Description Usage Arguments Details Value References Examples

Implements the fast Laplace algorithm given a measurement matrix and samples
from a signal or function. Note that this function is a convenient wrapper
for the function `FastLaplace`

.

1 | ```
FindSparse(PHI, y, eta = 1e-08, roundit = FALSE, verbose = FALSE)
``` |

`PHI` |
typically equals the product of a measurment matrix and basis
representation matrix, such as the wavelet basis.
The solution vector |

`y` |
CS measurements, samples from the signal or function. |

`eta` |
tolerance level in determining convergence of marginal likelihood. |

`roundit` |
whether or not to round the marginal likelihood, in order to avoid machine precision error when comparing across platforms. |

`verbose` |
print to screen which basis are added, re-estimated, or deleted. |

This code implements the fast Laplace algorithm. The fast Laplace algorithm is a method used to solve the compressive sensing problem, or in general, a highly underdetermined system of equations. It does this by taking the system of equations

*y = Φ w + n*

and converting it into a minimization problem
where we minimize the error with a constraint on *w*
(the vector we are solving for) that enforces
sparsity. The fast Laplace method uses a Bayesian framework, and in
particular, uses a Laplace prior to enforce sparsity on *w*.
See [1] for more information.

The sparse signal *w* as found by the fast Laplace algorithm.

[1] S. D. Babacan, R. Molina and A. K. Katsaggelos, "Bayesian Compressive Sensing Using Laplace Priors," in IEEE Transactions on Image Processing, vol. 19, no. 1, pp. 53-63, Jan. 2010.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | ```
# size of the basis function expansion
N <- 64
# generate sparse coefficient vector
w <- rep(0,N)
w[sample(1:N,10)] <- runif(10,-1,1)
# create wavelet basis trasform matrix
wavelet.basis <- WaveletBasis(N)
# generate actual signal
signal <- wavelet.basis%*%w
# now we try and recover 'w' and 'signal' from samples
num_samps <- 25
# create random measurement matrix
measure.mat <- matrix(runif(num_samps*N),num_samps,N)
measure.mat <- measure.mat/matrix(rep(sqrt(apply(measure.mat^2,2,sum)),
num_samps),num_samps,N,byrow=TRUE);
PHI <- measure.mat%*%wavelet.basis
# actual samples we see
y <- measure.mat%*%signal
# use fast Laplace algorithm
w_est <- FindSparse(PHI, y)
# compare plots of the sparse vector and the estimated sparse vector
plot(w,type='h')
lines(w_est,type='h',col='red')
# estimate signal
signal_est <- wavelet.basis%*%w_est
# Root mean squared error of estimate
error <- sqrt(mean((signal - signal_est)^2))
``` |

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