Edge Linking & Boundary Detection Ideal case: –Techniques detecting intensity discontinuities should yield pixels lying only on edges ( or the boundary.

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),(α2.

Ayxyx <′′− ),(),( αα

where A is an angle threshold.

(x’,y’) and (x,y) are similar if:

Image SegmentationImage 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 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

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:

ρθθ =+ 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

Global Processing viaGraph-Theoretic Techniques

(0)

(1)

(2)

Global Processing viaGraph-Theoretic Techniques

• Minimum cost path:

Image SegmentationImage Segmentation

Image SegmentationImage Segmentation

Image SegmentationImage Segmentation