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Final Review
Image Processing
CSE 166
Lecture 18
Topics covered
• Basis vectors• Matrix-based transforms• Basis images• Wavelet transform• Image compression• Image watermarking• Morphological image processing• Edge detection• Segmentation
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Matrix-based transforms
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Forward transform
Inverse transform
where
General inverse transform using basis vectors
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Matrix-based transforms using orthonormal basis vectors
• In vector form
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Matrix-based transforms using orthonormal basis vectors
• In matrix form
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Matrix-based transforms using biorthonormal basis vectors
• In matrix form
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Matrix-based transform
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Example: 8-point DFT of f(x) = sin(2πx)
real part + imaginary part
Matrix-based transforms
• Discrete Fourier transform (DFT)• Discrete Hartley transform (DHT)• Discrete cosine transform (DCT)• Discrete sine transform (DST)• Walsh-Hadamard (WHT)• Slant (SLT)• Haar (HAAR)• Daubechies (DB4)• Biorthogonal B-spline (BIOR3.1)
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Basis vectors of matrix-based 1D transforms
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N = 16
real part
imaginary part
Standard basis(for reference)
Basis vectors of matrix-based 1D transforms
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N = 16
basis dual Standard basis(for reference)
Matrix-based transformsin two dimensions
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Forward transform
Inverse transform
where
Matrix-based transformsin two dimensions
• If r and s are separable and symmetric, and M = N, then
– For orthonormal basis vectors
– For biorthonormal basis vectors
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Matrix-based transforms in two dimensions using basis images
• Inverse transform
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where
Each Su,v is a basis image
Basis images of matrix-based 2D transforms
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Standard basis images (for reference)
8-by-8 array of8-by-8
basis images
Basis images of matrix-based 2D transforms
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Discrete Fourier transform (DFT) basis images
real part imaginary part
Basis images of matrix-based 2D transforms
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Discrete Hartley transform (DHT) basis images
Basis images of matrix-based 2D transforms
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Discrete cosine transform (DCT) basis images
Basis images of matrix-based 2D transforms
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Discrete sine transform (DST) basis images
Basis images of matrix-based 2D transforms
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Walsh-Hadamard transform (WHT) basis images
Basis images of matrix-based 2D transforms
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Slant transform (SLT) basis images
Basis images of matrix-based 2D transforms
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Haar transform (HAAR) basis images
Wavelet transforms
• A scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 in resolution from its nearest neighboring approximations.
• Wavelet functions (wavelets) are then used to encode the differences between adjacent approximations.
• The discrete wavelet transform (DWT) uses those wavelets, together with a single scaling function, to represent a function or image as a linear combination of the wavelets and scaling function.
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Scaling functions and set of basis vectors
• Father scaling function
• Set of basis functions
– Integer translation k
– Binary scaling j
• Basis of the function space spanned by
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Scaling function, multiresolution analysis
1. The scaling function is orthogonal to its integer translates.
2. The function spaces spanned by the scaling function at low scales are nested within those spanned at higher scales.𝑉−∞⊂⋯⊂ 𝑉−1⊂ 𝑉0⊂ 𝑉1⊂⋯⊂ 𝑉∞
3. The only function representable at every scale (all 𝑉𝑗) is f(x) = 0.
4. All measureable, square-integrable functions can be represented as 𝑗 → ∞
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• Given father scaling function , there exists a mother wavelet function whose integer translations and binary scalings
span the difference between any two adjacent scaling spaces
• The orthogonal compliment of
Wavelet functions
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Relationship between scaling and wavelet function spaces
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Union
Scaling function coefficients and wavelet function coefficients
• Refinement (or dilation) equation
where are scaling function coefficients
• And
where are wavelet function coefficients
• Relationship
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1D discrete wavelet transform
• Forward
• Inverse
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Approximation
Details
2D discrete wavelet transform
• Forward
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Approximation
Details
where
Directionalwavelets
2D discrete wavelet transform
• Inverse
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2D discrete wavelet transform
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Decomposition
2D discrete wavelet transform
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3-levelwavelet
decomposition
Wavelet-based edge detection
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Zero horizontal
details
Zero lowest scale
approximation
Vertical edges
Edges
Wavelet-based noise removal
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Noisy imageThreshold
details
Zero highest
resolution details
Zero details for all levels
Data compression
• Data redundancy
where compression ratio
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where
Data redundancy in images
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Coding redundancy
Spatial redundancy
Irrelevant information
Does not need all 8 bits
Information is unnecessarily
replicated
Information is not useful
Image information
• Entropy
– It is not possible to encode input image with fewer than bits/pixel
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where
Fidelity criteria,objective (quantitative)
• Total error
• Root-mean-square error
• Mean-square signal to noise ratio (SNR)
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Fidelity criteriasubjective (qualitative)
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Approximations
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5.17 15.67 14.17
(a) (b) (c)
Objective (quantitative) qualityrms error (in intensity levels)
Subjective (qualitative) quality, relative
Lower is better
(a) is better than (b).
(b) is better than (c)
Compression system
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Compression methods
• Huffman coding• Golomb coding• Arithmetic coding• Lempel-Ziv-Welch (LZW) coding• Run-length coding• Symbol-based coding• Bit-plane coding• Block transform coding• Predictive coding• Wavelet coding
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Block-transform coding
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Encoder
Decoder
Block-transform coding
• Example: discrete cosine transform
• Inverse
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where
Block-transform coding
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Discrete cosine transformWalsh-Hadamard transform
4x4 subimages (4x4 basis images)
Block-transform coding
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Fourier transform
cosine transform
Walsh-Hadamardtransform
2.32 1.78 1.13rms error
Retain 32 largest
coefficients
Error image
8x8 subimages
Lower is better
Block-transform coding
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2x2 4x4 8x8
Reconstruction error versus subimage size
DCT subimage size:
JPEG uses block DCT-based coding
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Compression reconstruction
Scaled error image
Zoomed compression
reconstruction
25:1
52:1
Compression ratio
Predictive coding model
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Encoder
Decoder
Predictive coding
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Input image
Prediction error image
Example: previous pixel coding
Histograms
Wavelet coding
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Encoder
Decoder
Wavelet coding
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Detail coefficients below 25 are truncated to zero
JPEG-2000 uses wavelet-based coding
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Compression reconstruction
Scaled error image
Zoomed compression
reconstruction
25:1
52:1
Compression ratio
JPEG-2000 uses wavelet-based coding
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Compression reconstruction
Scaled error image
Zoomed compression
reconstruction
75:1
105:1
Compression ratio
Visible watermark
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Original image minus watermark
Watermarked image
Watermark
Invisible image watermarking system
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Encoder
Decoder
Invisible watermark
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Two least significant
bits
JPEGcompressed
Fragile invisible
watermark
Originalimage Extracted
watermark
Example: watermarking using two least significant bits
Invisible watermark
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Watermarked images
Example: DCT-based watermarking
Extracted robust
invisible watermark
Reflection and translation
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Translation
Reflection
Sets of pixels: objects and structuring elements (SEs)
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Border of background
pixels around objects
Tight border around SE
Reflection about the origin
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Origin
Don’t care elements
Erosion
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Example: square SE
Complement of A (i.e., set of elements not in A)
Erosion
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Example: elongated SE
Erosion
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11x11
45x4515x15
Shrinks
Dilation
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Square SEExamples
Elongated SE
Dilation
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Expands
Opening
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Structuring element rolls along inner boundary
Closing
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Structuring element rolls along outer boundary
Opening and closing
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Opening
Closing
Dilation
Erosion
Morphological image processing
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Noisy input
Dilation
Erosion
Opening
Dilation
Erosion
Closing
Boundary extraction
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Erosion Set difference
Boundary extraction
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Hole filling
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Given point in hole
Hole filling
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Given points in holes All holes filled
Connected components
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Given point in A
Connected components
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15 connected
components
X-ray image
Threshold (negative)
Image segmentation
• General approach
1. Spatial filtering
2. Additional processing
3. Thresholding
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Image segmentation
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Edge-based
Region-based
Input Edges Segmentation
Derivatives in 1D
• Forward difference
• Backward difference
• Central difference
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Image derivatives
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Detection of isolated points
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Laplacian(second derivative)
Input Segmentation
Threshold absolute
value
Line detection
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Threshold absolute
value
Threshold value
Laplacian(second
derivative)Input
Double lines
Line detection, specific directions
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Spatial filters
Line detection, specific directions
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+45°
Negative values set
to zero
Threshold
Edge models
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Step Ramp Roof edge
Edge models
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Step
RampRoofedge
Ramp edge
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First derivative
Second derivative
One point
Two points
Noise and image derivatives
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First derivative
Second derivativeInput
Noise
Edge detection
1. Image smoothing for noise reduction
2. Detection of image points (edge point candidates)
3. Edge localization (select from candidates, set of edge points)
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Gradient and edge direction
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Gradient direction is orthogonal to edge direction
Gradient operators
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Forward difference
Gradients
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Magnitude of
vertical gradient
Magnitude of
horizontal gradient
Magnitude of
gradient vector
Input
Gradients
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Magnitude of
vertical gradient
Magnitude of
horizontal gradient
Magnitude of
gradient vector
Input
Smooth image prior to computing gradients.Results in more selective edges
Edge detection
1. Smooth the input image
2. Compute the gradient magnitude image
3. Apply nonmaximal suppression to the gradient magnitude image
4. Threshold the resulting image
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Edge detection
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Without smoothing With smoothing
Threshold magnitude of gradient vector
Advanced edge detection
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Magnitude of gradient vector
(with smoothing)Input
Marr-Hildreth CannyFigure 10.25 in textbook looks better
Canny edge detector
1. Smooth the input image with a Gaussian filter
2. Compute the gradient magnitude and angle images
3. Apply nonmaximal suppression to the gradient magnitude image
4. Use double thresholding and connectivity analysis to detect and link edges
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Double thresholding
• Use a high threshold to start edge curves and a low threshold to continue them– Define two thresholds TH and TL– Starting with output of nonmaximal suppression, find a point q0,
which is a local maximum greater than TH– Start tracking an edge chain at pixel location q0 in one of the two
directions– Stop when gradient magnitude is less than TL
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TH
TL
q0
Double thresholding
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TH = 15 and TL = 5T = 5T = 15
Single threshold Double threshold
Thresholding
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Single threshold Dual thresholds
Histograms
Thresholding
• Single threshold
• Dual thresholds
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where
where
Noise and thresholding
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Noise
Varying background and thresholding
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Intensity ramp Product of input and intensity ramp
Input
Basic global thresholding
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Intensity ramp ThresholdInput
Basic global thresholding
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Value used to stop iterating
Optimum global thresholding
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Histogram
Basic global thresholding
Optimum global
thresholding using Otsu’s
method
Input
Otsu’s method
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Image smoothing to improve global thresholding
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Without smoothing
With smoothing
Otsu’s method
Image smoothing does not always improve global thresholding
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Without smoothing
With smoothing
Otsu’s method
Edges to improve global thresholding
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Mask image (thresholded
gradient magnitude)
Input
Masked input
Optimum global
thresholding using Otsu’s
method
Edges to improve global thresholding
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Input
Masked input
Optimum global
thresholding using Otsu’s
method
Mask image (thresholded absolute Laplacian)
Variable thresholding
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Global thresholding
Local standard
deviations
Local thresholding
using standard
deviations
Input
Variable thresholding
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Global thresholding
Local thresholding using moving
averages
Input(spot shading)
Variable thresholding
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Global thresholding
Local thresholding using moving
averages
Input(sinusoidal shading)
Segmentation by region growing
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Initial seed
image
InputX-ray image
Final seed
image
Output image
Segmentation by region growing
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Segmentation by region growing
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Difference image thresholded with the smallest of the dual
thresholds
Segmentation by region growing
Difference image thresholded using
dual thresholds
Difference image
Advanced segmentation methods
• k-means clustering
• Superpixels
• Graph cuts
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Segmentation using k-means clustering
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Input Segmentation using k-means, k = 3
Segmentation using k-means clustering
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Superpixels
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Image of 4,000 superpixels with
boundaries
Image of 4,000 superpixels
Input image of 480,000 pixels
Superpixels
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500 superpixels 250 superpixels1,000 superpixels
Superpixels for image segmentation
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Segmentation using k-means, k = 3
Superpixel image (100 superpixels)
Input image of 301,678 pixels
Segmentation using k-means, k = 3
Images as graphs
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Simple graph with edges only between
4-connected neighbors
Stronger (greater weight) edges are darker
Graph cuts for image segmentation
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Cut the weak edges
Graph cuts for image segmentation
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Smoothed input Graph cut segmentation
Input
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• Image acquisition
• Geometric transformations and image interpolation
• Intensity transformations
• Spatial filtering
• Fourier transform and filtering in the frequency domain
• Image restoration
• Color image processing
• Basis vectors
• Matrix-based transforms
• Basis images
• Wavelet transform
• Image compression
• Image watermarking
• Morphological image processing
• Edge detection
• Segmentation
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