Adam James Scholefield

Sampling and Exact Reconstruction of Pulses with Variable Width

Recent sampling results enable the reconstruction of signals composed of streams of fixed-shaped pulses. These results have found applications in topics as varied as channel estimation, biomedical imaging and radio astronomy. However, in many real signals, the pulse shapes vary throughout the signal. In this paper, we show how to sample and perfectly reconstruct Lorentzian pulses with variable width. In the noiseless case, perfect recovery is guaranteed by a set of theorems. In addition, we verify that our algorithm is robust to model mismatch and noise. This allows us to apply the technique to two practical applications: electrocardiogram (ECG) compression and bidirectional reflectance distribution function (BRDF) sampling. ECG signals are one dimensional, but the BRDF is a higher dimensional signal, which is more naturally expressed in a spherical coordinate system; this motivated us to extend the theory to the 2D and spherical cases. Experiments on real data demonstrate the viability of the proposed model for ECG acquisition and compression, as well as the efficient representation and low-rate sampling of specular BRDFs.

Quadtree Structured Image Approximation for Denoising and Interpolation

The success of many image restoration algorithms is often due to their ability to sparsely describe the original signal. Shukla proposed a compression algorithm, based on a sparse quadtree decomposition model, which could optimally represent piecewise polynomial images. In this paper, we adapt this model to the image restoration by changing the rate-distortion penalty to a description-length penalty. In addition, one of the major drawbacks of this type of approximation is the computational complexity required to find a suitable subspace for each node of the quadtree. We address this issue by searching for a suitable subspace much more efficiently using the mathematics of updating matrix factorisations. Algorithms are developed to tackle denoising and interpolation. Simulation results indicate that we beat state of the art results when the original signal is in the model (e.g., depth images) and are competitive for natural images when the degradation is high.

On the Accuracy of Point Localisation in a Circular Camera-Array

Although many advances have been made in light-field and camera-array image processing, there is still a lack of thorough analysis of the localisation accuracy of different multi-camera systems. By considering the problem from a frame-quantisation perspective, we are able to quantify the point localisation error of circular camera configurations. Specifically, we obtain closed form expressions bounding the localisation error in terms of the parameters describing the acquisition setup. These theoretical results are independent of the localisation algorithm and thus provide fundamental limits on performance. Furthermore, the new frame-quantisation perspective is general enough to be extended to more complex camera configurations.

Quadtree structured restoration algorithms for piecewise polynomial images

Iterative shrinkage of sparse and redundant representations are at the heart of many state of the art denoising and deconvolution algorithms. They assume the signal is well approximated by a few elements from an overcomplete basis of a linear space. If one instead selects the elements from a nonlinear manifold it is possible to more efficiently represent piecewise polynomial signals. This suggests that image restoration algorithms based around nonlinear transformations could provide better results for this class of signals. This paper uses iterative shrinkage ideas and a nonlinear quadtree decomposition to develop image restoration algorithms suitable for piecewise polynomial images.

Unlabeled sensing: Reconstruction algorithm and theoretical guarantees

It often happens that we are interested in reconstructing an unknown signal from partial measurements. Also, it is typically assumed that the location (temporal or spatial) of each sample is known and that the only distortion present in the observations is due to additive measurement noise. However, there are some applications where such location information is lost. In this paper, we consider the situation in which the order of noisy samples, taken from a linear measurement system, is missing. Previous work on this topic has only considered the noiseless case and exhaustive search combinatorial algorithms. We propose a much more efficient algorithm based on a geometrical viewpoint of the problem. We also study the uniqueness of the solution under different choices of the sampling matrix and its robustness to noise for the case of two-dimensional signals. Finally we provide simulation results to confirm the theoretical findings of the paper.

Shape from bandwidth: The 2-D orthogonal projection case

Could bandwidth—one of the most classic concepts in signal processing—have a new purpose? In this paper, we investigate the feasibility of using bandwidth to infer shape from a single image. As a first analysis, we limit our attention to orthographic projection and assume a 2-D world.

Image restoration using a sparse quadtree decomposition representation

Techniques based on sparse and redundant representations are at the heart of many state of the art denoising and deconvolution algorithms. A very sparse representation of piecewise polynomial images can be obtained by using a quadtree decomposition to adaptively select a basis. We have recently exploited this to restore images of this form, however the same model can also provide very good sparse approximations of real world images. In this paper we take advantage of this to develop both image denoising and deconvolution algorithms suitable for real world images. We present results on the cameraman image showing comparable performance with iterative soft thresholding using the undecimated wavelet transform.

Sampling at unknown locations, with an application in surface retrieval

We consider the problem of sampling at unknown locations. We prove that, in this setting, if we take arbitrarily many samples of a polynomial or real bandlimited signal, it is possible to find another function in the same class, arbitrarily far away from the original, that could have generated the same samples. In other words, the error can be arbitrarily large. Motivated by this, we prove that, for polynomials, if the sample positions are constrained such that they can be described by an unknown rational function, uniqueness can be achieved. In addition to our theoretical results, we show that, in 1-D, the problem of recovering a painted surface from a single image exactly fits this framework. Furthermore, we propose a simple iterative algorithm for recovering both the surface and the texture and test it with simple simulations.

Shape: Linear-time camera pose estimation with quadratic error-decay

We propose a novel camera pose estimation or perspective-n-point (PnP) algorithm, based on the idea of consistency regions and half-space intersections. Our algorithm has linear time-complexity and a squared reconstruction error that decreases at least quadratically, as the number of feature point correspondences increase., Inspired by ideas from triangulation and frame quantisation theory, we define consistent reconstruction and then present SHAPE, our proposed consistent pose estimation algorithm. We compare this algorithm with state-of-the-art pose estimation techniques in terms of accuracy and error decay rate. The experimental results verify our hypothesis on the optimal worst-case quadratic decay and demonstrate its promising performance compared to other approaches.