Topology-Based Method of Partition, Analysis, and Simplification of Dynamical Images and its Applications
First Claim
1. In a model of a video as a sequence of binary 2D images, wherein the improvement comprises a dynamical image:
- a time-dependent stream of pluralities of objects called frames, aligned with each other and an adjacency relation defined on these frames.
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Abstract
A dynamical image of is an array of black-and-white images, or frames, of arbitrary dimension. Dynamical images are constructed from gray scale and color images, video sequences etc. A method of topological analysis and decomposition of dynamical images through computation of homology groups of the frames is provided. Each frame is partitioned into a collection of components, which, in turn, have tunnels, voids, and other higher dimensional cycles. The cycles in each frame are linked to the cycles in each adjacent frame to record how they merge and split. Further, the dynamical image is simplified by removing from frames all cycles that are small in terms of length, area, volume, etc, or lifespan. Applications of the method lie in image enhancement and restoration, motion tracking, computer vision, surface and curve reconstruction, scientific image analysis, image recognition and matching.
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Citations
48 Claims
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1. In a model of a video as a sequence of binary 2D images, wherein the improvement comprises a dynamical image:
- a time-dependent stream of pluralities of objects called frames, aligned with each other and an adjacency relation defined on these frames.
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2. A method of analyzing time-dependent images wherein:
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homology groups of a filtration sequence of simplicial complexes are incrementally computed; and
comprising steps of;
A) processing an input image by providing a representation of the input image as a dynamical image comprising a set of frames;
B) analyzing the dynamical image by providing;
(1) a set of topological features of the set of frames of the dynamical image;
(2) a set of relations between the topological features of a pair of adjacent frames comprising at least one of;
matching, appearing, disappearing, merging, and splitting of topological features. - View Dependent Claims (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28)
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29. A method of creating a new subcomplex with Betti numbers within a specified range from a given subcomplex of an ambient cell complex by a topologically constrained transformation comprising the steps of:
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A) creating a target subcomplex with Betti numbers within the specified range;
B) creating a current subcomplex as a copy of the given subcomplex;
C) repeatedly morphing the current subcomplex toward the target subcomplex by adding and removing cells of the current subcomplex and stopping no later than the Betti numbers of the current subcomplex fall within the specified range; and
D) repeatedly morphing the current subcomplex toward the given subcomplex as a new target subcomplex by adding and removing cells of the current subcomplex, so that each adding or removing step is cancelled if the step results in the Betti numbers of the current subcomplex outside the specified range of Betti numbers, until the current subcomplex does not change; and
whereby the method cuts the given subcomplex into pieces and adds other topological features to the given subcomplex providing a resulting subcomplex. - View Dependent Claims (30, 31, 32, 33, 34, 35, 36, 37)
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38. A method of removing a plurality of frames from a dynamical image, comprising, for each frame A in the plurality of frames, the steps of:
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A) finding all topological features in the frame A that have a predetermined plurality of combinations of characteristics outside given ranges;
B) removing the topological features found in step (A) from the dynamical image;
C) choosing a set of one or several frames adjacent to the frame A and adding each cell of the frame A to the set of one or several frames;
D) updating an adjacency relation of the dynamical image by making each pair of frames adjacent to the frame A adjacent to each other; and
E) removing the frame A from the dynamical image. - View Dependent Claims (39)
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- 40. A method of measuring a similarity of two aligned images, computed as a difference of a total size of a collection A of objects in one of the two aligned images and a total size of the collection B of all objects in an other of the two aligned images overlapping A, or zero if the difference is negative, added over all collections A in a given plurality of collections of objects.
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42. A method of finding a point in each of two given images represented by density functions f(x) and g(y) respectively, where x is a position in a first image and y is a position in a second image, as a way to align the first image and the second image, comprising the steps of:
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A) finding the centers of mass a and b of the first image and the second image;
B) choosing a function of two variables p(z,r);
C) creating two new density functions F(x)=p(f(x),|x−
a|)) and G(y)=p(g(y),|y−
b|)), where |x−
a| is a distance from x to a, |y−
b| is a distance from y to b; and
D) computing centers of mass c and d of the first image and the second image with densities F(x) and G(y), respectively; and
whereby the method assigns a new density value to each pixel depending only on its current density and its distance to a in the first image and b in the second image, and provides a pair c, d of aligned points for each choice of the function p. - View Dependent Claims (43)
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- 44. A method of evaluating a saliency of a topological feature in an image of an object given by a density distribution function comprising computing the probability that a cycle corresponding to the topological feature is not homologous to zero, whereby the method evaluates a probability of a loss of integrity of the object.
Specification