Oversampling | EPFL Graph Search

In signal processing, oversampling is the process of sampling a signal at a sampling frequency significantly higher than the Nyquist rate. Theoretically, a bandwidth-limited signal can be perfectly reconstructed if sampled at the Nyquist rate or above it. The Nyquist rate is defined as twice the bandwidth of the signal. Oversampling is capable of improving resolution and signal-to-noise ratio, and can be helpful in avoiding aliasing and phase distortion by relaxing anti-aliasing filter performance requirements. A signal is said to be oversampled by a factor of N if it is sampled at N times the Nyquist rate. Motivation There are three main reasons for performing oversampling: to improve anti-aliasing performance, to increase resolution and to reduce noise. Anti-aliasing Oversampling can make it easier to realize analog anti-aliasing filters. Without oversampling, it is very difficult to implement filters with the sharp cutoff necessary to maximize use of the avail

Beyond Cartesian Representations for Local Descriptors

Patrick Winfried Josef Ebel, Pascal Fua, Anastasiia Mishchuk, Eduard Trulls Fortuny, Kwang Moo Yi

The dominant approach for learning local patch descriptors relies on small image regions whose scale must be properly estimated a priori by a keypoint detector. In other words, if two patches are not in correspondence, their descriptors will not match. A strategy often used to alleviate this problem is to "pool" the pixel-wise features over log-polar regions, rather than regularly spaced ones. By contrast, we propose to extract the "support region" directly with a log-polar sampling scheme. We show that this provides us with a better representation by simultaneously oversampling the immediate neighbourhood of the point and undersampling regions far away from it. We demonstrate that this representation is particularly amenable to learning descriptors with deep networks. Our models can match descriptors across a much wider range of scales than was possible before, and also leverage much larger support regions without suffering from occlusions. We report state-of-the-art results on three different datasets.

IEEE COMPUTER SOC2019

Bits from Photons

Feng Yang

The trends in the design of image sensors are to build sensors with low noise, high sensitivity, high dynamic range, and small pixel size. How can we benefit from pixels with small size and high sensitivity? In this dissertation, 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. The response function of the image sensor is non-linear and similar to a logarithmic function, which makes the sensor suitable for high dynamic range imaging. We first 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 Cramér-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 is concave, we guarantee the global optimality of iterative algorithms in finding the MLE. We study the performance of the oversampled binary sensing scheme in presence of dark current noise. The noise model is an additive Bernoulli noise with a known parameter, and the noise only flips the binary output from "0" to "1". We show that the binary sensor is quite robust with respect to noise and its dynamic range is only slightly reduced. The binary sensor first benefits from the increasing of the oversampling factor and then suffers in term of dynamic range. We again use the MLE to estimate the light intensity. When the threshold is a single photon, we show that the log-likelihood function is still concave. Thus, the global optimality can be achieved. But for thresholds larger than "1", this property does not hold true. By proving that when the light intensity is piecewise-constant, the likelihood function is a strictly pseudoconcave function, we guarantee to find the optimal solution of the MLE using iterative algorithms for arbitrary thresholds. For the general linear light field model, the log-likelihood function is not even quasiconcave when thresholds are larger than "1". In this circumstance, we find an initial solution by approximating the light intensity field with a piecewise-constant model, and then we use Newton's method to refine the estimation using the exact model. We then examine one of the most important parameters in the binary sensor, i.e., the threshold used to generate binary values. We prove the intuitive result that large thresholds achieve better estimation performance for strong light intensities, while small thresholds work better for low light intensities. To make a binary sensor that works in a larger range of light intensities, we propose to design a threshold array containing multiple thresholds instead of a single threshold for the binary sensing. The criterion is to minimize the average CRLB which is a good approximation of the mean squared error (MSE). The performance analysis on the new binary sensor verifies the effectiveness of our design. Again, the MLE is used for reconstructing the light intensity field from the binary measurements. By showing that the log-likelihood function is concave for arbitrary threshold arrays, we ensure that the iterative algorithms can find the optimal solution of the MLE. Finally, we study the reconstruction problem for the binary image sensor under a generalized piecewise-constant light intensity field model, which is quite useful when parameters like oversampling factors are unknown. We directly estimate light exposure values, i.e., the number of photons hitting on each pixel. We assume that the light exposure values are piecewise-constant and we use the MLE for the reconstruction. This optimization problem is solved by iteratively working out two subproblems. The first one is to find the optimal light exposure value for each segment, given the optimal segmentation of the binary measurements. The second one is to find the optimal segmentation of the binary measurements given the optimal light exposure values for each segment. Several algorithms are provided for solving this optimization problem. Dynamic programming can obtain the optimal solution for 1-D signals, but the computation is quite heavy. To reduce the burden of computation, we propose a greedy algorithm and a method based on pruning of binary trees or quadtrees.

EPFL2012

Oversampled A/D Conversion and Error-Rate Dependence of Non-Bandlimited Signals with Finite Rate of Innovation

Ivana Jovanovic

We study the problem of A/D conversion and error-rate dependence of a class of non-bandlimited signals which have a finite rate of innovation, particularly, a continuous periodic stream of Diracs, characterized by a finite set of time positions and weights. Previous research has only considered sampling of this type of signals, ignoring the presence of quantization, which is necessary for any practical application. We first define the concept of consistent reconstruction for these signals and introduce the operations of both: a) oversampling in frequency, determined by the bandwidth of the low pass filtering used in the signal acquisition, and b) oversampling in time, determined by the number of samples in time taken from the filtered signal. Accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constrains, defined by: the low-pass filtering operation, the quantization operation itself and the signal space of continuous periodic streams of Diracs. We provide two schemes to reconstruct the signal. For the first one, we prove that the mean squared error (MSE) of the time positions is of the order of O(1/R_t^2R_f^3), where R_t and R_f are the oversampling ratios in time and in frequency, respectively. For the second scheme, which has a higher complexity, it is experimentally observed that the MSE of the time positions is of the order of O(1/R_t^2R_f^5). Our experimental results show a clear advantage of consistent reconstruction over non-consistent reconstruction. Regarding the rate, we consider a threshold crossing based scheme where, as opposed to previous research, both oversampling in time and also in frequency influence the coding rate. We compare the error-rate dependence behavior that is obtained from both increasing the oversampling in time and in frequency, on the one hand, and on the other hand, from decreasing the quantization stepsize.