Solution Manual For Digital Image Processing, 4th Edition By Gonzalez

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Solution Manual For Digital Image Processing, 4th Edition By Rafael C. Gonzalez, Richard E. Woods, ISBN-13: 9780133356724

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Solution Manual For Digital Image Processing, 4th Edition By Gonzalez

Solution Manual For Digital Image Processing, 4th Edition By Rafael C. Gonzalez, Richard E. Woods, ISBN-13: 9780133356724

Table of Contents

1 Introduction

1.1 What is Digital Image Processing?

1.2 The Origins of Digital Image Processing

1.3 Examples of Fields that Use Digital Image Processing

Gamma-Ray Imaging

X-Ray Imaging

Imaging in the Ultraviolet Band

Imaging in the Visible and Infrared Bands

Imaging in the Microwave Band

Imaging in the Radio Band

Other Imaging Modalities

1.4 Fundamental Steps in Digital Image Processing

1.5 Components of an Image Processing System

2 Digital Image Fundamentals

2.1 Elements of Visual Perception

Structure of the Human Eye

Image Formation in the Eye

Brightness Adaptation and Discrimination

2.2 Light and the Electromagnetic Spectrum

2.3 Image Sensing and Acquisition

Image Acquisition Using a Single Sensing Element

Image Acquisition Using Sensor Strips

Image Acquisition Using Sensor Arrays

A Simple Image Formation Model

2.4 Image Sampling and Quantization

Basic Concepts in Sampling and Quantization

Representing Digital Images

Linear vs. Coordinate Indexing

Spatial and Intensity Resolution

Image Interpolation

2.5 Some Basic Relationships Between Pixels

Neighbors of a Pixel

Adjacency, Connectivity, Regions, and Boundaries

Distance Measures

2.6 Introduction to the Basic Mathematical Tools Used in Digital Image Processing

Elementwise versus Matrix Operations

Linear versus Nonlinear Operations

Arithmetic Operations

Set and Logical Operations

Basic Set Operations

Logical Operations

Fuzzy Sets

Spatial Operations

Single-Pixel Operations

            Neighborhood Operations

            Geometric Transformations

            Image Registration

Vector and Matrix Operations

Image Transforms

Probability and Random Variables

3 Intensity Transformations and Spatial Filtering

3.1 Background

The Basics of Intensity Transformations and Spatial Filtering

About the Examples in this Chapter

3.2 Some Basic Intensity Transformation Functions

Image Negatives

Log Transformations

Power-Law (Gamma) Transformations

Piecewise Linear Transformation Functions

Contrast Stretching

            Intensity-Level Slicing

            Bit-Plane Slicing

3.3 Histogram Processing

Histogram Equalization

Histogram Matching (Specification)

Exact Histogram Matching (Specification)



Computing the neighborhood averages and extracting the K-tuples:

                  Exact Histogram Specification Algorithm

Local Histogram Processing

Using Histogram Statistics for Image Enhancement

3.4 Fundamentals of Spatial Filtering

The Mechanics of Linear Spatial Filtering

Spatial Correlation and Convolution

Separable Filter Kernels

Some Important Comparisons Between Filtering in the Spatial and Frequency Domains

A Word about how Spatial Filter Kernels are Constructed

3.5 Smoothing (Lowpass) Spatial Filters

Box Filter Kernels

Lowpass Gaussian Filter Kernels

Order-Statistic (Nonlinear) Filters

3.6 Sharpening (Highpass) Spatial Filters


Using the Second Derivative for Image Sharpening—The Laplacian

Unsharp Masking and Highboost Filtering

Using First-Order Derivatives for Image Sharpening—The Gradient

3.7 Highpass, Bandreject, and Bandpass Filters from Lowpass Filters

3.8 Combining Spatial Enhancement Methods

3.9 Using Fuzzy Techniques for Intensity Transformations and Spatial Filtering


Principles of Fuzzy Set Theory


                  Some Common Membership Functions

Using Fuzzy Sets

Using Fuzzy Sets for Intensity Transformations

Using Fuzzy Sets for Spatial Filtering

4 Filtering in the Frequency Domain

4.1 Background

A Brief History of the Fourier Series and Transform

About the Examples in this Chapter

4.2 Preliminary Concepts

Complex Numbers

Fourier Series

Impulses and their Sifting Properties

The Fourier Transform of Functions of One Continuous Variable


4.3 Sampling and the Fourier Transform of Sampled Functions


The Fourier Transform of Sampled Functions

The Sampling Theorem


Function Reconstruction (Recovery) from Sampled Data

4.4 The Discrete Fourier Transform of One Variable

Obtaining the DFT from the Continuous Transform of a Sampled Function

Relationship Between the Sampling and Frequency Intervals

4.5 Extensions to Functions of Two Variables

The 2-D Impulse and Its Sifting Property

The 2-D Continuous Fourier Transform Pair

2-D Sampling and the 2-D Sampling Theorem

Aliasing in Images

                  Extensions from 1-D Aliasing

                  Image Resampling and Interpolation

                  Aliasing and Moiré Patterns

The 2-D Discrete Fourier Transform and Its Inverse

4.6 Some Properties of the 2-D DFT and IDFT

Relationships Between Spatial and Frequency Intervals

Translation and Rotation


Symmetry Properties

Fourier Spectrum and Phase Angle

The 2-D Discrete Convolution Theorem

Summary of 2-D Discrete Fourier Transform Properties

4.7 The Basics of Filtering in the Frequency Domain

Additional Characteristics of the Frequency Domain

Frequency Domain Filtering Fundamentals

Summary of Steps for Filtering in the Frequency Domain

Correspondence Between Filtering in the Spatial and

Frequency Domains

4.8 Image Smoothing Using Lowpass Frequency Domain Filters

Ideal Lowpass Filters

Gaussian Lowpass Filters

Butterworth Lowpass Filters

Additional Examples of Lowpass Filtering

4.9 Image Sharpening Using Highpass Filters

Ideal, Gaussian, and Butterworth Highpass Filters from Lowpass Filters

The Laplacian in the Frequency Domain

Unsharp Masking, High-boost Filtering, and High-Frequency-Emphasis Filtering

Homomorphic Filtering

4.10 Selective Filtering

Bandreject and Bandpass Filters

Notch Filters

4.11 The Fast Fourier Transform

Separability of the 2-D DFT

Computing the IDFT Using a DFT Algorithm

The Fast Fourier Transform (FFT)

5 Image Restoration and Reconstruction

5.1 A Model of the Image Degradation/Restoration Process

5.2 Noise Models

Spatial and Frequency Properties of Noise

Some Important Noise Probability Density Functions

                  Gaussian Noise

                  Rayleigh Noise

                  Erlang (Gamma) Noise

                  Exponential Noise

                  Uniform Noise

                  Salt-and-Pepper Noise

Periodic Noise

Estimating Noise Parameters

5.3 Restoration in the Presence of Noise Only—Spatial Filtering

Mean Filters

                  Arithmetic Mean Filter

                  Geometric Mean Filter

                  Harmonic Mean Filter

                  Contraharmonic Mean Filter

Order-Statistic Filters

                  Median Filter

                  Max and Min Filters

                  Midpoint Filter

                  Alpha-Trimmed Mean Filter

Adaptive Filters

                  Adaptive, Local Noise Reduction Filter

                  Adaptive Median Filter

5.4 Periodic Noise Reduction Using Frequency Domain Filtering

More on Notch Filtering

Optimum Notch Filtering

5.5 Linear, Position-Invariant Degradations

5.6 Estimating the Degradation Function

Estimation by Image Observation

Estimation by Experimentation

Estimation by Modeling

5.7 Inverse Filtering

5.8 Minimum Mean Square Error (Wiener) Filtering

5.9 Constrained Least Squares Filtering

5.10 Geometric Mean Filter

5.11 Image Reconstruction from Projections


Principles of X-ray Computed Tomography (CT)

Projections and the Radon Transform


The Fourier-Slice Theorem

Reconstruction Using Parallel-Beam Filtered Backprojections

Reconstruction Using Fan-Beam Filtered Backprojections

6 Wavelet and Other Image Transforms

6.1 Preliminaries

6.2 Matrix-based Transforms

Rectangular Arrays

Complex Orthonormal Basis Vectors

Biorthonormal Basis Vectors

6.3 Correlation

6.4 Basis Functions in the Time-Frequency Plane

6.5 Basis Images

6.6 Fourier-Related Transforms

The Discrete hartley Transform

The Discrete Cosine Transform

The Discrete Sine Transform

6.7 Walsh-Hadamard Transforms

6.8 Slant Transform

6.9 Haar Transform

6.10 Wavelet Transforms

Scaling Functions

Wavelet Functions

Wavelet Series Expansion

Discrete Wavelet Transform in One Dimension

                  The Fast Wavelet Transform

Wavelet Transforms in Two Dimensions

Wavelet Packets

7 Color Image Processing

7.1 Color Fundamentals

7.2 Color Models

The RGB Color Model

The CMY and CMYK Color Models

The HSI Color Model

                  Converting Colors from RGB to HSI

                  Converting Colors from HSI to RGB

                  Manipulating HSI Component Images

A Device Independent Color Model

7.3 Pseudocolor Image Processing

Intensity Slicing and Color Coding

Intensity to Color Transformations

7.4 Basics of Full-Color Image Processing

7.5 Color Transformations


Color Complements

Color Slicing

Tone and Color Corrections

Histogram Processing of Color Images

7.6 Color Image Smoothing and Sharpening

Color Image Smoothing

Color Image Sharpening

7.7 Using Color in Image Segmentation

Segmentation in HSI Color Space

Segmentation in RGB Space

Color Edge Detection

7.8 Noise in Color Images

7.9 Color Image Compression

8 Image Compression and Watermarking

8.1 Fundamentals

Coding Redundancy

Spatial and Temporal Redundancy

Irrelevant Information

Measuring Image Information

                  Shannon’s First Theorem

Fidelity Criteria

Image Compression Models

                  The Encoding or Compression Process

                  The Decoding or Decompression Process

Image Formats, Containers, and Compression Standards

8.2 Huffman Coding

8.3 Golomb Coding

8.4 Arithmetic Coding

Adaptive context dependent probability estimates

8.5 LZW Coding

8.6 Run-length Coding

One-dimensional CCITT compression

Two-dimensional CCITT compression

8.7 Symbol-based Coding

JBIG2 compression

8.8 Bit-plane Coding

8.9 Block Transform Coding

Transform selection

Subimage size selection

Bit allocation

                  Zonal Coding Implementation

                  Threshold Coding Implementation


8.10 Predictive Coding

Lossless predictive coding

Motion compensated prediction residuals

Lossy predictive coding

Optimal predictors

Optimal quantization

8.11 Wavelet Coding

Wavlet selection

Decomposition level selection

Quantizer design


8.12 Digital Image Watermarking

9 Morphological Image Processing

9.1 Preliminaries

9.2 Erosion and Dilation




9.3 Opening and Closing

9.4 The Hit-or-Miss Transform

9.5 Some Basic Morphological Algorithms

Boundary Extraction

Hole Filling

Extraction of Connected Components

Convex Hull





9.6 Morphological Reconstruction

Geodesic Dilation and Erosion

Morphological Reconstruction by Dilation and by Erosion

Sample Applications

                  Opening by Reconstruction

                  Automatic Algorithm for Filling Holes

                  Border Clearing

9.7 Summary of Morphological Operations on Binary Images

9.8 Grayscale Morphology

Grayscale Erosion and Dilation

Grayscale Opening and Closing

Some Basic Grayscale Morphological Algorithms

                  Morphological Smoothing

                  Morphological Gradient

                  Top-Hat and Bottom-Hat Transformations


                  Textural Segmentation

Grayscale Morphological Reconstruction

10 Image Segmentation I: Edge Detection,

   Thresholding, and Region Detection

10.1 Fundamentals

10.2 Point, Line, and Edge Detection


Detection of Isolated Points

Line Detection

Edge Models

Basic Edge Detection

                  The Image Gradient and Its Properties

                  Gradient Operators

                  Combining the Gradient with Thresholding

More Advanced Techniques for Edge Detection

                  The Marr-Hildreth Edge Detector

                  The Canny Edge Detector

Linking Edge Points

                  Local Processing

                  Global Processing Using the Hough Transform

10.3 Thresholding


                  The Basics of Intensity Thresholding

                  The Role of Noise in Image Thresholding

                  The Role of Illumination and Reflectance in Image Thresholding

Basic Global Thresholding

Optimum Global Thresholding Using Otsu’s Method

Using Image Smoothing to Improve Global Thresholding

Using Edges to Improve Global Thresholding

Multiple Thresholds

Variable Thresholding

                  Variable Thresholding Based on Local Image Properties

                  Variable Thresholding Based on Moving Averages

10.4 Segmentation by Region Growing and by Region Splitting and Merging

Region Growing

Region Splitting and Merging

10.5 Region Segmentation Using Clustering and Superpixels

Region Segmentation using K-Means Clustering

Region Segmentation using Superpixels

                  SLIC Superpixel Algorithm

                  Specifying the Distance Measure

10.6 Region Segmentation Using Graph Cuts

Images as Graphs

Minimum Graph Cuts

Computing Minimal Graph Cuts

Graph Cut Segmentation Algorithm

10.7 Segmentation Using Morphological Watersheds


Dam Construction

Watershed Segmentation Algorithm

The Use of Markers

10.8 The Use of Motion in Segmentation

Spatial Techniques

                  A Basic Approach

                  Accumulative Differences

                  Establishing a Reference Image

Frequency Domain Techniques

11 Image Segmentation II: Active Contours: Snakes and Level Sets

11.1 Background

11.2 Image Segmentation Using Snakes

Explicit (Parametric) Representation of Active Contours

Derivation of the Fundamental Snake Equation

Iterative Solution of the Snake Equation

External Force Based on the Magnitude of the Image

Gradient (MOG)

External Force Based on Gradient Vector Flow (GVF)

11.3 Segmentation Using Level Sets

Implicit Representation of Active Contours

Derivation of the Level Set Equation

Discrete (Iterative) Solution of The Level Set Equation


Specifying, Initializing, and Reinitializing Level Set Functions

Force Functions Based Only on Image Properties

Edge/Curvature-Based Forces

Region/Curvature-Based Forces

Improving the Computational Performance of Level Set Algorithms

12 Feature Extraction

12.1 Background

12.2 Boundary Preprocessing

Boundary Following (Tracing)

Chain Codes

                  Freeman Chain Codes

                  Slope Chain Codes

Boundary Approximations Using Minimum-Perimeter Polygons


                  MPP Algorithm


Skeletons, Medial Axes, and Distance Transforms

12.3 Boundary Feature Descriptors

Some Basic Boundary Descriptors

Shape Numbers

Fourier Descriptors

Statistical Moments

12.4 Region Feature Descriptors

Some Basic Descriptors

Topological Descriptors


                  Statistical Approaches

                  Spectral Approaches

Moment Invariants

12.5 Principal Components as Feature Descriptors

12.6 Whole-Image Features

The Harris-Stephens Corner Detector

Maximally Stable Extremal Regions (MSERs)

12.7 Scale-Invariant Feature Transform (SIFT)

Scale Space

Detecting Local Extrema

                  Finding the Initial Keypoints

                  Improving the Accuracy of Keypoint Locations

                  Eliminating Edge Responses

Keypoint Orientation

Keypoint Descriptors

Summary of the SIFT Algorithm

13 Image Pattern Classification

13.1 Background

13.2 Patterns and Pattern Classes

Pattern Vectors

Structural Patterns

13.3 Pattern Classification by Prototype Matching

Minimum-Distance Classifier

Using Correlation for 2-D prototype matching

Matching SIFT Features

Matching Structural Prototypes

                  Matching Shape Numbers

                  String Matching

13.4 Optimum (Bayes) Statistical Classifiers

Derivation of the Bayes Classifier

Bayes Classifier for Gaussian Pattern Classes

13.5 Neural Networks and Deep Learning


The Perceptron

Multilayer Feedforward Neural Networks

                  Model of an Artificial Neuron

                  Interconnecting Neurons to Form a Fully Connected Neural Network

                  Forward Pass Through a Feedforward Neural Network

                  The Equations of a Forward Pass

                  Matrix Formulation

Using Backpropagation to Train Deep Neural Networks

                  The Equations of Backpropagation

                  Matrix Formulation

13.6 Deep Convolutional Neural Networks

A Basic CNN Architecture

                  Basics of How a CNN Operates

                  Neural Computations in a CNN

                  Multiple Input Images

The Equations of a Forward Pass Through a CNN

The Equations of Backpropagation Used to Train CNNs

13.7 Some Additional Details of Implementation