Method and apparatus for distributed compressed sensing

• US 7,511,643 B2
• Filed: 08/08/2007
• Issued: 03/31/2009
• Est. Priority Date: 05/10/2005
• Status: Active Grant

• Current Assignee: Rice University
• Original Assignee: Rice University
• Inventors: Baron, Dror Z., Wakin, Michael B., Duarte, Marco F., Davenport, Mark, Sarvotham, Shriram, Baraniuk, Richard G.
• Primary Examiner(s): JEANGLAUDE, JEAN BRUNER
• Application Number:
  US11/836,086
• Publication Number:
  US 20080129560A1
• Time in Patent Office: 601 Days
• Field of Search: 341/87, 341/50, 348/418.1, 348/420.1, 375/148, 342/357.12, 706/20
• US Class Current: 341/87
• CPC Class Codes: G06F 18/213 Feature extraction, e.g. by transforming the feature space; Summarisation; Mappings, e.g. subspace methods
• CPC Class Codes: G06F 2218/22 Source localisation; Inverse modelling
• CPC Class Codes: H03M 7/30 Compression speech analysis-synthesis for redundancy reduction G10L19/00; for image communication H04N; Expansion; Suppression of unnecessary data, e.g. redundancy reduction
• CPC Class Codes: H04N 19/60 using transform coding

Abstract

A method for approximating a plurality of digital signals or images using compressed sensing. In a scheme where a common component x_c of said plurality of digital signals or images an innovative component x_i of each of said plurality of digital signals each are represented as a vector with m entries, the method comprises the steps of making a measurement y_c, where y_c comprises a vector with only n_i entries, where n_i is less than m, making a measurement y_i for each of said correlated digital signals, where y_i comprises a vector with only n_i entries, where n_i is less than m, and from each said innovation components y_i, producing an approximate reconstruction of each m-vector x_i using said common component y_c and said innovative component y_i.

Claims

1. A method for reconstructing a plurality of signals, images, or video sequences, comprising the steps of:

taking M_j measurements of each signal, image or video sequence x_j among the plurality of J signals, images, or video sequences;

reconstructing the plurality of signals, images or video sequences by seeking possible values for the plurality of signals, images or video sequences that explain the measurements and account for commonality and/or structure shared among the signals, images, or video sequences.

2. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1, wherein said method reconstructs said signals, images or video sequences approximately.

3. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1,

wherein in said scheme each of the plurality of signals, images or video sequences is measured using the same number M of measurements.

4. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1,

wherein in said scheme for each signal, image or video sequence x_j among the plurality of J signals, images or video sequences there exists an N-parameter representation and the M_j measurements of x_j are obtained by projecting x_j onto a plurality of measurement vectors.

5. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1,

wherein in said scheme for each signal, image or video sequence x_j among the plurality of J signals, images or video sequences there exists a discrete representation of length N and the M_j measurements of x_j are obtained by projecting x_j onto a plurality of measurement vectors.

6. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1,

wherein in said scheme for each signal, image or video sequence x_j among the plurality of J signals, images or video sequences there exists a discrete representation of length N and the number of measurements M_j taken of x_j is smaller than N.

7. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1,

wherein in said scheme the M_j measurements of each of said signals, images or video sequences x_j among the plurality of J signals, images or video sequences are encoded.

8. A method for reconstructing a plurality of signals, images, or video sequences according to claim 7, wherein said scheme the M_j measurements of each of said signals, images or video sequences x_j among the plurality of J signals, images or video sequences are encoded digitally.

9. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 4,

wherein in said scheme the M_j measurements of each signal, image or video sequence x_j among the plurality of J signals, images or video sequences is measured using different sets of measurement vectors.

10. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 3,

wherein in said scheme the M measurements of each of the plurality of signals, images or video sequences are measured using the same plurality of measurement vectors.

11. A method for reconstructing a plurality of signals, images, or video sequences according to claim 4,

wherein in said scheme the M_j measurements of each signal, image or video sequence x_j among a plurality of J signals, images or video sequences comprises a subset of measurements obtained from projections onto the same set of measurement vectors for each signal and another subset of measurements obtained from projections onto different measurement vectors.

12. A method according to claim 4 wherein said means for measuring measures said plurality of signals, images or video sequences using an incoherent measurement matrix.

13. A method for reconstructing a plurality of signals, images, or video sequences according to claim 4,

wherein in said scheme of obtaining M_j measurements of each signal, image or video sequence x_j among a plurality of J signals, images or video sequences comprises projecting x_j onto at least one of the following types of sequences:

random sequences, Bernoulli-distributed sequences, Gaussian-distributed sequences, coding scheme sequences, Fourier sequences, Hadamard sequences, cyclic sequences, Toeplitz sequences, and sparse sequences.

14. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 1,

wherein said reconstruction method reconstructs each signal, image or video sequence x_j among said plurality of J signals, images or video sequences independently of reconstruction of other signals among said plurality.

15. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 1,

wherein in said reconstruction method of seeking possible values for said plurality of signals, images or video sequences that explain the measurements relies on any possible mathematical model for said plurality of signals.

17. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 1,

wherein said reconstruction method relies on a mathematical model where each of said plurality of signals, images or video sequences comprises a linear or approximately linear combination of identical signal elements under different coefficients.

18. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 15,

wherein said reconstruction method relies on a mathematical model where each of said plurality of signals, images or video sequences comprises a linear or approximately linear combination of identical signal elements under different coefficients.

19. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 1,

wherein said reconstruction method relies on a mathematical model where each of said plurality of signals, images or video sequences comprises a sum of a common signal component and an additional component that comprises a linear or approximately linear combination of identical signal elements under different coefficients.

20. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 1,

wherein said reconstruction method relies on mathematical model where each of said plurality of signals, images or video sequences comprises a sum of a common signal component and a set of additional signal components being different for each of the plurality of signals.

21. A method for reconstructing a video sequence according to claim 20,

wherein said reconstruction method relies on a mathematical model where each of said plurality of different additional signal components represents motion between different image frames of the video sequence.

22. A method for reconstructing a plurality of signals, images, or video sequences according to claim 1,

wherein said reconstruction method relies on at least one of the following algorithms:

l₀, l₁, Basis Pursuit with Denoising, LARS, LASSO, Simultaneous Orthogonal Matching Pursuit or Trivial Pursuit.

23. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 17, wherein said reconstruction method relies on a non-iterative greedy algorithm based on matched filtering to identify the said identical signal elements.

24. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 19,

wherein said reconstruction method relies on the iterative sequential estimation of the common and additional components of each of said plurality of signals, images or video sequences.

25. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 20,

wherein said reconstruction method relies on the iterative sequential estimation of the common and additional components of each of said plurality of signals, images or video sequences.

26. A method according to claim 1 wherein said means for measuring comprises a plurality of sensors in any modality.

27. A method according to claim 15 wherein said plurality of sensors comprise at least one selected from the group of:

cameras, acoustic and temperature sensors.

16. A method for reconstructing exactly or approximately a plurality of digital signals, images, or video sequences comprising the steps of:

estimating a common component of said plurality of digital signals, images or video sequences;

estimating measurements generated by innovations of each of said plurality of digital signals, images or video sequences;

constructing approximations of said innovations of each of said plurality of digital signals, images or video sequences; and

obtaining an estimate of at least one signal, image or video sequence from said plurality of digital signals, images or video sequences as the sum of said estimate of said common component and estimates of at least one innovation component of said at least one signal, image or video sequence.

34. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 16,

wherein said estimation of the common component and at least one innovation component uses l₁ norm minimization, where the measurements of the plurality of signals, images or video sequences are constrained to explain all signal components while having minimal l₁ norm.

35. A method for reconstructing exactly or approximately a plurality of signals, images, or video sequences according to claim 34,

wherein said norm can be weighted differently for each among the single common component and the plurality of J innovation components.

28. A method for detecting the presence or absence of a signal of interest in at least one of a plurality of signals, images, or video sequences, comprising the steps of:

taking M_j measurements of each signal x_j among the plurality of J signals; and

determining if the measurements are more consistent with the presence or absence of the signal of interest, said determining step taking into account commonality and/or structure shared a month the signals, images or video sequences.

29. A method as in claim 28, wherein said step of determining if the measurements are more consistent with the presence or absence of the signal of interest comprises using any reconstruction method to obtain an approximation of the signal from which one can more easily test if the measurements are more consistent with the presence or absence of the signal of interest.

30. A method as in claim 28, wherein said step of determining if the measurements are more consistent with the presence or absence of the signal of interest comprises using a greedy reconstruction method such as Orthogonal Matching Pursuit or Matching Pursuit, but stopping the algorithm after a small number of iterations, to obtain an approximation of the signal from which one can more easily test if the measurements are more consistent with the presence or absence of the signal of interest.

31. A method for estimating the value of a function that depends on one or more of a plurality of signals, images, or video sequences, comprising the steps of:

taking M_j measurements of each signal, image or video sequence x_j among the plurality of J signals, images or video sequences; and

determining the value of the function that is most consistent with the measurements, said determining step taking into account commonality and/or structure shared among the signals, images or video sequences.

32. A method as in claim 31, wherein said step of determining the value of the function that is most consistent with the measurements comprises using any reconstruction method to obtain an approximation of the signal from which one can then directly estimate the function of interest.

33. A method as in claim 31, wherein said step of determining the value of the function that is most consistent with the measurements comprises using a greedy reconstruction method such as Orthogonal Matching Pursuit or Matching Pursuit, but stopping the algorithm after a small number of iterations, to obtain an approximation of the signal from which one can then directly estimate the function of interest.