Compressed Sensing: Introduction and Apps Achuta Kadambi Camera Culture, MIT
Exploiting Signals Not all signals are equal! Find a weakness then exploit.
Exploiting Signals Not all signals are equal! Find a weakness then exploit. Shannon-Nyquist Bandlimited signals can be sampled/reconstructed
Exploiting Signals Not all signals are equal! Find a weakness then exploit. Shannon-Nyquist Bandlimited signals can be sampled/reconstructed Rank-constrained Optimization Low Rank signals can be interpolated (Netflix Problem)
Exploiting Signals Not all signals are equal! Find a weakness then exploit. Shannon-Nyquist Bandlimited signals can be sampled/reconstructed Rank-constrained Optimization Low Rank signals can be interpolated. (Netflix Problem) Compressed Sensing Sparse signals can be undersampled and recovered.
Outline of this talk. Compressed Sensing overview. Very brief explanation on the why and how of Compressed Sensing. Apps that use compressed sensing. Practical strategies for implementation (e.g. pseudocode, libraries).
Motivation: JPEG Compression Our visual system is less sensitive to high (spatial) frequency detail. Can we throw away these frequencies and retain a similar image? This is the intuition behind JPEG. Spatial Frequency E.g.: High Hair, Blades of Grass, etc. Low Sky, Skin, etc. Compressed Sensing: If we are going to throw away stuff why spend time acquiring it?
Wired Magazine: Fill in the Blanks
1D Implementation in L1Magic Step 1: The original signal and its Fourier Transform. Original Signal (N = 256) Spectrum
Implementation in L1Magic Step 2: The subsampled signal Red Entries (80 samples) are observed. Blue Entries (176 samples) must be recovered. That means we observe only 30% of the original signal.
Implementation in L1Magic Step 3: Exact Recovery of the Signal. Original Signal (N = 256) Reconstruction (N = 256)
L1Magic for Images Original Image: 1 million pixels Reconstruction: from 100,000 random measurements.
Goes back to Fourier
Fourier Transform Intuition: Projection, or Inner Product, of Signal with Trigonometric Functions.
Sparsity goes back to Fourier (circa 1800) Superposition of Sinusoids Original Time Domain Function Frequency Domain Representation
Discrete Fourier Transform Example DFT: Time Signal is a Delta. Spectrum is Broadband.
DFT in Matrix Form
Nyquist-Shannon Sampling Theorem In Shannons words: How to Reconstruct? (Interpolation) Compressive Sensing: Can we do better?
Inverse Problem Example 1: Sinc Interpolation. Given the Data (a sufficiently sampled signal), how can we obtain the original signal? Example 2: Blurry Photos. Given a Blurry Photo, from a Camera, how can we go back to the original, sharp image? Example 3: Given a discrete time signal, how can we obtain its discrete spectrum? **DFT problem is a Linear Inverse Problem
Solving the DFT Problem Done?
Solving the DFT Problem via Optimization Done? Loss Function
Solving the DFT Problem via Optimization Done? PseudoInverse: Minimize MSE
Constraining our Solution via Regularization We can go beyond loss function, e.g., Tikhonov Regularization Additional Term allows for some prior on original signal. For instance if Tikhonov Matrix is a first order difference, then you are biasing x toward smooth solutions. Linked to the Lagrange problem, as well as Maximum A Posteriori from probability, and Weiner filter from Sig proc.
Compressed Sensing Structure Underdetermined system. y=Ax. Y is m-dimensional sampled vector A is mxn matrix X is n-dimensional original vector. And m