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## Computer vision

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**Computer vision**Seeks to generate intelligent and useful description of visual scenes and sequences • Examples: • Automatic face recognition • Visual guidance • Medical Image Analysis • Smart offices • Object based compression of video streams • etc.**Challenge of vision**• Signal to symbol converter • Physical signals Symbolic representations Manipulation of symbols allows machine or organism to interact intelligently with the world Simple for humans and animals, but very difficult to automate**Quantised image information**• Grey-scale (or colour) resolution and • Spatial resolution Nyquist’s theorem: highest spatial frequency component of information is equal to one-half the sampling density of the pixel array Consider amount of data in a video sequence: 768 x 576 pixels/frame x 25 frames/second = 11 million pixels per second. 3 colour planes each resolved to 8 bits finally gives 24x11 = 264 million bits per second.**Manipulation/analysis of image**Signal Modifed signal Transformation ”Sensible” representation ”Sensible” representation Inverse transformation Manipulation**Fourier analysis**Decompose image/signal into sinusoids Convolution is the basis for filtering Convolution in image domain is filtering in Fourier domain. With powerful and fast 2D-FFT algorithms this is very efficient**Fourier transform and non-stationary signals**• The Fourier transform does not provide information concerning the time interval(s) where a given frequency component exist. • Not a problem with stationary signals • (The frequency components of the signal then exist at “all times “) • Not suited for non-stationary signals • Neither suited for signals with discontinuities and sharp peaks • (like EKG or EEG signals)**Stationary**signal Non - stationary signal**From**• THE WAVELET TUTORIAL PART I • by ROBI POLIKAR**Windowed Fourier transform**• Sliding time window is used Heisenberg uncertainty relations are playing! One cannot know the exact time-frequency representation of a signal.**Alternative basis functions**Anyone heard about wavelets?**Wavelets**An Introduction**Wavelets**• Cut up data into different frequency components • Study each component with a resolution matched to its scale • Suitable for dealing with discontinuities and sharp peaks**Fourier versus wavelets**• Fourier basis functions are non-local • frequency resolution but no time resolution • Wavelet basis functions contained in finite domains • large scale is the big picture • small scale corresponds to the details**The Continuous Wavelet Transform (CWT)**• CWT decomposes f(t) into a set of basis wavelet functions: • Wavelets are generated from a mother wavelet**Wavelet properties**If satisfied then the wavelets can be used to analyze and then reconstruct a signal without loss of information • Admissibility condition: • Implication: Oscillatory ! Must be a wave**Regularityconditions and vanishing moments**Expand the wavelet transform into the Taylor series at t = 0 (we consider for simplicity) If moments up to order n is zero, then the coefficients will decay as fast as for decreasing scale s**Problems with the CWT**• Redundancy - The scaled wavelets will not be an orthogonal basis • An infinite number of wavelets in the transform • For most functions no analytical expressions exist for the wavelet transform • Fast algorithms are needed to exploit the power of the transform**Discrete wavelets**The wavelet itself is not discrete but it is scaled and translated in discrete steps • Usual choices: (dyadic sampling)**Is reconstruction from a wavelet decomposition possible?**• Necessary and sufficient condition for stable reconstruction: • If A=B the family (frame) of basis functions is tight and behave like an orthonormal basis**Orthogonal basis**• Wavelets that are orthogonal to their own dilations and translations demands special choices of the mother wavelets**How many scales are needed?**• Compression in time stretches the spectrum and shifts it upwards: • Using dilated wavelets we can cover the signal spectrum • The ratio between the center frequency of a wavelet spectrum and the width of the spectrum (the Q factor) is constant for all the wavelets**The scaling function**• An infinite number of wavelets is needed to cover the spectrum down to zero • Solution: Use a scaling function as cork plug Smaller scale**Iterated filter bank**• Iteratively split the signal in two • a low-pass and a high-pass signal • The wavelet transform is equivalent with a subband coding scheme**Closer look at the filter scheme**Twice as many data as we started! lowpass highpass Reconstruction:**Downsampling**Can the signal still be reconstructed ?**Reconstruction**Upsampling: lowpass highpass**Quadrature mirror filters**Analysing filters: Synthesis filters:**The discrete wavelet transform**• Often the signal will be discrete due to sampling • Will a digital filter bank do the job? • Two-scale relation: • Scaling function at one scale can be expressed from translated scaling functions at the next smaller scale!**Two-scale relation between scaling function and wavelets**As our signal f(t) can be expressed in terms of dilated and translated wavelets up to a scale j-1, we see from this two-scale relation:**Continued….**• If we step in scale from j to j-1 (less detail) we have to add wavelets to keep the same level of detail • One can show: We can regard our sampled signal as the output of a low-pass filter at a previous (imaginary) scale.**The coefficients are the key to everything!!**A particular wavelet is specified by a particular set of numbers, called wavelet filter coefficients By choosing a given number of filter coefficients, and demanding orthogonality as well as a given number of vanishing moments determines the shape of the wavelet. The two set of coefficient correspond to a high (g) and low pass filter (h) being applied recursively to the signal. Quadrature mirror filters !**To be noted**• Wavelets cut up data into different frequency components, and studies each component with a resolution matched to its scale ! • We do not have a unique set of wavelet functions There are an infinity of possible sets The sets can be tailored for different applications**Haar wavelet**Mother Father (or scaling function)**Daubechies 4 wavelet**Mother Father (or scaling function)**Coiflet 4 wavelet**Mother Father (or scaling function)**Denoising**• Discard wavelet coefficients of low significance • Two simple methods: • Hard thresholding • Soft thresholding • Determination of threshold values • Absolute threshold (given in advance) • Relative threshold (threshold is given as a fraction of the maximum coefficient) • Absolute quantitative threshold (A given number of coefficients (the largest) are kept) • Relative quantitative threshold ( z% of the total number of coefficients (the largest) are kept)**Denoising**Raw OCT image (false color coded) Denoising using wavelet transform and soft thresholding**Soft Skin on the Palm**low high Depth Averaging over ten scans per position One scan per position Speckle noise is not removed here! Denoising using wavelet decomposition “Hidden” structured regions are extracted !