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Publication# On the Bandwidth of the Plenoptic Function

Minh Do, Davy Marchand-Maillet, Martin Vetterli

*Institute of Electrical and Electronics Engineers, *2012

Journal paper

Journal paper

Abstract

The plenoptic function (POF) provides a powerful conceptual tool for describing a number of problems in image/video processing, vision, and graphics. For example, image based-rendering can be seen as sampling and interpolation of the POF. In such applications, it is important to characterize the bandwidth of the POF. We study a simple but representative model of the scene where bandlimited signals (e.g. texture images) are "painted" on smooth surfaces (e.g. of objects or walls). We show that in general the POF is not bandlimited unless the surfaces are flat. We then provide simple rules to estimate the essential bandwidth of the POF for this model. Our analysis reveals that, in addition to the maximum and minimum depths, the bandwidth of the POF also depend on the maximum surface slope and maximum frequency of painted signals. With a unifying formalism based on multidimensional signal processing, we can verify several key results in POF processing, such as induced filtering in space and depth correction interpolation, and quantify the necessary sampling rates.

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Interpolation

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.

Sampling (signal processing)

In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a val

Signal

In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages b

Sampling has always been at the heart of signal processing providing a bridge between the analogue world and discrete representations of it, as our ability to process data in continuous space is quite limited. Furthermore, sampling plays a key part in understanding how to efficiently capture, store and process signals. Shannon's sampling theorem states that if the original signal is known to have a limited bandwidth, we can retrieve the signal from uniformly-spaced samples, provided that the sampling rate is greater than twice the highest frequency in the signal. Here, we see two key attributes: prior knowledge on the original signal (limited bandwidth) and a constrained sampling setup (uniform samples at a particular rate). In this thesis, we make weaker assumptions on the sampling setup by assuming that some information, such as the sample positions, is lost. We show that under proper prior knowledge, we can reconstruct the signal from its samples uniquely or up to some equivalence class. We start by the problem of linear sampling of discrete signals, where the sample values are known, but their order is lost. In general, the original signal is impossible to retrieve from the samples, but we show that by taking out symmetry from the sampling vectors, we can reconstruct the signal uniquely. We provide an efficient algorithm to find the sample orders and thus reconstruct the original signal. We also study the problem of reconstructing a continuous signal from samples taken at unknown locations. The lost sample locations take away any hope of uniquely retrieving the signal without prior knowledge. We show that this problem is equivalent to reconstructing a composite of functions from uniformly spaced samples. Then we provide an efficient algorithm that can recover bandlimited signals warped by a linear function uniquely given enough sampling frequency. We then investigate a problem, dubbed shape from bandwidth, where we have uniform samples from a picture (projection) of an unknown surface that is painted with an unknown texture. The goal is to reconstruct the shape of the surface from these samples. We show that having prior knowledge of the texture bandwidth provides us with enough information to reconstruct the surface from its picture. We provide reconstruction algorithms for both orthogonal and central projections and provide equivalence classes of solutions in each case. Next, in two consecutive chapters, using techniques from geometrical signal processing, we offer new designs for 3-D barcodes, whose information can be retrieved from a single projection using penetrating waves from an unknown direction. Because of the unknown scan direction, the correct correspondence of the samples to the information bits in the barcodes is lost. In this case, we use the known shape of the barcode as prior knowledge to estimate the unknown scan direction from the samples, and then transform the reconstruction as a linear problem that can be solved efficiently. Finally, we cover the theory of coordinate difference matrices (CDMs): matrices that store mutual differences between coordinates of points (sensors, microphones, molecules, etc.) in space. We show how we can leverage specific properties of these matrices, such as their low rank, as prior knowledge in order to reconstruct the position of the points in space using CDMs. We use our reconstruction algorithm to solve many real-life signal processing problems.