edge linking & boundary detection ms.mona deshmukh
TRANSCRIPT
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:
1. The strength of the response of the gradient operator that produced the edge pixel.
2. The direction of the gradient vector.
Local Processing
• In other words:
(x’,y’) and (x,y) are similar if:
Tyxfyxf ),(),(
where T is a nonnegative threshold.
2/122)( yx GGFmagf or |||| yx GGf 1.
Local Processing
• In other words (cont.):
x
y
G
Gyx 1tan),(2.
Ayxyx ),(),(
where A is an angle threshold.
(x’,y’) and (x,y) are similar if:
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 linesn(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
• 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:
sincos yx
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:
23
22
21 )()( ccycx
Global Processing viaGraph-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 viaGraph-Theoretic Techniques
• Advantage: – performs well in the presence of noise
• Disadvantage: – complicated and requires more processing
time.
Global Processing viaGraph-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 viaGraph-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 viaGraph-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 viaGraph-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:
k
iii nncc
21 ),(
Global Processing viaGraph-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 viaGraph-Theoretic Techniques
)]()([),( qfpfHqpc
H: the highest intensity value in the image