A_Primer_on_Wavelets_and_Their

A_Primer_on_Wavelets_and_Their - DIGITAL VISION Imaging via...

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© DIGITAL VISION Digital Object Identifier 10.1109/MSP.2007.914729 IEEE SIGNAL PROCESSING MAGAZINE [ 14 ] MARCH 2008 1053-5888/08/$25.00©2008IEEE T he ease with which we store and transmit images in modern-day appli- cations would be unthinkable without compression. Image compression algorithms can reduce data sets by orders of magnitude, making sys- tems that acquire extremely high-resolution images (billions or even trillions of pixels) feasible. There is an extensive body of literature on image compression, but the cen- tral concept is straightforward: we transform the image into an appropriate basis and then code only the important expansion coefficients. The crux is finding a good transform, a problem that has been studied extensively from both a theo- retical [14] and practical [25] standpoint. The most notable product of this research is the wavelet transform [9], [16]; switching from sinusoid-based repre- sentations to wavelets marked a watershed in image compression and is the [ Justin Romberg ] Imaging via Compressive Sampling [ Introduction to compressive sampling and recovery via convex programming ] Authorized licensed use limited to: ULAKBIM UASL - DOGU AKDENIZ UNIV. Downloaded on April 04,2010 at 14:50:15 EDT from IEEE Xplore. Restrictions apply.
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IEEE SIGNAL PROCESSING MAGAZINE [ 15 ] MARCH 2008 essential difference between the classical JPEG [18] and modern JPEG-2000 [22] standards. Image compression algorithms convert high-resolution images into a relatively small bit streams (while keeping the essential features intact), in effect turning a large digital data set into a substantially smaller one. But is there a way to avoid the large digital data set to begin with? Is there a way we can build the data compression directly into the acquisition? The answer is yes, and is what compressive sampling (CS) is all about. To begin, we need to generalize our notion of “sampling” an image. Instead of collecting point evaluations of the image X at distinct locations, or averages over small areas (pixels), each measurement y k in our acquisition system is an inner product against a different test function φ k : y 1 =h X 1 i , y 2 X 2 i ,..., y m X m i .( 1 ) We note here that our entire discussion in this article (and the majority of the work to date in the field of CS) will revolve around finite dimensional signals and images. To make the transi- tion to acquisition of continuous- time (and -space) signals, we would choose a discretization space on which to apply the dis- crete theory. For example, we might assume that the image is (or can very closely approximated by) a gridded array of n pixels. The test functions φ k , which would also be pixelated, then give us measurements of the projection of the continuous image onto this discretization space. The choice of the φ k allows us to choose in which domain we gather information about the image. For example, if the φ k are sinusoids at different frequencies, we are essentially collecting Fourier coefficients (as in magnetic resonance imaging), if they are delta ridges, we are observing line integrals (as in tomogra-
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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A_Primer_on_Wavelets_and_Their - DIGITAL VISION Imaging via...

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