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Lectures 3: Image Transforms

Lectures 3: Image Transforms. Professor Heikki Kälviäinen Machine Vision and Pattern Recognition Laboratory Department of Information Technology Faculty of Technology Management Lappeenranta University of Technology (LUT) Heikki.Kalviainen@lut.fi http://www.lut.fi/~kalviai

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Lectures 3: Image Transforms

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  1. Lectures 3:Image Transforms • Professor Heikki Kälviäinen • Machine Vision and Pattern Recognition Laboratory • Department of Information Technology • Faculty of Technology Management • Lappeenranta University of Technology (LUT) • Heikki.Kalviainen@lut.fi • http://www.lut.fi/~kalviai • http://www.it.lut.fi/ip/research/mvpr/ Prof. Heikki Kälviäinen CT50A6100

  2. 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. Prof. Heikki Kälviäinen CT50A6100

  3. Image Transforms • Fourier Transform • Cosine • Sine • Hadamard • Haar • Slant • Karhunen-Loeve • Fast KL • Sinusoidal • SDV • etc. Prof. Heikki Kälviäinen CT50A6100

  4. 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. Fourier Transform From: R.C. Gonzalez and R.E. Woods: Digital Image Processing, 3rd edition, 2008. Prof. Heikki Kälviäinen CT50A6100

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  7. Images and Their Fourier Spectra F(k) also denoted F(u) i = sqrt(-1) F(x,y) also denoted F(u,v) Prof. Heikki Kälviäinen CT50A6100

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  9. Discrete Fourier Transform (DFT) Prof. Heikki Kälviäinen CT50A6100

  10. 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. Prof. Heikki Kälviäinen CT50A6100

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  15. 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. Prof. Heikki Kälviäinen CT50A6100

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  19. 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. Prof. Heikki Kälviäinen CT50A6100

  20. Fourier Transform: Feature Extraction • Fourier transforms of texture. • Regular patterns. • Feature extraction. Prof. Heikki Kälviäinen CT50A6100

  21. Fourier Transform: Image Compression • Not so many coefficients needed => image compression. • Lossy/lossless compression? • Radius (pixels) % Image power • 8 95 • 16 97 • 32 98 • 64 99.4 • 128 99.8 Prof. Heikki Kälviäinen CT50A6100

  22. 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). Prof. Heikki Kälviäinen CT50A6100

  23. Enhancement in the Frequency Domain • Ideal high pass filter: • Original (left) and filtered image (right). Prof. Heikki Kälviäinen CT50A6100

  24. 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. Prof. Heikki Kälviäinen CT50A6100

  25. Feature Detector: 2-D Gabor Filter Prof. Heikki Kälviäinen CT50A6100

  26. Gabor Features • Maximal joint localization in the spatial and frequency domain. • Smooth and noise tolerant. • Parameters for invariance manipulation: Frequency Envelope sharpness Orientation Prof. Heikki Kälviäinen CT50A6100

  27. columns represent orientations rows represent frequencies image rotation appears as a circular shift of the columns image scaling appears as a shift of the rows (high frequencies may vanish) A SCALE AND ROTATION INVARIANT TREATMENT OF THE RESPONSE MATRIX CAN BE ESTABLISHED, AND THUS, WE CAN CONCENTRATE ONLY HOW TO CLASSIFY THEM IN THE STANDARD POSE Constructing Response Matrix Filter responser(x,y; f,)can be calculated for various frequencies f and orientations  to construct a response matrix. Prof. Heikki Kälviäinen CT50A6100

  28. 2-D Gabor Features What do they ”see”? Prof. Heikki Kälviäinen CT50A6100

  29. Face Detection and Recognition • 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. Prof. Heikki Kälviäinen CT50A6100

  30. Proposed Algorithm • Avoiding a scanning window. • Using feature detectors. • Shape-free texture model for the final decision. Prof. Heikki Kälviäinen CT50A6100

  31. 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). Prof. Heikki Kälviäinen CT50A6100

  32. eye eye Bayesian classification of features Gaussian mixture model densities (EM estimation) nostril Classifier Construction • 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.). eye eye nostril Prof. Heikki Kälviäinen CT50A6100

  33. 1 2 3 4 5 6 Affine Learned Correspondences Aligned images of objects and manually selected features Variability and correspondences Prof. Heikki Kälviäinen CT50A6100

  34. 2 False alarms occur and hypothesis verification is needed 1 3 Instance approved 2 1 1 4 1 5 2 2 2 Affine Hypothesis Search • Evidence • extraction. 2. Affine search and match to correspon- dence model. Prof. Heikki Kälviäinen CT50A6100

  35. 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. Prof. Heikki Kälviäinen CT50A6100

  36. 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. Prof. Heikki Kälviäinen CT50A6100

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  42. 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. Prof. Heikki Kälviäinen CT50A6100

  43. 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. Prof. Heikki Kälviäinen CT50A6100

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