Valerio Cambareri

Low-Complexity Multiclass Encryption by Compressed Sensing

The idea that compressed sensing may be used to encrypt information from unauthorized receivers has already been envisioned but never explored in depth since its security may seem compromised by the linearity of its encoding process. In this paper, we apply this simple encoding to define a general private-key encryption scheme in which a transmitter distributes the same encoded measurements to receivers of different classes, which are provided partially corrupted encoding matrices and are thus allowed to decode the acquired signal at provably different levels of recovery quality. The security properties of this scheme are thoroughly analyzed: first, the properties of our multiclass encryption are theoretically investigated by deriving performance bounds on the recovery quality attained by lower-class receivers with respect to high-class ones. Then, we perform a statistical analysis of the measurements to show that, although not perfectly secure, compressed sensing grants some level of security that comes at almost-zero cost and thus may benefit resource-limited applications. In addition to this, we report some exemplary applications of multiclass encryption by compressed sensing of speech signals, electrocardiographic tracks and images, in which quality degradation is quantified as the impossibility of some feature extraction algorithms to obtain sensitive information from suitably degraded signal recoveries.

On Known-Plaintext Attacks to a Compressed Sensing-Based Encryption: A Quantitative Analysis

Despite its intrinsic linearity, compressed sensing may be exploited to at least partially encrypt acquired signals from unintentional receivers: in the companion paper we have shown that the simplicity of its encoding allows the definition of a general, lightweight scheme in which transmitters distribute the same information to receivers of different classes enabled to recover it with different quality levels. In this investigation we quantify the robustness of such a scheme with respect to known-plaintext attacks. The odds of such an attack are shown by theoretical means, proving that the number of candidate encoding matrices matching a typical plaintext-ciphertext pair is astronomically large, thus making the search for the true encoding infeasible. These attacks are also simulated by applying compressed sensing to a variety of signals (speech, images and electrocardiographic traces) showing how this difficulty in extracting information on the true encoding matrix from a plaintext-ciphertext pair is reflected on the quality of the signals recovered by the attacker. The results clarify that, although not perfectly secure, CS grants a noteworthy level of security that may come at almost-zero cost and especially benefit resource-limited applications.Despite the linearity of its encoding, compressed sensing (CS) may be used to provide a limited form of data protection when random encoding matrices are used to produce sets of low-dimensional measurements (ciphertexts). In this paper, we quantify by theoretical means the resistance of the least complex form of this kind of encoding against known-plaintext attacks. For both standard CS with antipodal random matrices and recent multiclass encryption schemes based on it, we show how the number of candidate encoding matrices that match a typical plaintext-ciphertext pair is so large that the search for the true encoding matrix inconclusive. Such results on the practical ineffectiveness of known-plaintext attacks underlie the fact that even closely related signal recovery under encoding matrix uncertainty is doomed to fail. Practical attacks are then exemplified by applying CS with antipodal random matrices as a multiclass encryption scheme to signals such as images and electrocardiographic tracks, showing that the extracted information on the true encoding matrix from a plaintext-ciphertext pair leads to no significant signal recovery quality increase. This theoretical and empirical evidence clarifies that, although not perfectly secure, both standard CS and multiclass encryption schemes feature a noteworthy level of security against known-plaintext attacks, therefore increasing its appeal as a negligible-cost encryption method for resource-limited sensing applications.

A rakeness-based design flow for Analog-to-Information conversion by Compressive Sensing

Classical design of Analog-to-Information converters based on Compressive Sensing uses random projection matrices made of independent and identically distributed entries. Leveraging on previous work, we define a complete and extremely simple design flow that quantifies the statistical dependencies in projection matrices allowing the exploitation of non-uniformities in the distribution of the energy of the input signal. The energy-driven reconstruction concept and the effect of this design technique are justified and demonstrated by simulations reporting conspicuous savings in the number of measurements needed for signal reconstruction that approach 50%.

A two-class information concealing system based on compressed sensing

We elaborate on the possibility of exploiting the (pseudo)random projection operator, which is at the heart of the most common architecture for compressed sensing, to prevent access to the acquired information by unauthorized receivers. In low-resource applications, this approach may make dedicated cryptographic layers unnecessary when the security requirement is not particularly high. Beyond proving that the proposed system is at least asymptotically immune to straightforward statistical attacks, we also exploit the sensitivity of compressed sensing recovery algorithms to the complete knowledge of the projection matrix to introduce two-class protection. The encoding is such that first-class decoders can retrieve the signal to its full resolution while second-class decoders are able to retrieve only a degraded version of the same signal. Examples are given with reference to ECG signal acquisition.

Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

This letter focuses on the estimation of low-complexity signals when they are observed through M uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like M - 1/4 when all parameters but M are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic “dimension”, as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as M is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.

A Case Study in Low-Complexity ECG Signal Encoding: How Compressing is Compressed Sensing?

When transmission or storage costs are an issue, lossy data compression enters the processing chain of resource-constrained sensor nodes. However, their limited computational power imposes the use of encoding strategies based on a small number of digital computations. In this case study, we propose the use of an embodiment of compressed sensing as a lossy digital signal compression, whose encoding stage only requires a number of fixed-point accumulations that is linear in the dimension of the encoded signal. We support this design with some evidence that for the task of compressing ECG signals, the simplicity of this scheme is well-balanced by its achieved code rates when its performances are compared against those of conventional signal compression techniques.

Rakeness-Based Design of Low-Complexity Compressed Sensing

Compressed Sensing (CS) can be introduced in the processing chain of a sensor node as a mean to globally reduce its operating cost, while maximizing the quality of the acquired signal. We exploit CS as a simple early-digital compression stage that performs a multiplication of the signal by a matrix. The operating costs (e.g., the consumed power) of such an encoding stage depend on the number of rows of the matrix, but also on the value and position of the rows’ coefficients. Our novel design flow yields optimized sparse matrices with very few rows. It is a non-trivial extension of the rakeness-based approach to CS and yields an extremely lightweight stage implemented by a very small number of possibly signed sums with an excellent compression performance. By means of a general signal model we explore different corners of the design space and show that, for example, our method is capable of compressing the signal by a factor larger than 2.5 while not considering 30% of the original samples (so that they may not be acquired at all, leaving the analog front-end and ADC stages inactive) and by processing each of the considered samples with not more than three signed sums.

A non-convex blind calibration method for randomised sensing strategies

The implementation of computational sensing strategies often faces calibration problems typically solved by means of multiple, accurately chosen training signals, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not require any training, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models, in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors each subject to an unknown multiplicative factor (gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable initialisation. An analysis of this algorithm allows us to show that it converges to the global optimum provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Finally, we present some numerical experiments in which our algorithm allows for a simple solution to blind calibration of sensor gains in computational sensing applications.

Generalized inpainting method for hyperspectral image acquisition

A recently designed hyperspectral imaging device enables multiplexed acquisition of an entire data volume in a single snapshot thanks to monolithically-integrated spectral filters. Such an agile imaging technique comes at the cost of a reduced spatial resolution and the need for a demosaicing procedure on its interleaved data. In this work, we address both issues and propose an approach inspired by recent developments in compressed sensing and analysis sparse models. We formulate our superresolution and demosaicing task as a 3-D generalized inpainting problem. Interestingly, the target spatial resolution can be adjusted for mitigating the compression level of our sensing. The reconstruction procedure uses a fast greedy method called Pseudo-inverse IHT. We also show on simulations that a random arrangement of the spectral filters on the sensor is preferable to regular mosaic layout as it improves the quality of the reconstruction. The efficiency of our technique is demonstrated through numerical experiments on both synthetic and real data as acquired by the snapshot imager.

A greedy blind calibration method for compressed sensing with unknown sensor gains

The realisation of sensing modalities based on the principles of compressed sensing is often hindered by discrepancies between the mathematical model of its sensing operator, which is necessary during signal recovery, and its actual physical implementation, which can amply differ from the assumed model. In this paper we tackle the bilinear inverse problem of recovering a sparse input signal and some unknown, unstructured multiplicative factors affecting the sensors that capture each compressive measurement. Our methodology relies on collecting a few snapshots under new draws of the sensing operator, and applying a greedy algorithm based on projected gradient descent and the principles of iterative hard thresholding. We explore empirically the sample complexity requirements of this algorithm by testing its phase transition, and show in a practically relevant instance of this problem for compressive imaging that the exact solution can be obtained with only a few snapshots.