Introduction to Linear Image Processing 1 Introduction to...
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1 Introduction to Linear Image Processing
IPAM - UCLA July 22, 2013
Introduction to Linear Image Processing
Iasonas Kokkinos
Center for Visual Computing
Ecole Centrale Paris / INRIA Saclay
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2 Introduction to Linear Image Processing
dA
dA’
Computer Vision
Image to Symbols
Computer Graphics
Symbols to Image
Imaging
Physics to Image
Image Processing
Image to Image
Image Sciences in a nutshell
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3 Introduction to Linear Image Processing
Images as functions
Continuous d=1: Gray
d=3: Color
Discrete
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4 Introduction to Linear Image Processing
Image Denoising
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5 Introduction to Linear Image Processing
Image Denoising
Key assumption: clean image is smooth
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6 Introduction to Linear Image Processing
Moving Average in 2D
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7 Introduction to Linear Image Processing
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8 Introduction to Linear Image Processing
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9 Introduction to Linear Image Processing
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10 Introduction to Linear Image Processing
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11 Introduction to Linear Image Processing
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12 Introduction to Linear Image Processing
Denoising: input
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13 Introduction to Linear Image Processing
Denoising: first application of averaring filter
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14 Introduction to Linear Image Processing
Denoising: tenth application of denoising filter
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15 Introduction to Linear Image Processing
Denoising: application of larger box filter
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16 Introduction to Linear Image Processing
Weighted averaging
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17 Introduction to Linear Image Processing
σ = 2 σ = 5
Weighting kernel
σ = 10
Standard deviation, σ: determines spatial support
Gaussian function:
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18 Introduction to Linear Image Processing
Moving average
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19 Introduction to Linear Image Processing
Gaussian blur
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20 Introduction to Linear Image Processing
Image Processing
image image filter
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21 Introduction to Linear Image Processing
Linearity
Translation Invariance
Linear, Translation-Invariant (LTI) system
Linear Image Processing
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22 Introduction to Linear Image Processing
Linear Image Processing
image image filter
From time-invariance: useful bases.
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23 Introduction to Linear Image Processing
Linear Image Processing
image image filter
From time-invariance: useful bases.
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24 Introduction to Linear Image Processing
Linear algebra reminder
Orthonormal basis:
Expansion coefficients:
Basis: N linearly independent vectors
Expansion on basis:
Expansion:
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25 Introduction to Linear Image Processing
Canonical basis
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26 Introduction to Linear Image Processing
Canonical basis for 2D signals
Kronecker delta
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27 Introduction to Linear Image Processing
Canonical basis for 2D signals
Kronecker delta
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28 Introduction to Linear Image Processing
Canonical basis for 2D signals
Kronecker delta
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29 Introduction to Linear Image Processing
Canonical basis for signals: expansion
Unit sample function
Signal expansion:
Identify terms:
Rewrite:
Sifting property:
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30 Introduction to Linear Image Processing
Canonical basis for signals and LTI filters
Any signal:
impulse response
Translation-invariance
Output of any LSI filter for any input:
convolution of input with filter’s impulse response
Convolution sum
unit
sample
By linearity:
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31 Introduction to Linear Image Processing
Convolution – discrete and continuous
2D convolution sum:
2D convolution integral:
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32 Introduction to Linear Image Processing
Linear Image Processing
image image filter
From time-invariance: useful bases.
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33 Introduction to Linear Image Processing
Associative Property:
Associative property & efficiency
Separability of Gaussian:
Slow Fast
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34 Introduction to Linear Image Processing
Associative Property:
Associative property & accuracy
Derivative of Gaussian:
approximate exact
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35 Introduction to Linear Image Processing
Associative Property:
Associative property & multi-scale processing
Semi-group property of Gaussian:
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36 Introduction to Linear Image Processing
Denoising: first application of averaging kernel
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37 Introduction to Linear Image Processing
Denoising: 10th application of denoising kernel
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38 Introduction to Linear Image Processing
Distributive property & efficiency
Distributive property:
Steerable fliter:
W. Freeman and E. Adelson, ‘The design and use of steerable filters’, PAMI, 1991
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39 Introduction to Linear Image Processing
Linear algebra reminder: eigenvectors
Eigenvectors:
Full-rank, real and symmetric: eigenbasis
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40 Introduction to Linear Image Processing
Eigenvectors and eigenfunctions
Eigenvector:
Eigenfunction:
Input:
Output:
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41 Introduction to Linear Image Processing
Eigenfunctions for LTI filters
LTI filter:
Let’s guess:
It works:
Frequency response:
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42 Introduction to Linear Image Processing
Expansion on harmonic basis
From orthonormality:
Inner product for complex functions:
Discrete-time:
Continuous-time:
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43 Introduction to Linear Image Processing
Change of basis
Canonical expansion:
Eigenbasis expansion:
Rotation matrix from eigenbasis:
Fourier transform: change of basis Rotation from canonical basis to eigenfunction basis
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44 Introduction to Linear Image Processing
Fourier Analysis
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45 Introduction to Linear Image Processing
Fourier synthesis equation
Continuous-time:
Discrete-time:
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46 Introduction to Linear Image Processing
Convolution theorem of Fourier transform
Input expansion:
Output:
Expansions:
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47 Introduction to Linear Image Processing
Linear Image Processing
image image filter
From time-invariance: useful bases.
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48 Introduction to Linear Image Processing
Convolution theorem
Fourier Analysis Fourier Synthesis
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49 Introduction to Linear Image Processing
Convolution theorem and efficiency
Fast Fourier Transform Fast Fourier Transform
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50 Introduction to Linear Image Processing
Gaussian blur Time
Frequency
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51 Introduction to Linear Image Processing
Moving average Time
Frequency
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52 Introduction to Linear Image Processing
Modulation property and Gabor filters
Modulation property:
Gaussian:
Time Frequency
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53 Introduction to Linear Image Processing
Modulation property and Gabor filters
Modulation property:
Gaussian:
Gabor:
Time Frequency
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54 Introduction to Linear Image Processing
• Consider many combinations of and
Frequency responce
isocurves
Increasing
Increasing
2D Gabor filterbank
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55 Introduction to Linear Image Processing
2D Gabor filterbank and texture analysis
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56 Introduction to Linear Image Processing
2D Gabor filterbank and texture analysis
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57 Introduction to Linear Image Processing
Thursday’s lecture: Pyramids, Scale-Invariant Blobs/Ridges, SIFT,
HOG, Log-polar features, Harmonic analysis on surfaces…
• Linear Time-Invariant filters
• Convolution
• Fourier Transform
• (Derivative-of) Gaussian filters
• Steerable filters
• Gabor filters
Summary
Further reading:
Fast recursive filters: Recursively implementing the Gaussian and its Derivatives - R. Deriche, 1993
Recursive implementation of the Gaussian filter. I. Young, L. Vliet, 1995
Fast IIR Isotropic 2D Complex Gabor Filters with Boundary Initialization,
A Bernardino, J. Santos-Victor, TIP, 2006
Wavelets: A Wavelet Tour of Signal Processing, S. Mallat, 2008