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The use of frames is analyzed in Compressed Sensing (CS) through proofs and experiments. First, a new generalized Dictionary-Restricted Isometry Property (D-RIP) sparsity bound constant for CS is established. Second, experiments with a tight frame to analyze sparsity and reconstruction quality using several signal and image types are shown. The constant is used in fulfilling the definition of D-RIP. It is proved that k-sparse signals can be reconstructed if by using a concise and transparent argument1. The approach could be extended to obtain other D-RIP bounds (i.e. ). Experiments contrast results of a Gabor tight frame with Total Variation minimization. In cases of practical interest, the use of a Gabor dictionary performs well when achieving a highly sparse representation and poorly when this sparsity is not achieved.

Let

with

for every k-sparse vector c (namely, c has at most k non-zero components), for some small constant

several sharp RIP bounds that cover the most interesting cases of δ_{k} and

A key requirement in this setting is a signal being sparse or approximately sparse. Indeed, Many families of integrating signals have sparse representations under suitable bases. Recently, an interesting sparsifying scheme was proposed by Candès et al. [

Let

The traditional RIP is no longer effective in the generalized setting. Candès et al. defined the D-restricted isometry property which extends RIP [

Definition 1. The measurement matrix

holds for all k-sparse vectors

The RIP is now a special case of D-RIP (when the dictionary D is the identity matrix). For D being a tight frame, Candès et al. [

The proof in Section 2 establishes an improved D-RIP bound which states that

previously available [

The practical application of this proof consists of experiments targeting the theory of using tight frames in this CS setting satisfying D-RIP. Similar experimental methods to those used by Candès et al. are followed [

This paper is organized into four main sections. Background information is presented in Section 1. Section 2 describes a proof of an improved D-RIP sparsity bound by using a concise and transparent approach. Section 3 details the methods of experimentation used to apply the tight frame in practical applications of signals and image recovery. Section 4 shows and discusses the reconstruction simulation results with an analysis of the robustness and shortcomings of using tight frames in CS.

Theorem 2. Let D be an arbitrary tight frame and let

where the constants

components (in magnitude) set to zero.

Before proving this theorem, some remarks are helpful. Firstly, Cai and Zhang have obtained a sharp bound

the ideas of Cai and Zhang [

In order to prove Theorem 2, the following

The following description is taken from Xu et al. [

Lemma 1. For positive integers

such that

for some nonnegative real numbers

Now Theorem 2 is proven.

Proof. This proof follows some ideas in the proofs of Theorems 1.1 and 2.1 by Cai et al. [

Let

For a subset

magnitude), i.e.,

[

Denote

By rearranging the columns of D if necessary, assume

Assume that the tight frame D is normalized, i.e.,

From the facts

Since

k-sparse decomposition of

with each

Cauchy-Schwartz inequality, there is

By the triangle inequality,

Note that

the theorem, it suffices to show that there are constants

In fact, assuming (17) there is

Now moving to the proof of (17). Denote

First, as

On the other hand, as each

Combining this with (23) shows

By making a perfect square, there is

which implies that

and finally have (17):

This demonstrates the use of Lemma 1 to get a good result. This could be pursued further to general cases for an even better bound. Indeed, this has been done recently by Wu and Li [

The focus in these experiments is to show practical applications of a sparsifying frame that satisfies the new lower bound proven in the previous section. A time-frequency Gabor dictionary is used as a sparsifying trans- form in re-weighted

One of the advantages of a Gabor dictionary is its characteristic of translational invariance both in time and frequency. In a similar context, a translational invariant wavelet transform was used with good results in MRI reconstruction [

Here comparison measurements of sparsity in Gabor and TV Coefficients of various signals are used to verify if good and robust results can be achieved. A selection of 5 different real valued signals and/or images with variants are used to simulate the complexity of practical signals.

・ Sinusoidal pulses (GHz range)

・ Shepp-Logan phantom image

・ Penguin image

・ T1 weighted MRI image

・ Time of Flight (TOF) Maximum Intensity Projection (MIP) MRI image

The original images and signals are sized to be a total length of 16k values-for images, this is a 128 × 128 gray-scale image, shown in part (a) of each figure. These signals are not sparse in their native domain, but can become sparse when a transform is applied.

The use of the Gabor dictionary to reconstruct these signals utilizes a core optimization algorithm solver for the primal-dual interior point method provided by

In

Test | L MSE | G MSE | TV MSE | %G Sparse | %TV Sparse |
---|---|---|---|---|---|

1 (Pulse 1) | 0.7045 | 0.0195 | 0.9411 | 99.5 | 0.19 |

2 (Pulse 2) | 0.7075 | 0.0192 | 0.8023 | 94.8 | 2.6 |

3 (Shepp-Logan) | 0.7062 | 0.2697 | 0.0 | 54.7 | 91.7 |

4 (Penguin) | 0.7077 | 0.3011 | 0.0 | 54.1 | 87.6 |

5 (Pulse + Shepp-Logan) | 0.7101 | 0.2541 | 0.2141 | 54.4 | 2.2 |

6 (T1 MRI) | 0.7013 | 0.1445 | 0.0637 | 82.9 | 7.8 |

7 (TOF MRI) | 0.7012 | 0.2555 | 0.1369 | 68.4 | 13.3 |

Sparsity. Measurements of normalized MSE are taken, where

A Linear (L) MSE of the reconstruction is used as a reference, shown as (b) in each figure. This is calculated by the transpose of the measurement matrix as the pseudo inverse. Gabor (G) MSE identifies the error in the use of that dictionary in the CS reconstruction, shown in parts (c) of the figures. TV MSE is a measure of the error when TV weighted

Sparsity measurements taken in the coefficient domain are based on a ratio of the count of values that are less than 1/256 of the maximum coefficient, divided by the total number of the coefficients. This ratio then is a percentage of sparsity. Two sparsity measures are taken:

・ % G Sparse-Sparsity of the Gabor transform coefficients of the fully sampled signal

・ % TV Sparse-Sparsity of the TV calculation of the fully sampled signal

The goal is to show how well a sparse tight frame representation of various signals performs in CS recon- struction. Analysis is done of the Gabor dictionary as a sparsifying transform on non-sparse signals and images. A large range of reconstruction errors and sparsity levels are observed for different image types and signals. The use of the Gabor frame with a reference of TV weighted

According to these measurements, sparsity in the coefficient domain will correlate to image reconstruction

success. For example, test 1 measures a Gabor coefficient sparsity of over 99% and a reconstruction success which reduces the MSE by 36 times compared to the linear reconstruction. Whereas, with the same signal, which is not sparse at all in the TV domain, TV minimization actually increases the MSE when compared with the linear reconstruction, see

In test 2, the complexity and sparsity are adjusted by adding additional sinusoidal pulses which may overlap. The complexity of the pulses significantly increases to 20 pulses and the Gabor dictionary is able to sparsely represent the signal very well compared to TV minimization, see

In tests 3 and 4, images are chosen which are sparse in the TV domain but not in the Gabor domain. The TV reconstruction reduces the MSE to zero compared to the Gabor reconstruction reducing by only a factor of 2.6 and 2.4 respectively. The penguin image is an example with a different background magnitude from the Shepp-Logan phantom, see

The signal in test 5 is a combination of a pulse from test 1 with the Shepp-Logan image from test 3. The same signal is plotted in the time domain for

In the last experiments, tests 6 and 7, MRI images of the brain as either T1 weighted or TOF MIP are used. They appear not to be sparsely represented in either the Gabor domain or in TV. The MSE result is poor in both reconstruction algorithms, see

It is also important to point out that the linear reconstructions, calculated with a pseudo-inverse, have a consistent MSE for all experiments, see L MSE in

The use of a new D-RIP sparsity bound constant for compressed sensing is proven using a transparent and concise approach. Practical numerical experiments for this setting are performed. The use of a Gabor tight frame in CS is contrasted with TV weighted

The author wishes to thank several people: Academic supervisor, Professor Guangwu Xu, for his assistance in the proofs of the results in Section 2, suggestions in setting up experiments, and help with analysis. Bing Gao, for pointing out an error in the earlier version of the proof [

ChristopherBaker, (2016) Sparse Representation by Frames with Signal Analysis. Journal of Signal and Information Processing,07,39-48. doi: 10.4236/jsip.2016.71006