By Peter G. Casazza, Gitta Kutyniok
Hilbert house frames have lengthy served as a helpful instrument for sign and photo processing because of their resilience to additive noise, quantization, and erasures, in addition to their skill to catch important sign features. extra lately, finite body idea has grown into an enormous examine subject in its personal correct, with a myriad of functions to natural and utilized arithmetic, engineering, computing device technology, and different components. The variety of learn guides, meetings, and workshops in this subject has elevated dramatically over the last few years, yet no survey paper or monograph has but seemed at the subject.
Edited via of the top specialists within the box, Finite Frames goals to fill this void within the literature by way of offering a accomplished, systematic learn of finite body conception and functions. With rigorously chosen contributions written by way of hugely skilled researchers, it covers themes including:
* Finite body Constructions;
* optimum Erasure Resilient Frames;
* Quantization of Finite Frames;
* Finite Frames and Compressed Sensing;
* workforce and Gabor Frames;
* Fusion Frames.
Despite the diversity of its chapters' resource and content material, the book's notation and terminology are unified all through and supply a definitive photograph of the present nation of body theory.
With a huge variety of purposes and a transparent, complete presentation, this booklet is a hugely worthy source for graduate scholars and researchers throughout disciplines comparable to utilized harmonic research, electric engineering, quantum computing, drugs, and extra. it really is designed for use as a supplemental textbook, self-study consultant, or reference book.
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Extra resources for Finite Frames: Theory and Applications
13. 5 Filter Banks Filter banks are the basis for most signal processing applications. We exemplarily mention the general books [125, 145] and those with a particular focus on wavelets [75, 134, 150], as well as the beautiful survey articles [109, 110]. Usually, several filters are applied in parallel to an input signal, followed by downsampling. This processing method is closely related to the decomposition with respect to finite frames provided that the frame consists of equally spaced translates of a fixed set of vectors, 1 Introduction to Finite Frame Theory 47 first observed in [19, 21, 71, 72] and later refined and extended in [62, 63, 90, 112].
Notice that the first formula can be interpreted as a reconstruction strategy, whereas the second formula has the flavor of a decomposition. 8. The next result shows that this sequence indeed also constitutes a frame. 13 Let (ϕi )M i=1 be a frame for H −1 N with frame with frame operator S. Then the sequence (S ϕi )M i=1 is a frame for H −1 −1 −1 bounds B and A and with frame operator S . 10, the sequence (S −1 ϕi )M i=1 forms a frame for H −1 −1 ∗ −1 associated frame operator S S(S ) = S . This in turn yields the frame bounds B −1 and A−1 .
M, we M have T ∗ ei = T ∗ P ei = ϕi , where P : M 2 → 2 denotes the orthogonal projection onto ran T . ∗ (ii) (ϕi )M i=1 is a frame if and only if T is surjective. 3 and the fact that ker T ∗ = (ran T )⊥ . 3(i). Often frames are modified by the application of an invertible operator. The next result shows not only the impact on the associated analysis operator, but also the fact that the new sequence again forms a frame. 9 Let Φ = (ϕi )M i=1 be a sequence of vectors in H N N ated analysis operator TΦ and let F : H → H be a linear operator.
Finite Frames: Theory and Applications by Peter G. Casazza, Gitta Kutyniok