4 research outputs found
Image = Structure + Few Colors
Topology plays an important role in computer vision by capturing
the structure of the objects. Nevertheless, its potential applications
have not been sufficiently developed yet. In this paper, we combine the
topological properties of an image with hierarchical approaches to build a
topology preserving irregular image pyramid (TIIP). The TIIP algorithm
uses combinatorial maps as data structure which implicitly capture the
structure of the image in terms of the critical points. Thus, we can achieve
a compact representation of an image, preserving the structure and topology
of its critical points (maxima, the minima and the saddles). The parallel
algorithmic complexity of building the pyramid is O(log d) where d is
the diameter of the largest object.We achieve promising results for image
reconstruction using only a few color values and the structure of the image,
although preserving fine details including the texture of the image
Characterizing slope regions
This paper provides a theoretical characterization of monotonically connected image
surface regions, called slope regions. The characterization is given by several topological
properties described in terms of critical points relative to the region.We formally
prove the necessary and sufficient conditions that a region needs to satisfy to be a slope
region.We also provide a prototype of slope regions which is general and contains, as
particular cases, the prototypes studied and published in previous conference papers.Ministerio de Ciencia e Innovación PID2019-107339GB-I0
Counting slope regions in the surface graphs
The discrete version of a continuous surface
sampled at optimum sampling rate can be well
expressed in form of a neighborhood graph containing
the critical points (maxima, minima, saddles) of
the surface. Basic operations on the graph such as
edge contraction and removal eliminate non-critical
points and collapse plateau regions resulting in the
formation of a graph pyramid. If the neighborhood
graph is well-composed, faces in the graph pyramid
are slope regions. In this paper we focus on the graph
on the top of the pyramid which will contain critical
points only, self-loops and multiple edges connecting
the same vertices. We enumerate the different
possible configurations of slope regions, forming a
catalogue of different configurations when combining
slope regions and studying the number of slope
regions on the top
A Step Towards Learning Contraction Kernels for Irregular Image Pyramid
A structure preserving irregular image pyramid can be computed by applying basic graph operations (contraction and removal of edges) on the 4 adjacent neighbourhood graph of an image. In this paper, we derive an objective function that classifies the edges as contractible or removable for building an irregular graph pyramid. The objective function is based on the cost of the edges in the contraction kernel (sub-graph selected for
contraction) together with the size of the contraction kernel. Based on the objective function, we also provide an algorithm that decomposes a 2D image into monotonically connected regions of the image surface, called slope regions. We proved that the proposed algorithm results in a graph-based irregular image pyramid that preserves the structure and the topology of the critical points (the local maxima, the local minima, and the saddles). Later we introduce the concept of the dictionary for the connected components of the contraction kernel, consisting of sub-graphs that can be combined together to form a set of contraction kernels. A favorable contraction kernel can be selected that best satisfies the objective function. Lastly, we show the experimental verification for the claims related to the objective function and the cost of the contraction kernel. The outcome of this paper can be envisioned as a step towards learning the contraction kernel for the construction of an irregular image pyrami