Edge Linking & Boundary Detection Ms.Mona Deshmukh

Edge Linking & Boundary Detection Ideal case: Techniques detecting intensity discontinuities should yield pixels lying only on edges ( or the boundary between regions). Real life: The detected set of pixels very rarely describes a complete edge due to effects from: noise, breaks in the edge due to non-uniform illumination.

Edge Linking & Boundary Detection Solution: Edge-detection techniques are followed by linking and other boundary detection procedures which assemble edge pixels into meaningful boundaries.

Local Processing Analyze the pixel characteristics in a small neighborhood (3x3, 5x5) about every (x,y) in an image. Link similar points to form a edge/boundary of pixels sharing common properties.

Local Processing Criteria used/Properties: The strength of the response of the gradient operator that produced the edge pixel. The direction of the gradient vector.

Local Processing In other words: or 1. (x’,y’) and (x,y) are similar if: where T is a nonnegative threshold.

Local Processing In other words (cont.): 2. (x’,y’) and (x,y) are similar if: where A is an angle threshold.

Image Segmentation

Global Processing via the Hough Transform Points are linked by determining whether they lie on a curve of specified shape. Problem: Find subsets of n points that lie on straight lines.

Global Processing via the Hough Transform Solution: Find all lines determined by every pair of points Find all subsets of points close to particular lines Involves: n(n-1)/2 ~ n2 lines n(n(n-1))/2 ~ n3 computations for comparing every point to all lines.

Global Processing via the Hough Transform Better solution: Hough Transform Equation of line passing through point (xi,yi): yi = axi + b (a,b varies) But: b = -xia + yi  equation of single line on ab plane

Global Processing via the Hough Transform

Global Processing via the Hough Transform A line in the (x,y) plane passes through several points of interest and has a set of specific (a,b) values. A line in parameter space [(a,b) plane] denotes all lines that pass through a certain point (xi,yi) and has an infinite number of (a,b) values.

Global Processing via the Hough Transform A specific line is represented by a point in the (a,b) plane. Two lines in parameter space that meet at a certain point show points belonging to the same line (in x,y plane).

Global Processing via the Hough Transform Since a,b approach infinity as a line approaches the vertical, we can use the normal representation of a line:

Global Processing via the Hough Transform Hough transform is applicable to any function of the form g(v,c) = 0. v: vector of coordinates, c: coefficients. e.g. points lying on a circle:

Global Processing via Graph-Theoretic Techniques A global approach based on representing edge segments in the form of a graph and searching the graph for low-cost paths that correspond to significant edges.

Global Processing via Graph-Theoretic Techniques Advantage: performs well in the presence of noise Disadvantage: complicated and requires more processing time.

Global Processing via Graph-Theoretic Techniques Graph G = (N,U): A finite, nonempty set of nodes N together with a set U of unordered pairs of distinct elements of N. (ni,nj) of U: arc

Global Processing via Graph-Theoretic Techniques Directed graph: a graph in which arcs are directed If ni to nj is directed, nj is a successor of its parent node ni.

Global Processing via Graph-Theoretic Techniques Expansion of node: To identify the successors of a node Level 0 consists of a single node (start node) Last level contains the goal nodes.

Global Processing via Graph-Theoretic Techniques A sequence of nodes n1,n2,…,nk (with each ni being a successor of ni-1) is called a path from n1 to nk and its cost is:

Global Processing via Graph-Theoretic Techniques Edge element: The boundary between two pixels p & q such that p and q are 4-neighbors. Edge: A sequence of connected edge elements.

Global Processing via Graph-Theoretic Techniques H: the highest intensity value in the image

Global Processing via Graph-Theoretic Techniques (0) (1) (2)

Global Processing via Graph-Theoretic Techniques Minimum cost path:

Image Segmentation

Image Segmentation

Image Segmentation