ࡱ> -/,4u7 .bjbjUU "F7|7|"lBBBBBBB 84 $X ,2 l  n/$S$# C jBgggcBB ;cccgB B cgc`c<BB t KTd0\, c cVtTBBBBHYPERLINK "..\\Mpeg\\MPEG.doc"Previous lectureHYPERLINK "Fractal.doc"Next Lecture HYPERLINK "..\\320Syllabus.doc" SyllabusWavelets Lets review what we did when we used the Cosine Transform for JPEG compression. We divided the image into 8x8 tiles, and treated the rows and columns separately. For each row of 8 pixels, we found the coefficients for which we could add up 8 different cosine waves and get back the original image. We can treat those 8 pixels as a vector space. Any set of 8 linearly independent functions can be used as a basis set. The easiest basis set is a set of functions B that are one at a pixel and zero elsewhere:  We can write the values of the 8 pixels as a function p(t), for t = 0, , 7. Each row in the graph represents a basis function bi(t), for i = 0, , 7. For example,  EMBED Equation.3  The basis set B = {b0(t), b1(t), , b7(t)} Any pixel can be expressed by adding up basis functions weighted by coefficients:  EMBED Equation.3  In this case, finding the values for the coefficients is trivial. We set bi = p(i). As a basis set, C, the DCT uses sampled values of cosines (values from  1 to 1):  EMBED Word.Picture.8  Each row in the graph represents a basis function ci(t), for i = 0, & , 7. For example, c2(t) = cos(p t 2 / 16) The basis set C = {c0(t), c1(t), & , c7(t)} Any pixel can be expressed by adding up basis functions weighted by coefficients:  EMBED Equation.3  We figure out how to select the coefficients gi by using the cosine transform. When we dont do any compression, we get the exact same values for p(t) using either the pixel basis B or the cosine basis C. We could use other basis sets, such as pairs of sines and cosines to get the Fourier transform. Basis sets and Compression The cosine transform basis C works better than the pixel basis B for compression. This is because pixels do not change rapidly in an image. The high frequency information tends to make textures. Most of the time we can reproduce the texture less faithfully and not notice it. However, when we have sharp edges, the high frequency information is needed at that spot to give the detail. We can select any 8 linearly independent functions to be a basis to represent an image. Is there a basis that works better than C for compression? The two basis sets that we have looked at are equivalent in that they both represent the same pixels, but they are different in what they make obvious. The pixel basis B tells us the exact location of each pixel, but tells us nothing about frequency. The cosine basis C [or the Fourier transform] tells us the frequencies, but says nothing about where they occur. It would be nice if we had a basis that gave us information about both location and frequency. This is the idea behind wavelets. Gabor Wavelets We could try using a set of basis functions like these (values from 1 to 1):  While the DCT used DC and 7 different frequencies, the wavelet basis uses only 3 different frequencies. The wavelet basis is generated from a mother wavelet. The mother wavelet is scaled by expanding or contracting it by a factor of 2. The smaller wavelets are translated to different positions. The 4 high frequency components in the DCT gave us 4 different frequencies, but only one location. The wavelet transform gives us just one high frequency, but it is localized to 4 different positions. The wavelet pictured is a Gabor function, which is a sinusoid windowed by a Gaussian: g(t) = exp(-s2 t2) cos (2 p t / N) Time  Frequency Resolution When we apply the Cosine Transform or the Fourier Transform to tiles of size NxN, we know the frequencies within each block, but our location information is uncertain: the frequency might be present everywhere (a texture) or it might be at only one point (an edge). The location of a vertical edge is uncertain by N. Thus the joint uncertainty in frequency and position is 1 * N. Using the pixel basis, we know the location, but the frequency is uncertain. Again the joint uncertainty is N * 1. With a Gabor function, the uncertainty in position from the highest frequency wavelets is 2/N. We might be able to refine that a bit by using information from the larger scale wavelets. It is also possible to find the uncertainty in frequency. It can be shown that for any linear system, there is a minimum for the joint uncertainty in position (space or time) versus frequency. The equation for a continuous system is Dt Df e" 1/(4p) where Dt is the uncertainty in time or position and Df is the uncertainty in frequency. The Gabor function produces the lowest possible joint uncertainty of 1/(4p). The time-frequency resolution limit is often confused with the Heisenberg Uncertainty Principle: A measurement of the energy of a particle performed during a time interval Dt must be uncertain by an amount DE, where the relationship between these two quantities is Dt DE e" h / (2p) The value for Plank s constant h is 6.63 * 10-27 erg-sec. One erg is 10-7 joules; One foot-pound is 1.356 joules. The Heisenberg Uncertainty Principle is a law of physics. The limitation on joint time-frequency resolution only applies to linear systems. It is entirely possible to build non-linear systems that do better than this. Mathematical considerations For digital compression we will always use discrete basis functions. We dont really have to worry much about the math behind the discrete functions, other than to insist that they be linearly independent. The main theory behind Fourier, Wavelet, Fractal and other transforms is based on continuous functions. These have more demanding requirements: Basis functions should be continuous Basis functions should be smooth (all derivatives should exist) Basis functions should be orthogonal: +"fi fj = 0 for i `" j. Basis functions should be normal: : +"fi fi = 1 Wavelets should be identically 0 outside of a finite region (compact support). There should be a computationally efficient algorithm to find the coefficients needed to express any function in terms of the basis functions. It should be possible to get a close approximation to any function (completeness). The representation should be stable: If you change an image just a little bit, its transform should change by just a little bit. The transform for an image should be unique. It is impossible to simultaneously satisfy all these conditions. Frequency transforms do not have compact support. Windowed Cosine or Fourier transforms are either not orthogonal or not complete. The Gabor wavelet transform is not orthogonal. Any non-orthogonal transform has more than one way to represent a function. This makes it impossible to do pattern recognition using the transformed image. Daubechies Wavelets There are complete orthogonal wavelet transforms with compact support. This was something of a mathematical surprise, with most of the initial work done in the 1980s. One of the most popular orthogonal wavelet transforms was invented by Ingrid Daubechies of AT&T Bell Labs. These are very jagged.  All orthogonal wavelet transforms are unstable. If you translate an image by a single pixel, you may get a completely different representation using a Daubechies wavelet transform. Wavelets for Image Compression Wavelets are very effective for image compression. JPEG is in the process of writing a new standard [JPEG 2000] that will replace the DCT with a wavelet transform. Wavelet transforms do not introduce the blockiness that can result with JPEG compression. The coefficients for a wavelet transform can be encoded very compactly using an algorithm such as the Embedded Zerotree Wavelet (EZW) coder or Set Partitioning in Hierarhical Trees (SPIHT). A commercial package for wavelet image compression is available from a Seattle company called LizardTech ( HYPERLINK "http://www.lizardtech.com" www.lizardtech.com). It uses a proprietary method developed at Los Alamos for satellite images. They call their format MrSID (Multiresolution Seamless Image Database). The technology is patented, but a wavelet compression program can be downloaded for free. It allows you do compressions ranging from 2:1 to 50:1. Reviewed by  HYPERLINK "MrSID--PC Magazine.htm" PC Magazine JPEG 2000 The current version of JPEG works well until you try to compress beyond about 32:1. Using the DCT on 8x8 tiles sometime produces a blocky looking image. A new standard, JPEG 2000, is under development. It will use wavelet compression and a scheme called Embedded Block Coding with Optimized Truncation (EBCOT). This starts with a 2x2 image decomposition, then breaks each section into four more 2x2 quads. 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