Abstract: A computer-implemented method includes causing a computer system to execute instructions for providing a first data set and a second data set, each derived from a common object, providing a first tomographic image object associated with the first data set providing a second tomographic image object associated with the second data set, generating a multimodal pixon map for pixon smoothing on the basis of the first data set, the first tomographic image object, the second data set, and the second tomographic image object, and outputting the multimodal pixon map.

Abstract: Computer-implemented methods of reconstructing an image object for a measured object in object space from image data in data space cause a computer system to execute instructions for providing zonal information separating the object space into at least two zones, providing at least two zonal image objects, each zonal image object being associated with one of the at least two zones, performing a zonal smoothing operation on at least one of the zonal image objects, thereby generating at least one smoothed zonal image object, reconstructing the image object on the basis of the at least one smoothed zonal image object, and outputting the image object.

Abstract: A computer-implemented method of presenting an image of an object includes causing a computer to execute instructions for providing a signal distribution of values N, generating a transformed distribution by calculating, for each value N, a transformed value X=?{square root over (N+3/8)}, and outputting the transformed distribution.

Abstract: Determining a pixon map for pixon smoothing of an object based on a data set includes receiving the data set and an input object associated to the data set. Determining a pixon map further includes determining, in a series of steps, statistical objects for a set of pixon kernel functions, wherein each step includes selecting a pixon kernel function from the set of pixon kernel functions, smoothing the input object on the basis of the selected pixon kernel function, thereby creating a smoothed object, and determining the statistical object for the selected pixon kernel function on the basis of the smoothed object, the data set, and a Mighell-like statistical weight. Determining a pixon map further includes determining contributions of the pixon kernel functions to the pixon map based on the statistical objects and assigning values to the pixon map corresponding to the contributions of the pixon kernel functions.

Abstract: In an aspect, tomographically reconstructing a 3D image object corresponding to a data set includes for each step in a series of iteration steps, determining an updated object by performing a combined operation, which includes performing an update operation for updating an input object and performing a smoothing operation, and following a last iteration, outputting one of the updated objects as the 3D image object.

Abstract: An iterative reconstruction method to reconstruct an object includes determining, in a series of iteration steps, updated objects, wherein each iteration step includes determining a data model from an input object, and determining a stop-criterion of the data model on the basis of a chi-square-gamma statistic. The method further includes determining that the stop-criterion of the data model has transitioned from being outside the limitation of a preset threshold value to being inside the limitation, ending the iterations, and selecting one of the updated objects to be the reconstructed object.

Abstract: In an aspect, tomographically reconstructing a 3D image object corresponding to a data set includes reconstructing a first reconstructed object from the data set, receiving a smoothing map, smoothing the first reconstructed object based on the smoothing map thereby creating a first smoothed object, and outputting the first smoothed object as the 3D image object.

Abstract: In an aspect, reconstructing an image object associated with an input data set, includes for each step, in a series of iteration steps, determining an updated object from an input object based on a data model of the input object, a weighting-factor data set, and a gradient object, wherein the weighting-factor data set includes modified entries of the input data set. Reconstructing the image object further includes following a last iteration, outputting one of the updated objects as the image object.

Abstract: A computer-based system and method are provided to determine the minimum number of factors required to account for input data by seeking an approximate minimum complexity model. In an exemplary embodiment, covariance in the daily returns of financial securities is estimated by generating a positive-definite estimate of the elements of a covariance matrix consistent with the input data. Complexity of the covariance matrix is minimized by assuming that the number of independent parameters is likely to be much smaller than the number of elements in the covariance matrix. Each variable is described as a linear combination of independent factors and a part that fluctuates independently. The simplest model for the covariance matrix is selected so that it fits the data to within a specified level as determined by the selected goodness-of-fit criterion.

Abstract: A curvature-preserving filter having a null covariance matrix is applied to an input image to produce a denoised output image for output to a graphic display device or to a machine analysis tool. In one embodiment, the input image is a small kernel consisting of a limited number of pixels and the filter is applied to the input image by direct summation. In another embodiment, a digital image is input into an image processor that executes a Fourier transform to produce a Fourier-transformed signal. The curvature-preserving filter is applied to the Fourier-transformed signal in Fourier space to produce a denoised signal, then the denoised signal is transformed by an inverse Fourier transform to generate a denoised output image In an alternate embodiment, the filter further produces a deblurred signal by including an inverse point-response function.

Abstract: Input data comprising a video signal is processed using a combination of a known image processing method to deblur, or sharpen, the image and convolution with Pixon™ kernels for smoothing. The smoothing process utilizes a plurality of different size Pixon™ kernels which operate in parallel so that the input data are convolved with each different Pixon™ kernel simultaneously. The smoothed image is convolved with the point response function (PRF) to form data models that are compared against the input data, then the broadest Pixon™ kernel that fits the input data within a predetermined criterion are selected to form a Pixon™ map. The data are smoothed and assembled according to the Pixon™ map, then are deconvolved and output to a video display or other appropriate device, providing a clearer image with less noise.

Abstract: The method identifies a Pixon element, which is a fundamental and indivisible unit of information, and a Pixon basis, which is the set of possible functions from which the Pixon elements are selected. The actual Pixon elements selected from this basis during the reconstruction process represents the smallest number of such units required to fit the data and representing the minimum number of parameters necessary to specify the image. The Pixon kernels can have arbitrary properties (e.g., shape, size, and/or position) as needed to best fit the data.

Abstract: The method identifies a Pixon element, which is a fundamental and indivisible unit of information, and a Pixon basis, which is the set of possible functions from which the Pixon elements are selected. The actual Pixon elements selected from this basis during the reconstruction process represents the smallest number of such units required to fit the data and representing the minimum number of parameters necessary to specify the image. The Pixon kernels can have arbitrary properties (e.g., shape, size, and/or position) as needed to best fit the data.

Abstract: The method identifies a Pixon element, which is a fundamental and indivisible unit of information, and a Pixon basis, which is the set of possible functions from which the Pixon elements are selected. The actual Pixon elements selected from this basis during the reconstruction process represents the smallest number of such units required to fit the data and representing the minimum number of parameters necessary to specify the image. The Pixon kernels can have arbitrary properties (e.g., shape size, and/or position) as needed to best fit the data.

Abstract: The method identifies a Pixon element, which is a fundamental and indivisible unit of information, and a Pixon basis, which is the set of possible functions from which the Pixon elements are selected. The actual Pixon elements selected from this basis during the reconstruction process represents the smallest number of such units required to fit the data and representing the minimum number of parameters necessary to specify the image. The Pixon kernels can have arbitrary properties (e.g., shape, size, and/or position) as needed to best fit the data.

Abstract: The method identifies a Pixon element, which is a fundamental and indivisible unit of information, and a Pixon basis, which is the set of possible functions from which the Pixon elements are selected. The actual Pixon elements selected from this basis during the reconstruction process represents the smallest number of such units required to fit the data and representing the minimum number of parameters necessary to specify the image. The Pixon kernels can have arbitrary properties (e.g., shape, size, and/or position) as needed to best fit the data.