Optimal Adaptive Wavelet Bases Mahdi Amiri Advisors Dr. H. R. Rabiee Dr. M. T. Manzuri February 2002 Sharif University of Technology

Optimal Adaptive Wavelet Bases Presentation Outline Motivation and Goals Road to the Optimal Adaptive Wavelet Bases Why Wavelet-domain detection? Wavelets for Object Detection Literature The Proposed System Conclusions and Future Works Page 1 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Motivation and Goals Our Focus: Image Understanding (IU) Applications Specifically: Object Detection Object Recovery Image Enhancement Page 2 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Motivation and Goals Key Applications Quality Control Fabric Defect Detection [LamB97] Page 3 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Motivation and Goals Key Applications Medical imaging Tumor detection, Microcalcification detection [WangK98] Page 4 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Motivation and Goals Key Applications Military Applications Automatic or Aided Target Detection [KimL98] Page 5 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Motivation and Goals Key Applications Military Applications An ultimate example [MesRK99] Page 6 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Motivation and Goals Key Applications Security Applications Face detection, Face recognition Autonomous Navigation Real-time object detection for autonomous robots Human-Computer Interaction Recognizing the emotion of users Data Compression Efficient communication and compact coding Cartography Geographical survey, classifying landscape Pattern Analysis Web searches of images Page 7 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB Road to the Optimal Adaptive Wavelet Bases (OAWB) Page 8 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB Fourier, 1807 Haar, 1910 Math World Orthogonal Bi-orthogonal Structural sets Page 9 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB What kind of Could be useful? Impulse Function (Haar): Best time resolution Sinusoids (Fourier): Best frequency resolution We want both of the best resolutions Heisenberg, 1930 Uncertainty Principle There is a lower bound for (An intuitive prove in [Mac91]) Page 10 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB Gabor, 1945 Short Time Fourier Transform (STFT) Disadvantage: Fixed window size Page 11 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB How to Construct A Wavelet? There was already some methods But not interesting for engineers Daubechies, 1988 Compactly Supported Wavelets Maximum number of vanishing moments for their support width Based on Filter Banks Page 12 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB Wavelet Construction Start with a pair of filters Iterate the filter bank The equivalent filter leads us to the “wavelet” Page 13 of 35 Optimal Adaptive Wavelet Bases

Theory of Multiresolution Signal Decomposition Road to the OAWB A Major Breakthrough Mallat, 1989: Theory of Multiresolution Signal Decomposition Fast Algorithm for the Computation of Wavelet Transform Coefficients using Filter Banks Page 14 of 35 Optimal Adaptive Wavelet Bases

Road to the OAWB Multiresolution Signal Representation Coarse version (Approximation) more useful than the Detail Browsing image databases on the web Signal transmission for communication Denoising Wavelet Tree Decomposition We may lose what is in the Detail Page 15 of 35 Optimal Adaptive Wavelet Bases

Road to the OAWB Full Tree Decomposition Which decomposition path could be the best choice? The answer leads us to the Best Basis Page 16 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Road to the OAWB Best Basis Selection Criterions Cut if: Entropy: Shannon, Norm, Log Entry, Threshold, SURE Coifman, Meyer, Wickerhauser (Patent 1995) Rate-Distortion Subband Coding (Vetterli, 1995) Page 17 of 35 Optimal Adaptive Wavelet Bases

Detection in Wavelet Domain So far we have reached to Optimal Adaptive Wavelet Bases Next Discussion Why Wavelet Domain Detection? Page 18 of 35 Optimal Adaptive Wavelet Bases

Find a test statistic and classify Detection in Wavelet Domain Signal Detection Find a test statistic and classify Matched Filter, Correlation Detector Each view of the object may require a unique template Mathematical Morphology A priori knowledge of the resolution level of the object Local Filtering Techniques Fine tuning of several parameters related to local image statistics, Result in large number of false positives Page 19 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Detection in Wavelet Domain Discrete Wavelet Transform (DWT) Rapid processing Adaptation to changing local image statistics Representation of abrupt changes Precise position information Adapts to high background noise Adapts to uncertainty about object properties Relative independence to object-to-sensor distance Page 20 of 35 Optimal Adaptive Wavelet Bases

Detection in Wavelet Domain Wavelets “frequency compact” energy into a small set of low frequency coefficients “spatially compact” energy into a small set of high frequency coefficients Compression, Denoising, “Feature Detection” Here, Compression is not among our desired applications Page 21 of 35 Optimal Adaptive Wavelet Bases

Wavelets for Object Detection Literature General Approach Forward Wavelet Transform Nonlinearly transform or adaptively weight the wavelet coefficients Inverse Wavelet Transform Page 22 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Wavelets for Object Detection Literature Early works Isolating singularities caused by edges in noisy data Denoising algorithm: Remove all maxima whose amplitude increase on average when the scale decreases [MalH92] Detecting faint optical targets (trucks) in multispectral satellite images [YuRK92] Uses feature vectors Parameters: Spectral, Orientation, Scale Page 23 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Wavelets for Object Detection Literature Early works Wavelet and Gabor transform coefficients Detection in infrared images [CasSY92] Wavelet transform = Bank of matched filters [SzuSc92] Ideal spatial adaptation via wavelet shrinkage A major breakthrough [DonJ94] Undecimated wavelet transform Wavelet thresholding Donoho Page 24 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Wavelets for Object Detection Literature State of the Art (Two Step) Classification of transient sonar signals [JinXB96] Adaptive Wavelet  Feature Vectors (FV) Classification: Neural net Microcalcification detection [StrH97] Biorthogonal spline wavelet  FV Linear FV Classification Adaptive texture representation for ATR [MesRK99] Wavelets and PCA  FV Model based classification Page 25 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Wavelets for Object Detection Literature Adaptive Thresholding Spatially adaptive thresholding in wavelet domain with context modeling Results are significantly superior to uniform thresholding [ChaYV00] Page 26 of 35 Optimal Adaptive Wavelet Bases

The Proposed System Contribution Adaptive thresholding Among all of the works two groups are interesting Adaptive thresholding [ZhaD01], [MadTR01] Adaptive wavelets [ZhiPY01], [LiBH01] We will take advantage from both of these adaptivities Page 27 of 35 Optimal Adaptive Wavelet Bases

The Proposed System Block Diagram Undecimated Wavelet Packet Tree Input Image Undecimated Wavelet Packet Tree Adaptive Best Basis Feature Selection Detection Classifier Threshold Nonlinear Mapping of Details Recovery Inverse Wavelet Transform Page 28 of 35 Optimal Adaptive Wavelet Bases

The Proposed System Biorthogonal Decomposition For Undecimated Wavelet Packet Tree Analysis Synthesis Page 29 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases The Proposed System Feature Vectors Feature Vector for each pixel: Components: A group of the following coefficients after Entropy based best basis selection Page 30 of 35 Optimal Adaptive Wavelet Bases

The Proposed System Classifier (Method 1) Hottelling Observer Fischer’s discriminant: Mean of “signal present” class FVs: Mean of “signal absent” class FVs: Provides maximum separation between distributions in an N class problem [DudH73] Page 31 of 35 Optimal Adaptive Wavelet Bases

Widely used in pattern classification applications The Proposed System Classifier (Method 2) Mahalanobis Distance Model FVs: Gaussian distribution Mean: Covariance matrix: Widely used in pattern classification applications Page 32 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases The Proposed System Threshold for detection Apply a weighting factor for the identified pixels Compute inverse wavelet transform for recovery Compare Performance ROC (Receiver Operating Characteristic) X axis: False Positive Fraction Y axis: 1 – True Positive Fraction Page 33 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases The Proposed System Disadvantages Logarithmic distance of decomposition scales Use “Voices” May need some pre/post processing Pre: Contrast Enhancement Post: Decrease false positive rate in conjunction with non-wavelet based methods (e.g. Spatial edge detection) Page 34 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Conclusions and Future Works Simulating Adaptive Best Basis Selection Methods Simulating Classifiers and Adaptive Thresholding Hardware Implementation Just for prove of concept, based on TMS320Cx FIND OUT MORE AT... 1. http://ce.sharif.edu/~m_amiri/ 2. http://www.aictct.com/dml/ (References may be found there in the accompanied report) Page 35 of 35 Optimal Adaptive Wavelet Bases

Optimal Adaptive Wavelet Bases Mahdi Amiri Advisors Dr. H. R. Rabiee Dr. M. T. Manzuri “You can't predict the future, but you can invent it.” Dennis Gabor