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# COMPRESSED SENSING Luis Mancera Visual Information Processing Group

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COMPRESSED SENSING Luis Mancera Visual Information Processing Group
Dep. Computer Science and AI Universidad de Granada

CONTENTS WHAT? HOW? FOR WHAT PURPOSE? AND THEN?
Introduction to Compressed Sensing (CS) HOW? Theory behind CS FOR WHAT PURPOSE? CS applications AND THEN? Active research and future lines

CONTENTS WHAT? HOW? FOR WHAT PURPOSE? AND THEN?
Introduction to Compressed Sensing (CS) HOW? Theory behind CS FOR WHAT PURPOSE? CS applications AND THEN? Active research and future lines

Transmission scheme Brick wall to performance N >> K Sample N
Compress K Transmit Why so many samples? N K Decompress Receive Natural signals (sparse/compressible)  no significant perceptual loss

Shannon/Nyquist theorem
Shannon/Nyquist theorem tell us to use a sampling rate of 1/(2W) seconds, if W is the highest frequency of the signal This is a worst-case bound for ANY band-limited signal Sparse / compressible signals is a favorable case CS solution: melt sampling and compression

Compressed Sensing (CS)
Receive Reconstruct M K < M << N Transmit N What do we need for CS to success? Recover sparse signals by directly acquiring compressed data Replace samples by measurements

We now how to Sense Compressively
I’m glad this battle is over. Finally my military period is over. I will now come back to Motril and get married, and then I will grow up pigs as I have always wanted to do Do you mean you’re glad this battle is over because now you’ve finished here and you will go back to Motril, get married, and grow up pigs as you always wanted to? Aye Cool!

What does CS need? Nice sensing dictionary Appropriate sensing
I know this guy so much that I know what he means Nice sensing dictionary Appropriate sensing A priori knowledge Recovery process Saint Roque’s dog has no tail Wie lange wird das nehmen? Cool! What? Words Idea

CS needs: Nice sensing dictionary Appropriate sensing
A priori knowledge Recovery process INCOHERENCE RANDOMNESS SPARSENESS OPTIMIZATION

Sparseness: less is more
Dictionary: “He was advancing by the valley, the only road traveled by a stranger approaching the Hut” Comments to Wyandotte Idea: A stranger approaching a hut by the only known road: the valley How to express it? Hummm, you could say the same using less words… Combining elements… SPARSER Combining elements… “He was advancing by the only road that was ever traveled by the stranger as he approached the Hut; or, he came up the valley” Wyandotte J.F. Cooper E.A. Poe

Sparseness: less is more
Sparseness: Property of being small in numbers or amount, often scattered over a large area [Cambridge Advanced Learner’s Dictionary] Cualidad de algo cuyos componentes están más separados de lo común en su clase A CERTAIN DISTRIBUTION A SPARSER DISTRIBUTION

Sparseness: less is more
Pixels: not sparse  A new domain can increase sparseness  Taking 10% pixels Original Einstein 10% Fourier coeffs. 10% Wavelet coeffs.

Sparseness: less is more
Dictionary: How to express it? X-lets elementary functions (atoms) Non-linear analysis SPARSER Linear analysis non-linear subband Synthesis-sense Sparseness: We can increase sparseness by non-linear analysis X-let-based representations are compressible, meaning that most of the energy is concentrated in few coefficients Analysis-sense Sparseness: Response of X-lets filters is sparse [Malllat 89, Olshausen & Field 96] linear subband

Sparseness: less is more
Idea: Dictionary: How to express it? X-lets elementary functions Combining other way… SPARSER Taking around 3.5% of total coeffs… non-linear subband Taking less coefficients we achieve strict sparseness, at the price of just approximating the image PSNR: dB

Incoherence Sparse signals in a given dictionary must be dense in another incoherent one Sampling dictionary should be incoherent w.r.t. that where the signal is sparse/compressible A time-sparse signal Its frequency-dense representation

Measurement and recovery processes
Measurement process: Sparseness + Incoherence  Random sampling will do Recovery process: Numerical non-linear optimization is able to exactly recover the signal given the measurements

CS relies on: A priori knowledge: Many natural signals are sparse or compressible in a proper basis Nice sensing dictionary: Signals should be dense when using the sampling waveforms Appropriate sensing: Random sampling have demonstrated to work well Recovery process: Bounds for exact recovery depends on the optimization method EL TERCER PASO DEPENDE DEL PRIMERO Y DEL SEGUNDO. SI NO HAY INCOHERENCIA HABRÍA QUE CREAR UN MÉTODO AD-HOC DE MUESTREO

Summary CS is a simple and efficient signal acquisition protocol which samples at a reduced rate and later use computational power for reconstruction from what appears to be an incomplete set of measurements CS is universal, democratic and asymmetrical

CONTENTS WHAT? HOW? FOR WHAT PURPOSE? AND THEN?
Introduction to Compressed Sensing (CS) HOW? Theory behind CS FOR WHAT PURPOSE? CS applications AND THEN? Active research and future lines

The sensing problem xt: Original discrete signal (vector)
F: Sampling dictionary (matrix) yk: Sampled signal (vector) sensing mechanisms in which information about a signal f(t) is obtained by linear functionals recording the values

The sensing problem Traditional sampling: N x 1 y N x N F = I
Sampled signal Sampling dictionary Original signal

The sensing problem When the signal is sparse/compressible, we can directly acquire a condensed representation with no/little information loss Random projection will work if M = O(K log(N/K)) [Candès et al., Donoho, 2004] y F x K nonzero entries K < M << N Random projection Phi is not full rank but 1) preserves structure and information, 2) is invertible, in sparse/compressible signal models with high probability M x 1 M x N N x 1

Universality Random measurements can be used if signal is sparse/compressible in any basis y F Y a K nonzero entries K < M << N M x 1 M x N N x N N x 1

Good sensing waveforms?
F and Y should be incoherent Measure the largest correlation between any two elements: Large correlation  low incoherence Examples Spike and Fourier basis (maximal incoherence) Random and any fixed basis The coherence between the sensing system Phi and the representation system Psi is

Solution: sensing randomly
M = O(K log(N/K)) Random measurements M Transmit N M Reconstruct Receive We have set up the encoder Let’s now study the decoder

CS recovery Assume a is K-sparse, and y = FYa
We can recover a by solving: This is a NP-hard problem (combinatorial) Use some tractable approximation Count number of active coefficients

Robust CS recovery What about a is only compressible and y = F(Ya + n), with n and unknown error term? Isometry constant of F: The smallest K such that, for all K-sparse vectors x: F obeys a Restricted Isometry Property (RIP) if dK is not too close to 1 F obeys a RIP  Any subset of K columns are nearly orthogonal To recover K-sparse signals we need d2K < 1 (unique solution)

Recovery techniques Minimization of L1-norm Greedy techniques
Iterative thresholding Total-variation minimization

Recovery by minimizing L1-norm
Sum of absolute values Convexity: tractable problem Solvable by Linear or Second-order programming For C > 0, â1 = â if:

Recovery by minimizing L1-norm
Noisy data: Solve the LASSO problem Convex problem solvable via 2nd order cone programming (SOCP) If d2K < 2 – 1, then: 0.4142

Example of L1 recovery A120X512: Random orthonormal matrix
Perfect recovery of x by L1-minimization

Recovery by Greedy Pursuit
Algorithm: New active component: that whose corresponding fi is most correlated with y Find best approximation, y’, to y using active components Substract y’ from y to form residual e Make y = e and repeat Very fast for small-scale problems Not as accurate/robust for large signals in the presence of noise

Recovery by Iterative Thresholding
Algorithm: Iterates between shrinkage/thresholding operation and projection onto perfect reconstruction If soft-thresholding is used, analogous theory to L1-minimization If hard-thresholding is used, the error is within a constant factor of the best attainable estimation error [Blumensath08]

Recovery by TV minimization
Sparseness: signals have few “jumps” Convexity: tractable problem Accurate and robust, but can be slow for large-scale problems Resoluble mediante IP methods of proyección de gradiente

Example of TV recovery F: Fourier transform
xLS = FTFx F: Fourier transform Perfect recovery of x by TV-minimization

Summary Sensing: Recovery:
Use random sampling in dictionaries with low coherence to that where the signal is sparse. Choose M wisely Recovery: A wide range of techniques are available L1-minimization seems to work well, but choose that best fitting your needs

CONTENTS WHAT? HOW? FOR WHAT PURPOSE? AND THEN?
Introduction to Compressed Sensing (CS) HOW? Theory behind CS FOR WHAT PURPOSE? CS applications AND THEN? Active research and future lines

Some CS applications Data compression Compressive imaging
Detection, classification, estimation, learning… Medical imaging Analog-to-information conversion Biosensing Geophysical data analysis Hyperspectral imaging Compressive radar Astronomy Comunications Surface metrology Spectrum analysis

Data compression The sparse basis Y may be unknown or impractical to implement at the encoder A randomly designed F can be considered a universal encoding strategy This may be helpful for distributed source coding in multi-signal settings [Baron et al. 05, Haupt and Nowak 06,…]

Magnetic resonance imaging

Rice Single-Pixel CS Camera

Rice Analog-to-Information conversion
Analog input signal into discrete digital measurements Extension of A2D converter that samples at signal’s information rate rather than its Nyquist rate

CS in Astronomy [Bobin et al 08]
Desperate need for data compression Resolution, Sensitivity and photometry are important Herschel satellite (ESA, 2009): conventional compression cannot be used CS can help with: New compressive sensors A flexible compression/decompression scheme Computational cost (Fx): O(t) vs. JPEG 2000’s O(t log(t)) Decoupling of compression and decompression CS outperforms conventional compression Herschel: necesita mucha compresión y poco gasto de CPU simultáneamente. USA HARD-THRESHOLDING Y DESCENSO DE UMBRAL

CONTENTS WHAT? HOW? FOR WHAT PURPOSE? AND THEN?
Introduction to Compressed Sensing (CS) HOW? Theory behind CS FOR WHAT PURPOSE? CS applications AND THEN? Active research and future lines

CS is a very active area

CS is a very active area More than seventy 2008 papers in CS repository Most active areas: New applications (de-noising, learning, video, New recovery methods (non-convex, variational, CoSamp,…) ICIP 08: COMPRESSED SENSING FOR MULTI-VIEW TRACKING AND 3-D VOXEL RECONSTRUCTION COMPRESSIVE IMAGE FUSION IMAGE REPRESENTATION BY COMPRESSED SENSING KALMAN FILTERED COMPRESSED SENSING NONCONVEX COMPRESSIVE SENSING AND RECONSTRUCTION OF GRADIENT-SPARSE IMAGES: RANDOM VS. TOMOGRAPHIC FOURIER SAMPLING

Conclusions CS is a new technique for acquiring and compressing images simultaneously Sparseness + Incoherence + random sampling allows perfect reconstruction under some conditions A wide range of applications are possible Big research effort now on recovery techniques

Our future lines? Convex CS: Non-convex CS: CS Applications:
TV-regularization Non-convex CS: L0-GM for CS Intermediate norms (0 < p < 1) for CS CS Applications: Super-resolved sampling? Detection, estimation, classification,… Non-convex cost function seems to work well in practice and there are also some theoretical results

See references and software here: http://www.dsp.ece.rice.edu/cs/
Thank you See references and software here:

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