Chapter 2. Image Analysis. Image Analysis Domains Frequency Domain Spatial Domain

Chapter 2. Image AnalysisChapter 2. Image Analysis

Image Analysis DomainsImage Analysis Domains

Frequency Domain

Spatial Domain

Image AlgebraImage Algebra

• AdditionAddition Morphing Morphing

• SubtractionSubtraction Segmentation Segmentation

• Multiplication by constant Multiplication by constant brighter brighter

• Division by constantDivision by constant darker darker

• ANDAND mask mask

• OROR mask mask

• NOTNOT negative negative

ExampleExample

Image GeometryImage Geometry

• ScalingScaling

• TranslationTranslation

• RotationRotation

How to enlarge an imageHow to enlarge an image(Scaling or Sampling)(Scaling or Sampling)

• Zero-order hold (expand & duplicate)Zero-order hold (expand & duplicate)

• First-order hold (linear interpolation)First-order hold (linear interpolation)

Two methodsTwo methods1.1. Expand rows, then expand columnsExpand rows, then expand columns2.2. Extend with zeros, then perform Extend with zeros, then perform convolutionconvolution

process (support by hardware)process (support by hardware)

First Method (Method I)First Method (Method I)

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How to reduce # of gray levelsHow to reduce # of gray levels(Quantization)(Quantization)

• Converting the lower bits to 0 via an Converting the lower bits to 0 via an AND operation.AND operation.

• Converting the lower bits to 1 via an Converting the lower bits to 1 via an OR operation.OR operation.

• Improved gray-scale (IGS) Improved gray-scale (IGS) quantizationquantization remove remove false contourfalse contour

• Variable bin size quantizationVariable bin size quantization

Example of Example of IGSIGS

Example of IGSExample 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) Improved Gray-Scale (IGS) Quantization 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 ExampleAn Example

IGS PracticeIGS 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.

SmoothingSmoothing

Just like IntegrationJust like Integration

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Image FilteringImage Filtering

• Linear filterLinear filter

• Non-linear filterNon-linear filter

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Image SmoothingImage Smoothing

• Mean Filtering

• Gaussian Filtering

• Median Filtering• Smoothing uniform regions• Preserve edge structure

Mean Filtering ExampleMean Filtering Example

GaussiaGaussian n Filtering Filtering MasksMasks

Properties of smoothing masksProperties 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 Median Filtering Filtering ExamplExamplee

ExampleExample

Edge DetectionEdge Detection

Just like DifferentiationJust like Differentiation

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Edge Detection MasksEdge Detection Masks

Properties of derivative Properties of derivative masksmasks

• 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 & Edge Magnitude & OrientationOrientation

Laplacian Of Gaussian (LOG)Laplacian Of Gaussian (LOG)

Zero crossing detectionZero 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/down2. left/right3. up-left/down-right4. up-right/down-left

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Two equivalent methodsTwo 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.

1 2

Gaussian EquationsGaussian Equations

GaussiaGaussian Plotsn Plots

Gaussian PropertiesGaussian 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 MasksLOG Masks

LOG ExampleLOG Example

Frei-Chen Edge DetectionFrei-Chen Edge Detection

• Represent any 3x3 subimage as a weighted Represent any 3x3 subimage as a weighted sum of the nine Frei-Chen masks.sum of the nine Frei-Chen masks.

• Weights are found by projecting a 3x3 Weights are found by projecting a 3x3 subimage onto each of these masks.subimage onto each of these masks.

• The projection is performed through The projection is performed through convolution.convolution.

Frei-Frei-Chen Chen MasksMasks

Projection of vectorsProjection of vectors

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• IINN = maximum( = maximum(III I , , IIFF))

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• IIFF = # of found edge points = # of found edge points

• αα = a scaling constant to adjust the penalty for offset edges = a scaling constant to adjust the penalty for offset edges

• ddii = the distance of a found edge point to an ideal edge point = the distance of a found edge point to an ideal edge point

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Noise RemovalNoise Removal

Pepper & Salt Noise Pepper & Salt Noise ReductionReduction

• 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 & ShrinkingExpanding & Shrinking

Example 1Example 1

Example Example 22

Image SegmentationImage Segmentation

• Region Based

• Clustering

• Region Growing

• Edge based

• Boundary Detection

Space of ClusteringSpace of Clustering

• Histogram spaceHistogram space Thresholding Thresholding

• Color spaceColor space K-Means K-Means ClusteringClustering

• Spatial spaceSpatial space Region Growing Region Growing

Histogram & ThresholdingHistogram & Thresholding

P-Tile ThresholdingP-Tile Thresholding

Mode ThresholdingMode Thresholding

Mode AlgorithmMode Algorithm

Iterative ThresholdingIterative Thresholding

Adaptive Adaptive ThresholdinThresholding Exampleg Example

Adaptive ThresholdingAdaptive Thresholding

Variable Variable ThresholdinThresholding Exampleg Example

Double Thresholding Double Thresholding MethodMethod

Double Double Thresholding Thresholding ExampleExample

RecursivRecursive e HistograHistogram m ClusterinClusteringg

ClusteringClustering

Iterative K-Means ClusteringIterative K-Means Clustering

Example of Region GrowingExample of Region Growing

Region GrowingRegion Growing(Split & Merge Algorithm)(Split & Merge Algorithm)

1. Split the image into equally sized regions.2. Calculate the gray level variance for each region3. 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.

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Boundary DetectionBoundary Detection

1. Canny Edge Detector

2. Hough Transform

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Canny Canny Edge Edge DetectoDetector r ExamplExamplee

Hough TransformHough Transform

Accumulator array for Accumulator array for Hough TransformHough Transform

Hough Hough Transform Transform

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Hough Hough TransforTransform m ExampleExample

Hough Hough Transform Transform for for Extracting Extracting Straight Straight LinesLines

Example of Example of Hough Hough TransformTransform

Morphological FilterMorphological Filter

MorphologicMorphological Filteral Filter

ExamplExamplee

Example Example

Closing & OpeningClosing & Opening

Opening ExampleOpening Example

Morphological Morphological Filter Example Filter Example 11

Structure Element Structure Element Example 1Example 1

MorphMorpho-o-logical logical Filter Filter ExamplExample 2e 2

Structure Element Example Structure Element Example 22

Conditional DilationConditional Dilation

Conditional Conditional Dilation Dilation ExampleExample

Image TransformImage Transform

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Basis VectorsBasis Vectors

Transform Transform CoefficientsCoefficients

Fourier TransformFourier Transform

1. Remove high frequency noise2. Extract texture features3. Image compression

Discrete Fourier TransformDiscrete Fourier Transform

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Walsh-Hadamard TransformWalsh-Hadamard Transform

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Construction of Walsh-Hadamard Construction of Walsh-Hadamard Basis ImagesBasis Images

Frequency Domain Image Frequency Domain Image FilteringFiltering

Bandpass FilteringBandpass Filtering

Symmetry Symmetry of the of the Fourier Fourier TransformTransform

SymmetrSymmetry of the y of the Discrete Discrete Cosine Cosine TransforTransformm

Ideal Lowpass FilterIdeal Lowpass Filter

Nonideal Lowpass FilterNonideal Lowpass Filter

Highpass FilterHighpass Filter

Bandpass & Bandreject FilterBandpass & Bandreject Filter

Convolution TheoremConvolution Theorem

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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