Measurement Quantization in Compressive Imaging

Abstract

In compressive imaging the measurement quantization and its impact on the overall system performance is an important problem. This work considers several challenges that derive from quantization of compressive measurements. We investigate the design of scalar quantizer (SQ), vector quantizer (VQ), and tree-structured vector quantizer (TSVQ) for information-optimal compressive imaging. The performance of these quantizer designs is quantified for a variety of compression rates and measurement signal-to-noise-ratio (SNR) using simulation studies. Our simulation results show that in the low SNR regime a low bit-depth (3 bit per measurement) SQ is sufficient to minimize the degradation due to measurement quantization. However, in mid-to-high SNR regime, quantizer design requires higher bit-depth to preserve the information in the measurements. Simulation results also confirm the superior performance of VQ over SQ. As expected, TSVQ provides a good tradeoff between complexity and performance, bounded by VQ and SQ designs on either side of performance/complexity limits. In compressive image the size of final measurement data (i.e. in bits) is also an important system design metric. In this work, we also optimize the compressive imaging system using this design metric, and investigate how to optimally allocate the number of measurement and bits per measurement, i.e. the rate allocation problem. This problem is solved using both an empirical data driven approach and a model-based approach. As a function of compression rate (bits per pixel), our simulation results show that compressive imaging can outperform traditional (non-compressive) imaging followed by image compression (JPEG 2000) in low-to-mid SNR regime. However, in high SNR regime traditional imaging (with image compression) offers a higher image fidelity compare to compressive imaging for a given data rate. Compressive imaging using blockwise measurements is partly limited due to its inability to perform global rate allocation. We also develop an optimal minimum mean-square error (MMSE) reconstruction algorithm for quantized compressed measurements. The algorithm employs Monte-Carlo Markov Chain (MCMC) sampling technique to estimate the posterior mean. Simulation results show significant improvement over approximate MMSE algorithms.