Read e-book online Discrete Fourier Analysis and Wavelets: Applications to PDF

By S. Allen Broughton

ISBN-10: 0470294663

ISBN-13: 9780470294666

A thorough advisor to the classical and modern mathematical tools of recent sign and photo processing

Discrete Fourier research and Wavelets offers a radical creation to the mathematical foundations of sign and photo processing. Key techniques and purposes are addressed in a thought-provoking demeanour and are carried out utilizing vector, matrix, and linear algebra equipment. With a balanced concentrate on mathematical concept and computational thoughts, this self-contained ebook equips readers with the fundamental wisdom had to transition easily from mathematical types to useful electronic facts applications.

The e-book first establishes an entire vector house and matrix framework for studying signs and pictures. Classical equipment reminiscent of the discrete Fourier remodel, the discrete cosine remodel, and their program to JPEG compression are defined by way of insurance of the Fourier sequence and the overall conception of internal product areas and orthogonal bases. The ebook then addresses convolution, filtering, and windowing innovations for indications and photographs. ultimately, sleek ways are brought, together with wavelets and the idea of filter out banks as a way of realizing the multiscale localized research underlying the JPEG 2000 compression standard.

Throughout the publication, examples utilizing picture compression reveal how mathematical conception interprets into program. extra functions akin to innovative transmission of pictures, snapshot denoising, spectrographic research, and area detection are mentioned. each one bankruptcy presents a chain of workouts in addition to a MATLAB undertaking that enables readers to use mathematical options to fixing actual difficulties. extra MATLAB workouts can be found through the book's comparable internet site.

With its insightful remedy of the underlying arithmetic in photo compression and sign processing, Discrete Fourier Analysis and Wavelets is a perfect publication for arithmetic, engineering, and computing device technological know-how classes on the upper-undergraduate and starting graduate degrees. it's also a helpful source for mathematicians, engineers, and different practitioners who wish to study extra in regards to the relevance of arithmetic in electronic information processing.

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Extra info for Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing

Example text

1, certain other arithmetic properties of vector spaces are worth noting. 1 / / V is-a vector space over R or C, then 1. the vector 0 is unique; 2. Ou = 0 for any vector u; 3. the additive inverse of any vector u is unique, and is given by (—l)u. 9. They also hold in any vector space, and can be shown directly from the eight arithmetic properties for a vector space. The careful proofs can be a bit tricky though! 12. 5). The vector space structure will be indispensable when we discuss the decomposition of signals and images into linear combinations of more "basic" components.

More precisely, the frequency (in Hertz) of the waveform e ' \%\q \ Hertz, since frequency is by convention nonnegative. In real-valued terms; we can use cos(2nqt) and sin(27T^f) with q > 0 in place of cos(ci;i) and sin(a)i) with ω > 0. 16) k=—oo for an appropriate choice of the constants c*. The natural frequency parameter ω assumes the values nk/T for k € Z, or equivalently the frequency q assumes values k/2T. 16) also exists using the sine/cosine waveforms. 2 2D Basic Waveforms The 2D basic waveforms are governed by a pair of frequency parameters a and β.

However, no matter which range in k we use to index, we'll always be using the same set of Ν vectors since E/v,* = EN^-NWhen there is no potential confusion, we will omit the Ν index and write simply E*, rather than E # , t . _1 — = 4 1" —I "1" E , 4 -1 1 = 0 1 i_ " 1" " -1 Ε'4,1 E4,2 = _1_ E43 1 = 1" —i -1 /_ Note the aliasing relations E 4 , _ 2 = E 2 and E ,_i = £ 4 , 3 . In particular, the sets {E4,_i, E , E i , E , } and {E4,o, E4,i, £ 4 , 2 , E 4 3 } (corresponding to E on the range —N/2 < k < Ν/2 or 0 < k < Ν - 1) are identical.

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Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing by S. Allen Broughton

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