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- 1. Compressed Sensing: Introduction and Apps Achuta Kadambi Camera Culture, MIT
- 2. Exploiting Signals • Not all signals are equal! Find a weakness then exploit.
- 3. Exploiting Signals • Not all signals are equal! Find a weakness then exploit. • Shannon-Nyquist Bandlimited signals can be sampled/reconstructed
- 4. 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)
- 5. 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.
- 6. 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).
- 7. 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?
- 8. E.g.
- 9. Wired Magazine: “Fill in the Blanks…”
- 10. 1D Implementation in L1Magic Step 1: The original signal and its Fourier Transform. Original Signal (N = 256) Spectrum
- 11. 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.
- 12. Implementation in L1Magic Step 3: Exact Recovery of the Signal. Original Signal (N = 256) Reconstruction (N = 256)
- 13. L1Magic for Images Original Image: 1 million pixels Reconstruction: from 100,000 random measurements.
- 14. Goes back to Fourier
- 15. Fourier Transform Intuition: Projection, or Inner Product, of Signal with Trigonometric Functions.
- 16. Sparsity goes back to Fourier (circa 1800) Superposition of Sinusoids Original Time Domain Function Frequency Domain Representation
- 17. Discrete Fourier Transform Example DFT: Time Signal is a Delta. Spectrum is Broadband.
- 18. DFT in Matrix Form
- 19. Nyquist-Shannon Sampling Theorem In Shannon’s words: How to Reconstruct? (Interpolation) Compressive Sensing: Can we do better?
- 20. 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
- 21. Solving the DFT Problem Done?
- 22. Solving the DFT Problem via Optimization Done? Loss Function
- 23. Solving the DFT Problem via Optimization Done? PseudoInverse: Minimize MSE
- 24. 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.
- 25. Compressed Sensing
- 26. 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 << n y A x
- 27. Simply Solving y=Ax not good enough • This gives you an affine space with many solutions to y=Ax. • So we must constrain our problem to look for the sparse solution to y=Ax.
- 28. Occams Razor Occam's Razor: among otherwise equal explanations, the simplest is best
- 29. Occams Razor Occam's Razor: among otherwise equal explanations, the simplest is best CS Occam's Razor: among otherwise equal solutions, the sparsest is best Unfortunately, this optimization is not tractable
- 30. Geometric Property of Norms
- 31. The l1 Optimization Problem
- 32. RIP/Spark/Coherence • The sensing matrix A must be carefully chosen. • For compressed sensing to work, the matrix A must satisfy the Restricted Isometry Property (RIP): • Calculating RIP is NP-hard. We can work with easier quantities than the RIP, such as spark and mutual incoherence.
- 33. App1: Single-Pixel Camera
- 34. App1: Single-Pixel Camera Design Advantage: A MP camera with just a single-pixel.
- 35. App2: Single-Pixel THz imaging. Design Advantage: CS allows for single-pixel THz sensors, which are much easier to fabricate than pixel array. In general, you can buy amazing things at single-pixel level, e.g., picosecond detectors, thermal IR sensor, etc. Chan et al. Applied Physics 2008
- 36. App2: Monitoring Breathing via Smartphone Very similar to OMP, DFT formulation. Basically finding sparse spectral components that characterize the audio signal of breathing. Design Advantage: Compressed sensing allows for low power acquisition and reduced streaming. Oletic, Skrapec, and Bilas MobiHealth 2012
- 37. App4: Biometrics … Face Recognition Design Advantage: Using compressed sensing to handle the small sample size problem. Before, the number of samples in the database Is less than the degrees of freedom of each sample.
- 38. App5: Fast MRI Design Advantage: Less samples means less time for an MRI Scan, which means less time a sick or disabled patient lies in the scanner.
- 39. App5: Fast MRI
- 40. App6: Compressive Sensing of High Speed Periodic Videos Design Advantage: Exploit sparsity of Periodic Videos to obtain a high speed video without using a high speed camera. Veeraraghavan A, Reddy D, Raskar R. IEEE PAMI
- 41. App7: Compressive Light Field Photography Marwah, Wetzstein, Bando, Ra skar. ACM SIGGRAPH 2013. Design Advantage: Obtain High-Resolution Light Field photos by placing a coded mask in front of the sensor.
- 42. App8: Sparsity-Induced Time of Flight Cameras Kadambi et al. ACM SIGGRAPH Asia E.g., Light Sweep Movies from Refael’s talk last week. Goal is to obtain a well-conditioned deconvolution problem. Design Advantage: Deconvolve to obtain bounces of light and construct a light sweep video.
- 43. App9 YOUR App! Design Advantage: <insert here>
- 44. Practical Strategies Many libraries are available for C++/Matlab/etc. • Recommended: L1magic (http://users.ece.gatech.edu/~justin/l1magic/) • SPGL1 (http://www.cs.ubc.ca/~mpf/spgl1/) • CVX (http://cvxr.com/cvx/) • On phone, nothing exists yet, but you can use Eﬃcient Java Matrix Library (EJML) to implement solely in linear algebra.
- 45. L1Magic Pseudocode For 1D signal. x = original_signal; R = randn(m,n); A = orth(R’)’; // read in your original signal. // create a random matrix of dimension mxn // sensing matrix with orthogonal columns. y = A*x; // create subsampled signal y of only m entries. X0 = A’*y; // Initial guess by taking matrix inverse. x_hat = l1eq_pd(x0, A, [], y, 1e-3); // run l1 solver. norm(x_hat – x); // error; should be zero in ideal case
- 46. Take-home Messages • Opportunity to integrate cutting-edge mathematical techniques into your camera apps. • Compressed Sensing is lightweight in terms of coding. The key is correctly identifying the sparsity in your engineering problem. • Not all signals are equal… find a weakness, e.g., sparsity/rank, and exploit it. • Exploit in Hardware, Exploit in Software…

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