Compressed Sensing - Achuta Kadambi

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  • 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?
  • E.g.
  • 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
  • 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