压缩感知PPT课件

Low-rate
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Concept
Goal: Identify the bucket with fake coins.
Nyquist:
Weigh a coin from each bucket
Compression numbers
Bucket # 1 number
Compressed Sensing:
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上次课内容回顾
Lecture 1： 压缩感知概述
• 为什么研究压缩感知 • 压缩感知的涵义 • 压缩感知的过程 • 压缩感知的关键问题
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From Nyquist to CS
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Compression
“Can we not just directly measure the part that will not end up being thrown
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Vector space
Unit spheres in for the norms with quasinorm with
is uniquely determined by
Donoho and Elad, 2003
with high probability
Donoho, 2006 and Candès et. al., 2006
Convex and tractable
Donoho, 2006 and Candès et. al., 2006
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CS theory
Compressed sensing (2003/4 and on) – Main results
is uniquely determined by
Donoho and Elad, 2003
Maximal cardinality of linearly independent column subsets
object from a small number of randomly selected
observations”
Candès et. al.
Analog Audio Signal
Nyquist rate Sampling
CoHmipgrhe-srsaetde Sensing
Compression (e.g. MP3)
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Our Point-Of-View
Compressed Sensing(CS) must be based on sparsity and compressibility. The signals must be sparse in time-domain or in frquency-domain.
5ห้องสมุดไป่ตู้
Sparse in wavelet-domain
Sparse approximation of a natural image. (a) Original image.(b) Approximation of image obtained by keeping only the largest 10% of the wavelet coefficients.
Smallest number of columns that are linearly-dependent. Hard to compute ! 11
CS theory
Compressed sensing (2003/4 and on) – Main results
is random NP-hard
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Compressed Sensing
“Can we not just directly measure the part that will not
end up being thrown away ?”
Donoho
“sensing … as a way of extracting information about an
XIDIAN University
压缩感知理论与应用
智能感知与图像理解教育部重点实验室
2011年8月
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away ?”
Donoho
Original 2500 KB 100%
Compressed 319945280 KB 13658%%
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Sparse in wavelet-domain
Sparse representation of an image via a multiscale wavelet transform. (a) Original image. (b) Wavelet representation. Large coefficients are represented by light pixels, while small coefficients are represented by dark pixels. Observe that most of the wavelet coefficients are close to zero.
Greedy algorithms: OMP, FOCUSS, etc.
Tropp, Cotter et. al. Chen et. al. and many other
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RIP criterion
(a)The measurements can maintain the energy of the original timedomain signal . (b)If is sparse, the must be dense to maintain the energy.
Weigh a linear combination of coins from all buckets
Bucket #
1 number 9
Mathematical Tools
y
Ax
non-zero entries at least measurements
Recovery: brute-force, convex optimization, greedy algorithms, and more…