Sparse signals can be recovered from a reduced set of samples by using compressive sensing algorithms. In common compressive sensing methods the signal is recovered in the sparsity domain. A method for the reconstruction of sparse signals which reconstructs the missing/unavailable samples/measurements is recently proposed. This method can be efficiently used in signal processing applications where a complete set of signal samples exists. The missing samples are considered as the minimization variables, while the available samples are fixed. Reconstruction of the unavailable signal samples/measurements is preformed using a gradient-based algorithm in the time domain, with an adaptive step. Performance of this algorithm with respect to the step-size and convergence are analyzed and a criterion for the step-size adaptation is proposed in this paper. The step adaptation is based on the gradient direction angles. Illustrative examples and statistical study are presented. Computational efficiency of this algorithm is compared with other two commonly used gradient algorithms that reconstruct signal in the sparsity domain. Uniqueness of the recovered signal is checked using a recently introduced theorem. The algorithm application to the reconstruction of highly corrupted images is presented as well.
A signal is sparse in a transformation domain if the number of nonzero coefficients is much lower than the number of signal samples. For linear signal transforms the signal samples can be considered as linear combinations (measurements) of the signal transform coefficients. Sparse signals can be fully recovered from a reduced set of samples/measurements. Compressive sensing theory is dealing with sparse signal reconstruction [
Various algorithms for reconstruction of sparse signals from a reduced set of observations are introduced [
The topic of this paper is the reconstruction of signals with some arbitrary positioned unavailable samples. These samples may result from a compressive sensing strategy or from their unavailability due to various reasons. A method for the unavailable signal samples reconstruction, considering them as variables, has been proposed in [
The paper is organized as follows. After the definitions in the next section, the adaptive gradient algorithm is presented. In the comments to the algorithm the step-size and bias are considered and illustrated. A new criterion for the algorithm parameter adaptation is proposed in Section
Discrete-time signal
Although there are some approaches to reconstruct the signal using the
A gradient-based algorithm that minimizes the sparsity measure by varying the missing sample values is presented next. In this gradient-based minimization approach the missing samples are considered as variables [
The signal
Through statistical study [
The algorithm inputs are the signal length For the DFT instead of using signals defined by (
with
The influence of step-size
This simple relation between the bias and step-size has been confirmed through statistical analysis on more complex cases, with a large number of missing samples.
The gradient-based algorithm convergence improvement by changing steps is analyzed in detail in [
When the minimum of the sparsity measure is reached with a sufficiently small
The precision of the result in iterative algorithms is commonly estimated based on the change of the result values in the last iterations. Therefore an average of changes in a large number of variables is a good estimate of the achieved precision. The value of
A possible divergence of a gradient-based algorithm is related to large steps
A pseudocode of this algorithm is presented in Algorithm
(i) Set of missing/omitted sample positions (ii) Available samples ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
(i) Reconstructed signal
(a) Illustration of a signal reconstruction using variable step gradient algorithm. (b) Original signal. (c) Signal with two missing samples. (d) Reconstructed signal. (e) Reconstructed signal MSE as a function of the iteration number.
The bias upper limit in the stationary state for step
Gradient-based reconstruction of a sparse signal with reconstructed MSE (a). Angle between successive gradient estimations
Consider a signal
Signal independent uniqueness corresponds to the worst case signal form, when
The answer is obtained almost immediately, since the computational complexity of the Theorem is of order
For the considered set of missing samples
Consider a general real-valued form of a signal sparse in the DFT domain
Signal-to-reconstruction-error (SRR) obtained by using Algorithm
Bright colors indicate the region where the algorithm had fully recovered missing samples (compared to the original samples) in all realizations, while dark colors indicate the region where the algorithm could not recover missing samples in any realization. In the transition region for
The average reconstruction error in the noise-free cases is related to the number of the full recovery events. For
((a), (b)) The percentage of the full recovery events as a function of the number of available samples
The efficiency of the presented algorithm is compared with the standard routines used for the
MAE and computational time for the L1-magic (LP-DP), LASSO-ISTA, and the proposed algorithm.
|
|
L1-magic | LASSO | Proposed | |||
---|---|---|---|---|---|---|---|
MAE | Time | MAE | Time | MAE | Time | ||
8 | 8 |
|
29.7 |
|
25.9 |
|
7.6 |
16 | 8 |
|
28.4 |
|
25.9 |
|
7.9 |
32 | 8 |
|
29.8 |
|
31.5 |
|
13.1 |
8 | 16 |
|
31.2 |
|
30.1 |
|
15.2 |
16 | 16 |
|
25.4 |
|
30.6 |
|
18.9 |
32 | 16 |
|
22.7 |
|
26.4 |
|
21.5 |
8 | 24 |
|
23.2 |
|
25.7 |
|
20.6 |
16 | 24 |
|
21.6 |
|
25.4 |
|
22.8 |
32 | 24 |
|
24.2 |
|
25.1 |
|
35.8 |
8 | 32 |
|
20.4 |
|
25.4 |
|
28.1 |
16 | 32 |
|
22.4 |
|
25.0 |
|
32.8 |
32 | 32 |
|
25.0 |
|
25.1 |
|
74.1 |
We can conclude that in most of the considered cases the proposed algorithm outperforms other two algorithms in both the calculation time and accuracy.
Original MATLAB image canoe.tif (a). Corrupted image with salt-and-pepper noise in 50% of the pixels (b). Reconstructed images after 4 and 50 iterations (c).
Performance of the recently proposed gradient-based signal reconstruction algorithm in the domain of the signal samples/measurements is studied. Application of this algorithm is very efficient in signal processing problems with reduced set of data, since it is easy to define a set of missing signal samples/measurements. The values of signal samples are considered as variables, while the available samples are fixed and not changed in the algorithm. An efficient criterion, based on the gradient directions, is proposed for the algorithm step-size adaptation. Statistical analysis confirms calculation efficiency. The algorithm can easily be adapted to the cases on nonuniform signal samples, when only a reduced set of linear combinations of the signal samples are available.
One of the forms for formulation of minimization problem (
The
The authors declare that they have no competing interests.