Restricted Structural Random Matrix for compressive sensing

Highlights

• Propose a novel sampling matrix that improves the sampling efficiency without scarifying the democracy.

• Combine partial sampling, multi-images super-resolution, coded imaging, and compressive sensing.

• Proposed matrix satisfies the Restricted Isometry Property with competitive reconstruction performance.

Abstract

Compressive sensing (CS) is well-known for its unique functionalities of sensing, compressing, and security (i.e. equal importance of CS measurements). However, there is a tradeoff. Improving sensing and compressing efficiency with prior signal information tends to favour particular measurements, thus decreasing security. This work aimed to improve the sensing and compressing efficiency without compromising security with a novel sampling matrix, named Restricted Structural Random Matrix (RSRM). RSRM unified the advantages of frame-based and block-based sensing together with the global smoothness prior (i.e. low-resolution signals are highly correlated). RSRM acquired compressive measurements with random projection of multiple randomly sub-sampled signals, which was restricted to low-resolution signals (equal in energy), thereby its observations are equally important. RSRM was proven to satisfy the Restricted Isometry Property and showed comparable reconstruction performance with recent state-of-the-art compressive sensing and deep learning-based methods.

Introduction

Compressive Sensing (CS) [1], [2], [3], [4] is an emerging sampling technique that directly captures a sparse or compressible signal $\mathit{x}\in {\mathbb{R}}^n$ in a compressed form $\mathit{y}\in {\mathbb{R}}^m,m\ll n$ via random projection:

$$\mathit{y}=\mathbf{\Phi }\mathit{x},$$

where $\mathbf{\Phi }\in {\mathbb{R}}^{m\times n}$ denotes the sampling matrix. CS is well known for its unique functionalities of simultaneous sampling, compressing, and security (or democracy).

Sensing. With linear projection sampling, CS guarantees to reconstruct a signal at a high probability if the sampling matrix $\mathbf{\Phi }$ satisfies some condition, like the Restricted Isometry Property (RIP) [3], [4], as

Definition 1

A matrix $\mathbf{\Phi }$ is said to satisfy the RIP condition with an isometric constant $\delta \in (0,1)$ if the inequality

$$\left(1-\delta \right){\Vert \mathit{\alpha }\Vert }_2^2\le {\Vert \mathbf{\Phi }\mathit{\alpha }\Vert }_2^2\le \left(1+\delta \right){\Vert \mathit{\alpha }\Vert }_2^2,$$

is held for all $s$-sparse signals $\mathit{\alpha }\in {\mathbb{R}}^n$. The smaller $\delta$ becomes, the better the RIP condition.

The asymmetric RIP condition was developed [5], [6] to allow a difference in upper and lower bound.

In the literature, CS widely uses the Gaussian Random Matrix (GRM) due to its theoretical guarantee. Unfortunately, GRM requires significant computation and storage for its fully random structure. Over the past decades, researchers have sought to reduce its computational complexity. These studies can be classified as (i) block-based CS (BCS), which relies on processing signals in a block-based manner [7], [8], [9]; (ii) frame-based sensing, which samples each signal dimension separately in Kronecker CS (KCS) [10], [11], [12], [13], random sample signals in fast-transform domains [14], [15], circulant random matrix [16], or sparse random matrices [17]. In general, frame-based CS is preferred for high sampling efficiency [10], [11], [12], [13], [14], [15] and BCS for coding efficiency [18], [19], [20] and parallel sampling [21].

Compressing. As the number of acquired measurements is significantly smaller than the signal dimension, $m\ll n$, CS achieves compressing capabilities. As a sampling technique, CS assumes no information about the to-be-sampled signal, except for the sparsity prior. Meanwhile, standard compression methods (e.g., JPEG and HEVC) know the to-be-compressed signals, and thereby encode them at higher coding efficiencies than CS [18], [22]. To overcome this drawback, researchers have utilized other signal priors, such as the low-frequency prior (e.g. the human visual system is more sensitive to lower frequencies) [23], [24], [25], [26], [27], [28], [29], structural of signals (e.g. tree-sparse in wavelet [30] or zig-zag order in DCT [31]), perceptual prior (in image/video applications [32]), and smoothness prior (nearby samples are highly correlated) [18], [19], [20], [22]. Multi-scale CS [23], [24], [25], [26], [27] and hybrid CS [28], [29] captured more low-frequency components to utilize the low-frequency prior. Weighted sampling exploited the structure of signals [30], [31], [32] to capture more important components without influencing the perceptual quality. The smoothness prior [18], [19], [20], [22] was used for predictive coding in BCS because nearby block measurements are highly correlated.

Security. CS measurements are equally important due to the randomness of the sampling matrix, named the democracy property [33]. It is possible to recover signals from CS acquisitions despite damaged/degraded or lost measurements, which enables error-resilient applications [34]. Additionally, if GRM is used as a sampling matrix, then CS measurements also follow the Gaussian distribution. Therefore, despite revealing the energy of the signal, the sampling matrix was used as the private key providing a certain level of security [35], [36], [37], [38], [39]. As a result, numerous studies have attempted to analyze the theoretical security performance [35], [36], practical applications in multimedia capturing [40], [41] and/or transmission including cloud computing (e.g., for error resilience [34], to improve visual privacy protection [37], or for encryption [38], [39] or cloud server-side protection [40], [41], [42]). CS was further combined with conventional encryption techniques like random scrambling/flipping, chaos methods [41], etc.

There is, however, a trade-off between these functionalities. Researchers often used prior information to reduce sampling computation or improve sampling efficiency without considering security. Utilizing prior information leads to reducing the randomness of the projection, and tends to favour particular measurements. As a result, CS measurements are no longer equally important. For instance, reducing complexity with BCS reveals the signal structure by evaluating the block measurement’s energy. Alternatively, CS-based security methods reduce sampling and compressing performance and may even increase the sampling complexity.

In this work, we develop Generalized Structural Sensing Matrix (GSSM) and derive a special version named Restricted Structural Random Matrix (RSRM). RSRM is targeted to improve sensing efficiency and reduce complexity while maintaining the security/democracy property. Theoretically, RSRM unifies the advantages of BCS, the structural frame-based CS, and utilizes the smoothness prior which is held for compressible and structure sparse signals. We prove the asymmetric RIP condition of GSSM and symmetric RIP condition of RSRM together with RIP supported experiments. Practically, RSRM is a combination of CS with conventional under-sampling methods such as partial sub-sampling, multiple-images super-resolution, and coded imaging. RSRM randomly sub-samples multiple LR signals, which are equal in energy, then applies random block-based sampling for each. Therefore, each measurement of RSRM is equally important, and thus democracy is preserved. Note that our work focuses on the security approach in the encoder part in which security, sampling, and compression are performed simultaneously.

The contributions of this paper are summarized as following:

• Propose a Generalized Structural Sensing Matrix from block-based and structural frame-based sampling.

• Present the asymmetric RIP of GSSM and derive the conditions to improve its RIP condition.

• Propose a Restricted Structural Random Matrix as a special case of GSSM by utilizing the local smoothness prior via global manner with Restricted Permutation.

We briefly review the related works in Section 2. Section 3 introduces the generalized structural sensing matrix and its asymmetric RIP condition. Section 4 proposes the restricted structure random matrix, a special version of GSSM, to achieve the best RIP condition taking the advantage of the global smoothness prior, especially for compressible and structure sparse signals. We present support experiments and reconstruction performance in Section 5. Finally, we draw conclusions in Section 6.

For notation convention, a scalar value is written in lower case form as $m$, a vector is represented in normal bold as $\mathit{x}$, while a matrix is denoted in capital bold notation as $\mathit{X}$. Other notations are given in Table 1.