Yll Haxhimusa

Segmentation graph hierarchies

The region’s internal properties (color, texture, ...) help to identify them and their external relations (adjacency, inclusion, ...) are used to build groups of regions having a particular consistent meaning in a more abstract context. Low-level cue image segmentation in a bottom-up way, cannot and should not produce a complete final “good” segmentation. We present a hierarchical partitioning of images using a pairwise similarity function on a graph-based representation of an image. The aim of this paper is to build a minimum weight spanning tree (MST) of an image in order to find region borders quickly in a bottom-up ’stimulus-driven’ way based on local differences in a specific feature.

TRAVELING SALESMAN PROBLEM: A FOVEATING PYRAMID MODEL

We tested human performance on the Euclidean Traveling Salesman Problem using problems with 6–50 cities. Results confirmed our earlier findings that: (a) the time of solving a problem is proportional to the number of cities, and (b) the solution error grows very slowly with the number of cities. We formulated a new version of a pyramid model. The new model has an adaptive spatial structure, and it simulates visual acuity and visual attention. Specifically, the model solves the E-TSP problem sequentially by moving attention from city to city, the same way human subjects do. The model includes a parameter representing the magnitude of local search. This parameter allows modeling individual differences among the subjects. The computational complexity of the current implementation of the model is O(n 2 ), but this can most likely be improved to O[nlog(n)]. Simulation experiments demonstrated psychological plausibility of the new model.

Vision pyramids that do not grow too high

In irregular pyramids, their vertical structure is not determined beforehand as in regular pyramids. We present three methods, all based on maximal independent sets from graph theory, with the aim to simulate the major sampling properties of the regular counterparts: good coverage of the higher resolution level, not too large sampling gaps and, most importantly, the resulting height, e.g. the number of levels to reach the apex. We show both theoretically and experimentally that the number of vertices can be reduced by a factor of 2.0 at each level. The plausibility of log (diameter) pyramids is supported by psychological and psychophysical considerations. Their technical relevance is demonstrated by enhancing appearance-based object recognition. An irregular pyramid hypothesis generation for robust PCA through top-down attention mechanisms achieves higher speed and quality than regular pyramids and non-pyramidal approaches.

Multiresolution Image Segmentations in Graph Pyramids

”How do we bridge the representational gap between image features and coarse model features?” is the question asked by the authors of [47] when referring to several contemporary research issues. They identify the one-to-one correspondence between salient image features (pixels, edges, corners,...) and salient model features (generalized cylinders, polyhedrons, invariant models,...) as a limiting assumption that makes prototypical or generic object recognition impossible. They suggested to bridge and not to eliminate the representational gap, as it is done in the computer vision community for quite long, and to focus efforts on: i) region segmentation, ii) perceptual grouping, and iii) image abstraction. Let us take these goals as a guideline to consider multiresolution representations under the special viewpoint of segmentation and grouping. In [34] multiresolution representation is considered under the abstraction viewpoint. Wertheimer [51] has formulated the importance of wholes (Ganzen) and not of its individual elements and introduced the importance of perceptual grouping and organization in visual perception. Regions as aggregations of primitive pixels play an extremely important role in nearly every image analysis task. Their internal properties (color, texture, shape, ...) help to identify them, and their external relations (adjacency, inclusion, similarity of properties) are used to build groups of regions having a particular meaning in a more abstract context. The union of regions forming the group is again a region with both internal and external properties and relations. Low-level cue image segmentation can not and should not produce a complete final ’good’ segmentation, because there is no general ’good’ segmentation. Without prior knowledge, segmentation based on low-level cues will not be able to extract semantics in generic images. Using some similarity measures, the segmentation process results in ‘homogeneity’ regions with respect to the low-level cues. Problems emerge because i) homogeneity of low-level cues will not map to the semantics [28] and ii) the degree of homogeneity of a region is in general quantified by threshold(s) for a given measure [12]. Even though segmentation methods (including ours) that do not take the context of the image into consideration can not produce a ’good’ segmentation, they can be valuable tools in image analysis in the same sense as efficient edge detectors are. Note that efficient edge detectors do not consider the context of the image, too. Thus, the low-level coherence of brightness, color, texture or motion attributes should be used to sequentially come up with hierarchical partitions [46]. Mid and high level knowledge can be used to either confirm these groups or select some further attention. A wide range of computational vision problems could make use of segmented images, were such segmentation rely on efficient computation, e.g. motion estimation requires an appropriate region of support for finding correspondences; higher-level problems such as recognition and image indexing can also make use of segmentation results in the problem of matching. It is important for a grouping method to have the following properties [10]:

Logarithmic Tapering Graph Pyramid

We present a new method to determine contraction kernels for the construction of graph pyramids. The new method works with undirected graphs and yields a reduction factor of at least 2.0. This means that with our method the number of vertices in the subgraph induced by any set of contractible edges is reduced to half or less by a single parallel contraction. Our method yields better reduction factors than the stochastic decimation algorithm, in all tests. The lower bound of the reduction factor becomes crucial with large images.

Hierarchy of Partitions with Dual Graph Contraction

We present a hierarchical partitioning of images using a pairwise similarity function on a graph-based representation of an image. This function measures the difference along the boundary of two components relative to a measure of differences of component’s internal differences. This definition attempts to encapsulate the intuitive notion of contrast. Two components are merged if there is a low-cost connection between them. Each component’s internal difference is represented by the maximum edge weight of its minimum spanning tree. External differences are the cheapest weight of edges connecting components. We use this idea to find region borders quickly and effortlessly in a bottom-up ’stimulus-driven’ way based on local differences in a specific feature, like as in preattentive vision. The components are merged ignoring the details in regions of high-variability, and preserving the details in low-variability ones.

Directly computing the generators of image homology using graph pyramids

We introduce a method for computing homology groups and their generators of a 2D image, using a hierarchical structure, i.e. irregular graph pyramid. Starting from an image, a hierarchy of the image is built by two operations that preserve homology of each region. Instead of computing homology generators in the base where the number of entities (cells) is large, we first reduce the number of cells by a graph pyramid. Then homology generators are computed efficiently on the top level of the pyramid, since the number of cells is small. A top down process is then used to deduce homology generators in any level of the pyramid, including the base level, i.e. the initial image. The produced generators fit on the object boundaries. A unique set of generators called the minimal set, is defined and its computation is discussed. We show that the new method produces valid homology generators and present some experimental results.

Computing homology group generators of images using irregular graph pyramids

We introduce a method for computing homology groups and their generators of a 2D image, using a hierarchical structure i.e. irregular graph pyramid. Starting from an image, a hierarchy of the image is built, by two operations that preserve homology of each region. Instead of computing homology generators in the base where the number of entities (cells) is large, we first reduce the number of cells by a graph pyramid. Then homology generators are computed efficiently on the top level of the pyramid, since the number of cells is small, and a top down process is then used to deduce homology generators in any level of the pyramid, including the base level i.e. the initial image. We show that the new method produces valid homology generators and present some experimental results.

The eccentricity transform (of a digital shape)

Eccentricity measures the shortest length of the paths from a given vertex v to reach any other vertex w of a connected graph Computed for every vertex v it transforms the connectivity structure of the graph into a set of values For a connected region of a digital image it is defined through its neighbourhood graph and the given metric This transform assigns to each element of a region a value that depends on its location inside the region and the regions shape The definition and several properties are given Presented experimental results verify its robustness against noise, and its increased stability compared to the distance transform Future work will include using it for shape decomposition, representation, and matching.

Constructing stochastic pyramids by MIDES: maximal independent directed edge set

We present a new method (MIDES) to determine contraction kernels for the construction of graph pyramids. Experimentally the new method has a reduction factor higher than 2.0. Thus, the new method yields a higher reduction factor than the stochastic decimation algorithm (MIS) and maximal independent edge set (MIES), in all tests. This means the number of vertices in the subgraph induced by any set of contractible edges is reduced to half or less by a single parallel contraction. The lower bound of the reduction factor becomes crucial with large images.