Chapter 2. Image Analysis

Image Analysis Domains Frequency Domain Spatial Domain

Image Algebra Addition  Morphing Addition  Morphing Subtraction  Segmentation Subtraction  Segmentation Multiplication by constant  brighter Multiplication by constant  brighter Division by constant  darker Division by constant  darker AND  mask AND  mask OR  mask OR  mask NOT  negative NOT  negative

Example

Image Geometry Scaling Scaling Translation Translation Rotation Rotation

How to enlarge an image (Scaling or Sampling) Zero-order hold (expand & duplicate) Zero-order hold (expand & duplicate) First-order hold (linear interpolation) First-order hold (linear interpolation)  Two methods 1.Expand rows, then expand columns 2.Extend with zeros, then perform convolution process (support by hardware)

First Method (Method I) 

Convolution process Kernel or Mask

Convolution

First Order (method II)

How to reduce # of gray levels (Quantization) Converting the lower bits to 0 via an AND operation. Converting the lower bits to 0 via an AND operation. Converting the lower bits to 1 via an OR operation. Converting the lower bits to 1 via an OR operation. Improved gray-scale (IGS) quantization Improved gray-scale (IGS) quantization  remove false contour Variable bin size quantization Variable bin size quantization

Example of IGS

 IGS Quantization recognizes the eye’s inherent sensitivity to edges and breaks them up by adding to each pixel a random number, which is generated from the low-order (Least Significant Bits) of neighboring pixels. Improved Gray-Scale (IGS) Quantization

 A sum is formed from the current 8-bit gray-level value and the four least significant bits of a previously generated sum. If the four most significant bits of the current value are 1111, however, 0000 is added instead. An Example

IGS Practice Consider an 8-pixel line of gray-scale data, {12, 12, 13, 13, 10, 13, 57, 54}, which has been uniformly quantized with 6-bit accuracy. Construct its 3-bit IGS (Improved Gray-Scale) code.

Smoothing Just like Integration

Image Filtering Linear filter Linear filter Non-linear filter Non-linear filter

Image Smoothing Mean Filtering Gaussian Filtering Median Filtering Smoothing uniform regions Preserve edge structure

Mean Filtering Example

Gaussian Filtering Masks

Properties of smoothing masks The amount of smoothing and noise reduction is proportional to the mask size. Step edges are blurred in proportion to the mask size.

Median Filtering Example

Example

Edge Detection Just like Differentiation

Detecting Edges

Edge Detection Masks

Properties of derivative masks The sum of coordinates of derivative masks is zero so that a zero response is obtained on constant regions. First derivative masks produce high absolute values at point of high contrast. Second derivative masks produce zero-crossings at points of high contrast.

Edge Magnitude & Orientation

Laplacian Of Gaussian (LOG)

Zero crossing detection A zero crossing at a pixel implies that the values of the two opposing neighboring pixels in some direction have different signs. There four cases to test: 1.up/down 2.left/right 3.up-left/down-right 4.up-right/down-left

Two equivalent methods 1.Convolve the image with a Gaussian smoothing filter and compute the Laplacian of the result. 2.Convolve the image with the linear filter that is the Laplacian of the Gaussian filter. 12

Gaussian Equations

Gaussian Plots

Gaussian Properties Symmetry matrix 95% of the total weight is contained within 2  of the center. In the first derivative of 1D Gaussian, extreme points are located at –  and + . In the second derivative of 1D Gaussian, zero crossings are located at –  and + . The LOG filter responds well to: 1. small blobs coinciding with the center lobe. 2. large step edges very close to the center lobe.

LOG Masks

LOG Example

Frei-Chen Edge Detection Represent any 3x3 subimage as a weighted sum of the nine Frei-Chen masks. Represent any 3x3 subimage as a weighted sum of the nine Frei-Chen masks. Weights are found by projecting a 3x3 subimage onto each of these masks. Weights are found by projecting a 3x3 subimage onto each of these masks. The projection is performed through convolution. The projection is performed through convolution.

Frei- Chen Masks

Projection of vectors Since f 1, f 2, …, f 9 are nine 9D orthonormal vectors

Errors in Edge Detection

Pratt Figure of Merit Rating Factor I N = maximum(I I, I F ) I N = maximum(I I, I F ) I I = # of ideal edge points I I = # of ideal edge points I F = # of found edge points I F = # of found edge points α = a scaling constant to adjust the penalty for offset edges α = a scaling constant to adjust the penalty for offset edges d i = the distance of a found edge point to an ideal edge point d i = the distance of a found edge point to an ideal edge point

Noise Removal

Pepper & Salt Noise Reduction Change a pixel from 0 to 1 if all neighborhood pixels of the pixel is 1 Change a pixel from 1 to 0 if all neighborhood pixels of the pixel is 0

Expanding & Shrinking

Example 1

Example 2

Image Segmentation Region Based Clustering Region Growing Edge based Boundary Detection

Space of Clustering Histogram space  Thresholding Histogram space  Thresholding Color space  K-Means Clustering Color space  K-Means Clustering Spatial space  Region Growing Spatial space  Region Growing

Histogram & Thresholding

P-Tile Thresholding

Mode Thresholding

Mode Algorithm

Iterative Thresholding

Adaptive Thresholding Example

Adaptive Thresholding

Variable Thresholding Example

Double Thresholding Method

Double Thresholding Example

Recursive Histogram Clustering

Clustering

Iterative K-Means Clustering

Example of Region Growing

Region Growing (Split & Merge Algorithm) 1.Split the image into equally sized regions. 2.Calculate the gray level variance for each region 3.If the gray level variance is larger than a threshold, then split the region. Otherwise, an effort is made to merge the region with its neighbors. 4.Repeat Step 2 & 3. Gray level variance :

Boundary Detection 1.Canny Edge Detector 2.Hough Transform

Canney Edge Detector

Canny Edge Detector Example

Hough Transform

Accumulator array for Hough Transform

Hough Transform for Accumulating Straight Lines

Hough Transform Example

Hough Transform for Extracting Straight Lines

Example of Hough Transform

Morphological Filter

Example

Example

Closing & Opening

Opening Example

Morphological Filter Example 1

Structure Element Example 1

Morpho- logical Filter Example 2

Structure Element Example 2

Conditional Dilation

Conditional Dilation Example

Image Transform

Basis Vectors

Transform Coefficients

Fourier Transform 1.Remove high frequency noise 2.Extract texture features 3.Image compression

Discrete Fourier Transform

Magnitude & Phase of Discrete Fourier Transform

Separability of Fourier Transform

Properties of Fourier Transform Translation Brightness Scaling Rotation

Discrete Cosine Transform

Discrete Cosine Transform Basis Images

Walsh-Hadamard Transform

Walsh- Hadamard Basis Images

Construction of Walsh-Hadamard Basis Images

Frequency Domain Image Filtering

Bandpass Filtering

Symmetry of the Fourier Transform

Symmetry of the Discrete Cosine Transform

Ideal Lowpass Filter

Nonideal Lowpass Filter

Highpass Filter

Bandpass & Bandreject Filter

Convolution Theorem 1.Fourier transform the image g(x,y) to obtain its frequency representation G(u,v) 2.Fourier transform the mask h(x,y) to obtain its frequency representation H(u,v) 3.Multiply G(u,v) and H(u,v) pointwise 4.Apply the inverse Fourier transform to obtain the filtered image