Compressed sensing

1. Compressed sensing 3D Digitization Course Carlos Becker, Guillaume Lemaître & Peter Rennert

2. Outline Introduction and motivation Compressed sensing and reconstruction workflow Applications: MRI and single-pixel camera Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

3. What is compressed sensing?Signal sparsity Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

4. What is compressed sensing?Why do we care about sparsity? Original 1 Megapixel image Non-sparse values Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

5. What is compressed sensing?Why do we care about sparsity? But, in the wavelet domain we get these coefficients: Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

6. What is compressed sensing?Why do we care about sparsity? The image is a nearly sparse in the wavelet domain… And the histogram of those coefficients is: Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

7. Reconstructed image • (only 50k highest wavelet coefficients) Original image 95% of the original image data was discarded What happens if we only keep the 50000 highest coefficients in the wavelet domain, set the rest to zero and reconstruct the image ? Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

8. What is compressed sensing? Full-resolution image (N pixels/measurements) Lossy compression Candès et al. showed that it is possible to subsample a signal if it is sparse in some domain, being able to obtain a perfect reconstruction if certain conditions are met. Random sampling (M << N measurements) Image reconstruction • Classic approach to compression: • Measure everything (ie: all pixels) • Apply some compression algorithm (ie: JPEG2000) • But, why would we sample 1 million pixels if we are going to throw away 90% of image data when compressing the image in JPEG? • Compressed sensing approach: if signal is sparse in some domain • Sample M << N random measurements • Reconstruct original signal by L1 minimization Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

9. Compressed SensingMotivation from mri • 2004, Candes came to results that people of his time could not believe • For a simple phantom(a) its possible to sample at only 22 radial lines (b) (equal to a sampling rate of π / 22, about 50 times below the Nyquist rate of 2 π) to retrieve a perfect reconstruction (d) • What does the trick? Simply setting the unknown Fourier coefficients to 0 leads to a very bad result (c) Candès, E.J.; Romberg, J.; Tao, T.: “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information” (2004) Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

10. Compressed sensingReconstruction Workflow • Sparse signal gets randomly sampled in another non-sparse domain (k-space) • Reconstruction leads to noisy non-sparse signal with significant peaks where original signal was high • After thresholding of significant peaks the strongest components of the original signal are detected • Using the noisy reconstruction of the newly sampled strongest components in k-space, the impact of this strongest components on the first reconstruction are determined and subtracted, leaving peaks of weaker components • With this iterative strategy weaker and weaker components can be retrieved Michael Lustig, David Donoho, John M. Pauly: “Sparse MRI: The application of compressed sensing for rapid MR imaging” (2007) DL Donoho, I Drori, Y Tsaig : Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit” (2006) Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

11. Compressed sensingRapid MRI – non-sparse Signal (Signal here means the underlying image that is sensed in the Fourier space) • Non- sparse signal sampled in sparse domain • That means: reconstruction of samples will produce no significant peaks (since there are no outstanding peaks in the signal domain) • Solution: use other sparse domain of signal for “reconstruction” and filtering of significant peaks Michael Lustig, David Donoho, John M. Pauly: “Sparse MRI: The application of compressed sensing for rapid MR imaging” Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

12. Single pixel cameraGeneral principle Object Photodiode Σ Memory SeveralMeasurements Reconstruction DMD M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, "Single-Pixel Imaging via Compressive Sampling," IEEE Signal Processing Magazine, Vol. 25, No. 2, pp. 83-91, March 2008 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

13. Single pixel cameraResults 2 % measurements 5 % measurements Original: 16384 pixels 10 % measurements 20 % measurements M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, "Single-Pixel Imaging via Compressive Sampling," IEEE Signal Processing Magazine, Vol. 25, No. 2, pp. 83-91, March 2008 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert

14. Compressed sensingconclusion • Compressed sensing lets us sub-sample a signal w.r.t. Nyquist rate and reconstruct it perfectly, if this signal is known to be sparse in some domain and some conditions are met • Compressed sensing is promising for a wide range of future technologies, specially for high-frequency signals • Speeds up acquisition process: specially interesting for MRI • Cheaper hardware (ie: IR cameras with only a few sensors) • Sparsity can also be exploited for classification and image processing tasks[Huang, K. and Aviyente, S., Sparse representation for signal classification] Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert