Transport-based techniques for signal and data analysis have received
increased attention recently. Given their ability to provide accurate generative
models for signal intensities and other data distributions, they have been used
in a variety of applications including content-based retrieval, cancer
detection, image super-resolution, and statistical machine learning, to name a
few, and shown to produce state of the art results in several applications.
Moreover, the geometric characteristics of transport-related metrics have
inspired new kinds of algorithms for interpreting the meaning of data
distributions. Here we provide a practical overview of the mathematical
underpinnings of mass transport-related methods, including numerical
implementation, as well as a review, with demonstrations, of several
applications. Software accompanying this tutorial is available at [43].
Discriminating data classes emanating from sensors is an important problem with many applications in science and technology. We describe a new transform for pattern identification that interprets patterns as probability density functions, and has special properties with regards to classification. The transform, which we denote as the Cumulative Distribution Transform (CDT) is invertible, with well defined forward and inverse operations. We show that it can be useful in 'parsing out' variations (confounds) that are 'Lagrangian' (displacement and intensity variations) by converting these to 'Eulerian' (intensity variations) in transform space. This conversion is the basis for our main result that describes when the CDT can allow for linear classification to be possible in transform space. We also describe several properties of the transform and show, with computational experiments that used both real and simulated data, that the CDT can help render a variety of real world problems simpler to solve.
Invertible image representation methods (transforms) are routinely
employed as low-level image processing operations based on which feature
extraction and recognition algorithms are developed. Most transforms in current
use (e.g. Fourier, Wavelet, etc.) are linear transforms, and, by themselves, are
unable to substantially simplify the representation of image classes for
classification. Here we describe a nonlinear, invertible, low-level image
processing transform based on combining the well known Radon transform for image
data, and the 1D Cumulative Distribution Transform proposed earlier. We describe
a few of the properties of this new transform, and with both theoretical and
experimental results show that it can often render certain problems linearly
separable in transform space.
Free space optical communications utilizing orbital angular momentum beams have recently emerged as a new technique for communications with potential for increased channel capacity. Turbulence due to changes in the index of refraction emanating from temperature, humidity, and air flow patterns, however, add nonlinear effects to the received patterns, thus making the demultiplexing task more difficult. Deep learning techniques have been previously been applied to solve the demultiplexing problem as an image classification task. Here we make use of a newly developed theory suggesting a link between image turbulence and photon transport through the continuity equation to describe a method that utilizes a "shallow" learning method instead. The decoding technique is tested and compared against previous approaches using deep convolutional neural networks. Results show that the new method can obtain similar classification accuracies (bit error ratio) at a small fraction (1/90) of the computational cost, thus enabling higher bit rates.
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