Hardware Implementation of Enhancement Embedded Reduced Runtime Recovery Algorithm for Compressively Sensed Images

Abstract

Compressed sensing is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from fewer samples than required by the criterion. In compressed sensing, data is compressed and converted into fewer measurements and transmitted through wireless channel which is reconstructed at the receiver. Since very few samples are used for reconstruction, there are possibilities of degradation in quality of reconstruction. Unlike traditional methods, enhancement can now be done using the recent technique of compressed sensing by embedding image enhancement techniques like edge detection, histogram, filtering and their combinations with CS recovery procedure. This work proposes such a method by binding the image enhancement techniques along with the compressed sensing process. Filter of Gaussian filter (FGF), a combinational filter proposed in this study enables an increase in PSNR of 1 dB when compared to other filtering techniques besides using least number of measurements and maintaining minimum time consumption. The runtime difference with and without the FGF is ~ 3 s, which is affordable even in hardware with minimum specifications. Real time experimentation of embedded enhancement CS was carried out in WINGZ board to prove the feasibility of enhanced recovery process with lower end hardware.

Appendix

1.1 Mathematical Model of Error Bound

The error bound is the limit up to which the error is tolerable in a process. This section gives an idea about the error bound posed by FGF-R3A with the help of the following theorem.

Theorem

Consider a noisy observation $y = \phi x + e$, where $x$ is the input sparse vector with $k$ non-zero elements and $e$ the common error. Then, FGF-R3A would recover the enhanced estimation such that

$$\left\| {x - x_{e}} \right\|_{2} \le \left\| {x - x_{r}} \right\|_{2}$$

(5)

where $\left\| {x - x_{e}} \right\|_{2}$ is the error between input and FGF-R3A recovery and $\left\| {x - x_{r}} \right\|_{2}$ is the error between input and R3A recovery.

Proof

In order to prove Eq. (5), the concept of triangular inequality is considered with $x_{r}$ as the third vertex of a triangle. $x$ and $x_{r}$ form the other two vertices of the triangle. Then, the left hand side of Eq. (5) can be represented as

$$\left\| {x - x_{e}} \right\|_{2} \le \left\| {x_{r} - x_{e}} \right\|_{2} + \left\| {x - x_{r}} \right\|_{2}$$

(6)

where, $x_{r}$ is algorithmically the recovered signal without enhancement. From Theorem 1 in [6], it is clear that

$$\left\| {x - x_{r}} \right\|_{2} \le \left\| {x_{k}} \right\|_{2} + \tilde{\varepsilon}$$

(7)

where, $\tilde{\varepsilon}$ is the error that is dependent upon $x_{k}$. In [5] Needel and Tropp express this error as in Eq. (8).

$$\tilde{\varepsilon} = \left\| {x - x_{k}} \right\|_{2} + \frac{1}{\sqrt k}\left\| {x - x_{k}} \right\|_{1} + \left\| {\tilde{e}} \right\|_{2}$$

(8)

Substituting (8) in (7), (9) is obtained.

$$\left\| {x - x_{r}} \right\|_{2} \le \left\| {x_{k}} \right\|_{2} + \left\| {x - x_{k}} \right\|_{2} + \frac{1}{\sqrt k}\left\| {x - x_{k}} \right\|_{1} + \left\| {\tilde{e}} \right\|_{2}$$

(9)

Substituting (9) in the inequality expression (6),

$$\left\| {x - x_{e}} \right\|_{2} \le \left\| {x_{r} - x_{e}} \right\|_{2} + \left\| {x_{k}} \right\|_{2} + \left\| {x - x_{k}} \right\|_{2} + \frac{1}{\sqrt k}\left\| {x - x_{k}} \right\|_{1} + \left\| {\tilde{e}} \right\|_{2}$$

(10)

In the above equation, $\left\| {\tilde{e}} \right\|_{2}$ is the observation error which is almost zero $\left({\left\| {\tilde{e}} \right\|_{2} \cong 0} \right)$ and hence can be neglected. Also, $x_{r}$ is the recovered signal and $x_{e}$ is the enhanced signal and hence the difference between these two quantities will be close to zero compared to the other terms in the equation. Thus $\left\| {x_{r} - x_{e}} \right\|_{2} \cong 0$. After the elimination of these two elements, Eq. (10) can be written as

$$\left\| {x - x_{e}} \right\|_{2} \le \left\| {x_{k}} \right\|_{2} + \left\| {x - x_{k}} \right\|_{2} + \frac{1}{\sqrt k}\left\| {x - x_{k}} \right\|_{1}$$

(11)

Using Eq. (9), the left hand terms of (11) can be replaced as

$$\left\| {x - x_{e}} \right\|_{2} \le \left\| {x - x_{r}} \right\|_{2}$$

(12)

Thus the error between the input signal $x$ and the enhanced signal $x_{e}$ is less than the error between $x$ and the recovered signal $x_{r}$. Hence the error bound of FGF-R3A is limited to $\left\| {x - x_{r}} \right\|_{2}$.