Bits from Photons: Oversampled Image Acquisition Using Binary Poisson Statistics

We study a new image sensor that is reminiscent of traditional photographic film. Each pixel in the sensor has a binary response, giving only a one-bit quantized measurement of the local light intensity. To analyze its performance, we formulate the oversampled binary sensing scheme as a parameter estimation problem based on quantized Poisson statistics. We show that, with a single-photon quantization threshold and large oversampling factors, the Cram\'er-Rao lower bound (CRLB) of the estimation variance approaches that of an ideal unquantized sensor, that is, as if there were no quantization in the sensor measurements. Furthermore, the CRLB is shown to be asymptotically achievable by the maximum likelihood estimator (MLE). By showing that the log-likelihood function of our problem is concave, we guarantee the global optimality of iterative algorithms in finding the MLE. Numerical results on both synthetic data and images taken by a prototype sensor verify our theoretical analysis and demonstrate the effectiveness of our image reconstruction algorithm. They also suggest the potential application of the oversampled binary sensing scheme in high dynamic range photography

Frame Permutation Quantization

Frame permutation quantization (FPQ) is a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented, and reconstruction algorithms based on linear programming, quadratic programming, and recursive orthogonal projection are derived. Implementations of the linear and quadratic programming algorithms for uniform and Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate. Monte Carlo evaluation of the recursive algorithm shows that mean-squared error (MSE) decays as 1/M^4 for an M-element frame, which is consistent with previous results on optimal decay of MSE. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions.Comment: 29 pages, 5 figures; detailed added to proof of Theorem 4.3 and a few minor correction

Filters and Matrix Factorization

We give a number of explicit matrix-algorithms for analysis/synthesis in multi-phase filtering; i.e., the operation on discrete-time signals which allow a separation into frequency-band components, one for each of the ranges of bands, say N , starting with low-pass, and then corresponding filtering in the other band-ranges. If there are N bands, the individual filters will be combined into a single matrix action; so a representation of the combined operation on all N bands by an N x N matrix, where the corresponding matrix-entries are periodic functions; or their extensions to functions of a complex variable. Hence our setting entails a fixed N x N matrix over a prescribed algebra of functions of a complex variable. In the case of polynomial filters, the factorizations will always be finite. A novelty here is that we allow for a wide family of non-polynomial filter-banks. Working modulo N in the time domain, our approach also allows for a natural matrix-representation of both down-sampling and up-sampling. The implementation encompasses the combined operation on input, filtering, down-sampling, transmission, up-sampling, an action by dual filters, and synthesis, merges into a single matrix operation. Hence our matrixfactorizations break down the global filtering-process into elementary steps. To accomplish this, we offer a number of adapted matrix factorizationalgorithms, such that each factor in our product representation implements in a succession of steps the filtering across pairs of frequency-bands; and so it is of practical significance in implementing signal processing, including filtering of digitized images. Our matrix-factorizations are especially useful in the case of the processing a fixed, but large, number of bands

Quantization and erasures in frame representations

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 123-126).Frame representations, which correspond to overcomplete generalizations to basis expansions, are often used in signal processing to provide robustness to errors. In this thesis robustness is provided through the use of projections to compensate for errors in the representation coefficients, with specific focus on quantization and erasure errors. The projections are implemented by modifying the unaffected coefficients using an additive term, which is linear in the error. This low-complexity implementation only assumes linear reconstruction using a pre-determined synthesis frame, and makes no assumption on how the representation coefficients are generated. In the context of quantization, the limits of scalar quantization of frame representations are first examined, assuming the analysis is using inner products with the frame vectors. Bounds on the error and the bit-efficiency are derived, demonstrating that scalar quantization of the coefficients is suboptimal. As an alternative to scalar quantization, a generalization of Sigma-Delta noise shaping to arbitrary frame representations is developed by reformulating noise shaping as a sequence of compensations for the quantization error using projections.(cont.) The total error is quantified using both the additive noise model of quantization, and a deterministic upper bound based on the triangle inequality. It is thus shown that the average and the worst-case error is reduced compared to scalar quantization of the coefficients. The projection principle is also used to provide robustness to erasures. Specifically, the case of a transmitter that is aware of the erasure occurrence is considered, which compensates for the erasure error by projecting it to the subsequent frame vectors. It is further demonstrated that the transmitter can be split to a transmitter/receiver combination that performs the same compensation, but in which only the receiver is aware of the erasure occurrence. Furthermore, an algorithm to puncture dense representations in order to produce sparse approximate ones is introduced. In this algorithm the error due to the puncturing is also projected to the span of the remaining coefficients. The algorithm can be combined with quantization to produce quantized sparse representations approximating the original dense representation.by Petros T. Boufounos.Sc.D

Study and simulation of low rate video coding schemes

The semiannual report is included. Topics covered include communication, information science, data compression, remote sensing, color mapped images, robust coding scheme for packet video, recursively indexed differential pulse code modulation, image compression technique for use on token ring networks, and joint source/channel coder design

Generalizations of permutation source codes

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 87-90).Permutation source codes are a class of structured vector quantizers with a computationally- simple encoding procedure. In this thesis, we provide two extensions that preserve the computational simplicity but yield improved operational rate-distortion performance. In the first approach, the new class of vector quantizers has a codebook comprising several permutation codes as subcodes. Methods for designing good code parameters are given. One method depends on optimizing the rate allocation in a shape-gain vector quantizer with gain-dependent wrapped spherical shape codebook. In the second approach, we introduce frame permutation quantization (FPQ), a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented and reconstruction algorithms based on linear programming and quadratic programming are derived. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions. Simulations for uniform and Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate.by Ha Quy Nguyen.S.M

Digital Color Imaging

This paper surveys current technology and research in the area of digital color imaging. In order to establish the background and lay down terminology, fundamental concepts of color perception and measurement are first presented us-ing vector-space notation and terminology. Present-day color recording and reproduction systems are reviewed along with the common mathematical models used for representing these devices. Algorithms for processing color images for display and communication are surveyed, and a forecast of research trends is attempted. An extensive bibliography is provided

Low Power Analog to Digital Converters in Advanced CMOS Technology Nodes

The dissertation presents system and circuit solutions to improve the power efficiency and address high-speed design issues of ADCs in advanced CMOS technologies. For image sensor applications, a high-performance digitizer prototype based on column-parallel single-slope ADC (SS-ADC) topology for readout of a back-illuminated 3D-stacked CMOS image sensor is presented. To address the high power consumption issue in high-speed digital counters, a passing window (PW) based hybrid counter topology is proposed. To address the high column FPN under bright illumination conditions, a double auto-zeroing (AZ) scheme is proposed. The proposed techniques are experimentally verified in a prototype chip designed and fabricated in the TSMC 40 nm low-power CMOS process. The PW technique saves 52.8% of power consumption in the hybrid digital counters. Dark/bright column fixed pattern noise (FPN) of 0.0024%/0.028% is achieved employing the proposed double AZ technique for digital correlated double sampling (CDS). A single-column digitizer consumes total power of 66.8μW and occupies an area of 5.4 µm x 610 µm. For mobile/wireless receiver applications, this dissertation presents a low-power wide-bandwidth multistage noise-shaping (MASH) continuous-time delta-sigma modulator (CT-ΔΣM) employing finite impulse response (FIR) digital-to-analog converters (DACs) and encoder-embedded loop-unrolling (EELU) quantizers. The proposed MASH 1-1-1 topology is a cascade of three single-loop first-order CT-ΔΣM stages, each of which consists of an active-RC integrator, a current-steering DAC, and an EELU quantizer. An FIR filter in the main 1.5-bit DAC improves the modulator’s jitter sensitivity performance. FIR’s effect on the noise transfer function (NTF) of the modulator is compensated in the digital domain thanks to the MASH topology. Instead of employing a conventional analog direct feedback path, a 1.5-bit EELU quantizer based on multiplexing comparator outputs is proposed; this approach is suitable for highspeed operation together with power and area benefits. Fabricated in a 40-nm low-power CMOS technology, the modulator’s prototype achieves a 67.3 dB of signal-to-noise and distortion ratio (SNDR), 68 dB of signal-to-noise ratio (SNR), and 68.2 dB of dynamic range (DR) within 50.5 MHz of bandwidth (BW), while consuming 19 mW of total power (P). The proposed modulator features 161.5 dB of figure-of-merit (FOM), defined as FOM = SNDR + 10 log10 (BW/P)