visualized as a shade of gray denoted as the grayscale or ‐level value ranging from black (0) to white (), such that the higher the intensity value, the brighter the image pixel. Figure 1.2(b) shows the pixel values of an extract from the image .
Figure 1.2 Representation of the digital image Sculpture : (a) a grayscale printout of Sculpture, which is described by an 2D array within the computation system with each matrix element representing the intensity of a pixel taking a value in the quantizer (in this case, it is [0,255] as Sculpture is an 8‐bit quantized image); (b) a pixel intensity map of the selected region in the image, where the pixel intensity at [62,369] is 211; and the intensity variation across the complete image by viewing (c) the 2D vector mesh of the image on a plane with the height (‐axis) being the pixel intensity or through (d) the contour map, where the pixel with the same intensities are located to the same contour lines.
The discrete image is arranged with each pixel being located at the th row and th column starting from the top‐left image origin (as shown in Figure 1.2(a)) with respect to the MATLAB convention. For simplification, we shall also use the vector to represent the pixel location, such that . Now, the readers may have already noticed from Figure 1.2(a) that the matrix indices in the figure are different from those in Equation 1.1. This is one of the irritating features of MATLAB. Notwithstanding the similarity between the arithmetic and the language of MATLAB, all matrices within MATLAB are indexed with the top left‐hand entry as instead of , and hence the discrepancy between Figure 1.2(a) and Equation 1.1. The rest of the book will assume this difference to be natural and will no longer discuss the difference between the MATLAB implementation and the analytical analysis with respect to the indexing problem.
1.1 Color Image
As pointed out by Sir Isaac Newton, color is perceived by the mind to resolve the interaction of light sources, objects, and the visual system, which adds a subjective layer on top of the underlying objective physical properties – the wavelength of the electromagnetic radiation carried by color signal. The color signal is received by light‐sensitive cells in human eye. Hering's experimental results and the discovery of three different types of photosensitive molecules in human eyes [52] led us to the modern color perception theory, where color is perceived through a luminance (grayscale) and two chrominance (color) components. This is the basis of trichromacy, the ability to match any color with a mixture of three suitably chosen primaries. The basic principle of color additivity has led to a number of useful trichromatic descriptions of color, which is also known as the color space.
Among various color spaces, the RGB, and the YCrCb are the most popular. In particular, the RGB color space has been widely employed in digital cameras and monitors to capture and display digital color images. This is because the RGB space conveniently corresponds to the three primary colors which are mixed for display on a monitor or similar devices. A digital color image in the RGB space is similar to a digital monochrome (grayscale) image, except that it requires a three‐dimensional vector to represent each pixel, and thus three arrays are required to represent the whole image. Each of these array represents one of the RED, GREEN, and BLUE primitive color components. The RED, GREEN, and BLUE components of an RGB image can be viewed separately as a monochrome image by considering the corresponding array alone, as shown in Figure 1.3. When the three color components are superposed, it produces the rightmost color image in Figure 1.3. As a result, if each component image is encoded with the data type uint8 in MATLAB, the total number of bits required to represent each pixel will be 8 bits 3 = 24 bits. This is also the default representation adopted by MATLAB for the three color triplets, and such type of image is known as the True Color image. Disregarding the digital color image format, the MATLAB function imread can be used to import the image directly from the image file stored in the hard disk, as shown in Listing 1.1.1.
Figure 1.3 Three separate RED, GREEN, and BLUE channels are combined to create a final, full‐color image.
Although only grayscale image denoising algorithms are discussed in this book, the algorithms can be easily extended to color images by treating the spectral components of the color images as independent grayscale images. Actually the grayscale image contains a lot of information, and this is the reason why black‐and‐white television receivers have been perfectly acceptable to the public for many years, and black‐and‐white photographs are still popular. Nevertheless, color is an important property, and so we shall examine its role in this section.
1.1.1 Color Filter Array and Demosaicing
To capture a digital image in color, three sensors with each sensor measuring one of the three colors, respectively, are required to capture the RED, GREEN, and BLUE component images. A cheaper alternative to the three‐sensors camera system is to have one sensor only. In this case, each photo sensor in the sensor array is made to be sensitive to one of the three colors (ranges of wavelengths). This can be done in a number of different ways. A popular method in modern camera is to cover the photo sensor array with a Bayer pattern color filter array (CFA) [3], as shown in Figure 1.1. Besides the Bayer pattern CFA, the readers may have also noticed that there is a color demosaicing block by the end of the camera in Figure 1.1
