machine vision and dig. image analysis 1 prof. heikki kälviäinen ct50a6100 lectures 3: image...
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Machine Vision and Dig. Image Analysis
1 Prof. Heikki Kälviäinen CT50A6100
Lectures 3:Image Transforms
Professor Heikki Kälviäinen
Machine Vision and Pattern Recognition Laboratory Department of Information Technology
Faculty of Technology ManagementLappeenranta University of Technology (LUT)
Heikki.Kalviainen@lut.fihttp://www.lut.fi/~kalviai
http://www.it.lut.fi/ip/research/mvpr/
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Content
• Fourier Transform. – Discrete Fourier Transform. – Properties of 2-D Fourier Transform.
• Separability, periodicity, translation, rotation, scaling, convolution, correlation.
– Fast Fourier Transform. – Fourier Transform for image processing and analysis.
• Gabor filtering. – 2-D Gabor filter.– Applications.
• Other transforms.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Image Transforms
Fourier TransformCosineSineHadamardHaarSlantKarhunen-LoeveFast KLSinusoidal SDVetc.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Fourier Transform
• Jean Baptiste Joseph Fourier. • In 1807:
Any periodic function can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient. • The sum is nowadays called a Fourier series.• An image is a 2D signal.
From: R.C. Gonzalez and R.E. Woods: Digital Image Processing, 3rd edition, 2008.
Machine Vision and Dig. Image Analysis
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Images and Their Fourier Spectra
i = sqrt(-1)F(k) also denoted F(u)
F(x,y) also denoted F(u,v)
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Discrete Fourier Transform (DFT)
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Properties of 2-D Fourier Transform
• Separability.– 2-D can be computed as series of 1-D.
• Periodicity.• Translation, rotation, and scaling.
– Invariant features. • Convolution and correlation.
– Spatial filtering in the frequency domain. • Etc.
Machine Vision and Dig. Image Analysis
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Fast Fourier Transform (FFT)
• Brute-force implementation of Fourier Transform requires on the order of (MN)^2 summations and additions.
• 1024 x 1024 pixels => the order of a trillion multiplications and summations for one DFT
=> computationally too heavy. • FFT => the order of MN log_2 MN. • Based on the successive doubling method.
Machine Vision and Dig. Image Analysis
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Fourier Transform for Image Processing and Analysis
Properties of Fourier Transform offer for example the following use:• Feature extraction.
– Frequency domain features. • Image compression. • Image filtering.
– Image enhancement in the frequency domain. – Preprocessing.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Fourier Transform: Feature Extraction
• Fourier transforms of texture.• Regular patterns. • Feature extraction.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Fourier Transform: Image Compression
Radius (pixels) % Image power
8 95
16 97
32 98
64 99.4
128 99.8
Not so many coefficients needed => image compression.
Lossy/lossless compression?
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Fourier Transform: Image Filtering
• Enhancement in the frequency domain. • Convolution: f(x)*g(x) F(u) G(u).• Ideal low pass filter: Original (left) and filtered image (right).
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Enhancement in the Frequency Domain
• Ideal high pass filter:– Original (left) and filtered image (right).
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Image Processing Using Gabor Filtering
• For local and global feature extraction. • Filtering in time (spatial) space and frequency space.• For image processing and analysis two important parameters:
– Frequency f. – Orientation theta.
• Application example: – Face detection and recognition.– FACEDETECT project (http://www.it.lut.fi/project/facedetect/):
• Machine Vision and Pattern Recognition Laboratory (MVPR), Department of Information Technology, LUT, Finland.
• Centre for Vision, Speech and Signal Processing (CVSSP), University of Surrey, UK.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Feature Detector: 2-D Gabor Filter
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sincos'
),( '2''22
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fx
f
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Gabor Features
• Maximal joint localization in the spatial and frequency domain.• Smooth and noise tolerant.• Parameters for invariance manipulation:Frequency Envelope sharpness Orientation
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Constructing Response Matrix
Filter response r(x,y; f,) can be calculated for variousfrequencies f and orientations to construct a response matrix.
columns represent orientationsrows represent frequenciesimage rotation appears as acircular shift of the columns
image scaling appears as ashift of the rows (highfrequencies may vanish)
A SCALE AND ROTATION INVARIANTTREATMENT OF THE RESPONSE MATRIXCAN BE ESTABLISHED, AND THUS, WECAN CONCENTRATE ONLY HOW TOCLASSIFY THEM IN THE STANDARD POSE
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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2-D Gabor Features
discrete frequency [u]
dis
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te fr
eq
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ncy
[v]
-1/2 -1/4 0 1/4 1/2
-1/2
-1/4
0
1/4
1/2
What do they ”see”?
Machine Vision and Dig. Image Analysis
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Face verification (authentication) Validating a claimed identity based on the image of a face: are you Mr./Ms. X?
Face recognition (identification)Identifying a person based on an image of his/her face: who are you?
Face detection/localizationLocation of human faces in images at different positions, scales, orientations, and lighting conditions.
Face Detection and Recognition
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Proposed Algorithm
• Avoiding a scanning window.• Using feature detectors.• Shape-free texture model for the final decision.
Machine Vision and Dig. Image Analysis
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Evidence Extraction
Requirements
• Scale invariant extraction.• Rotation invariant extraction.• Provides sufficiently small amount of correct candidate points. (n best points from each class; needs confidence measure).
Preferred
• Estimation of evidence scale and orientation.• Fast extraction (scalability).
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Classifier Construction
eye
eye
nostrilnostril
eyeeye
Gaussian mixturemodel densities(EM estimation)
• Stability property guarantees approximately the Gaussian form of classes in the feature space.
• One class may still consist of several sub-clusters (open eye, closed eye, etc.).
Bayesianclassificationof features
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Affine Learned Correspondences
Aligned images of objects andmanually selected features Variability and correspondences
1 2
3 4
5 6
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Affine Hypothesis Search
2
2
3 11 12
4
5
2
2
1
1. Evidence extraction.
2. Affine search and match to correspon- dence model.
Instanceapproved
False alarms occur and hypothesisverification is needed
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Face Space
• Normalization of space where shape variations and capture effects are removed from patterns.
• Based on three points on the face -> affine registration.
• Optimal with regard to the photometric variance over a big set of faces.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Features & Feature Detectors
• Features = salient parts of face.
• Small localization variance and frequent occurrence over population.
• Illumination, scale, rotation, and translation invariance.
• Automatic analysis using the face space desirable.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Other transforms
• DFT, Cosine, Sine, Hadamar, Haar, Slant, Karhunen-Loeve, Fast KL, Sinusoidal transform, SVD transform.
• Basis images: Haar (wavelets) (left), Hadamard (right).– Haar values: positive-black, negative-white, zero-gray– Hadamard values: one – black, minus one – white.
Machine Vision and Dig. Image Analysis
Prof. Heikki Kälviäinen CT50A6100
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Summary
• Image transforms from the spatial domain into the frequency domain.• Fourier Transform.
– For overall image processing and analysis. – Feature extraction, image compression, and image filtering
(enhancement). – Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT). – Properties of 2-D Fourier Transform.
• Separability, periodicity, translation, rotation, scaling, convolution, correlation.
• Gabor filtering. – For local and global feature extraction. – Orientation and frequency.– 2-D Gabor filter.
• Other transforms. – For example, Cosine Transform for image compression. – More detailed information later in the course.
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