Chapter 10, Part IV

Definitions:◦ Let M1, M2 ….MR be sets denoting the coordinates

of the points in the regional.◦ Let C(Mi) be a set denoting the coordinates of the

points in the catchment basin associated with regional minimum Mi .

M1 M2

M3

C(M1)C(M2)

C(M3)

◦ Let T[n] represent the set of coordinates (s, t) for which g(s,t)<n. i.e., T[n]={(g,s)|g(s,t)<n}

The topology will be flooded in integer flood increments from n=min+1 to n=max +1.

T [n], n = 1

◦ Let Cn(Mi) denote the set of coordinates of points in the catchment basin associated with Mi that are flooded at stage n.

◦ Let C[n] denote the union of the flooded catchment basins portion at stage n as:

R

iin MCnC

1

)(][

C1(M1)C1(M2)

C1(M3)

C[1]={C1(M1), C1(M2), C1(M3)}

◦ Let Q[n] denote the set of connected components in T[n].

Q[1] = {Q1, Q2, Q3}

Q1 Q2

Q3

Obtain C[n] from C[n-1] as follows:◦ For each q Q[n], consider p = qC[n-1]. If

(a) p is empty : a new minimum is encountered. Add q into C[n-1] to form C[n].

Q[1] = {Q1, Q2, Q3}

Q1 Q2

Q3

Q4 is a new catchment basin. Add Q4 to C[2].

Q1 Q2

Q3 Q4

Q[2] = {Q1, Q2, Q3, Q4}

(b) p contains one connected component: q lies within the catchment basin of some regional minimum. Add q into C[n-1] to form C[n]

Q[1] = {Q1, Q2, Q3}

Q1 Q2

Q3

Q1, Q2, Q3 belong to existing catchment basins. Add Q1, Q2, Q3 to C[2].

Q1 Q2

Q3 Q4

Q[2] = {Q1, Q2, Q3, Q4}

(c) p contains more than one component: Flooding would cause water level in these catchment basin to merge. Therefore, a dam must built within q by dilating of p.

C[n-1] C[n] A region in C[n] more than one component

Dilate each region by the structure element.

◦ The dilation has to be constrained to q.

Second dilationFirst dilation

◦ When the dilation is performed on points and result to merge, mark the point as one of dam points.

Illustration of Dam ConstructionIllustration of Dam Construction

Illustration of Morphological WatershedIllustration of Morphological Watershed

Motion is a powerful cue used by human to extract objects from a background.

The detection in motion pictures is important in systems like robotic applications, autonomous navigation, and dynamic scene analysis.

Detection methods can be classified into two categories:Spatial techniques. ◦ Detection in frequency domain.

Basic approach:◦ Detect changes between two image frames

f(x, y, ti) and f(x, y, tj) taken at time ti and tj

◦ Form a difference image dij(x,y), which is define as

The result of difference cancels the stationary element, leaving nonzero parts corresponding to moving objects.

Noise could be an issue.

otherwise0

),,(),,(1),(

Ttyxftyxfifyxd ji

ij

Motivation:◦ The removal of noise could cause the

elimination of small moving objects.◦ To address this problem, consider changes at

a pixel over several frames.

Accumulate differences: ◦ Consider a sequence of image frame f(x, y,

t1)… f(x, y, tn). ◦ Let f(x, y, t1) is the reference image.◦ An accumulative difference image (ADI) is

formed by comparing this reference with every subsequence image.

◦ Each pixel in ADI is essentially a counter that accumulate the difference at that pixel.

Three types of ADIs can be constructed: ◦ absolute, positive and negative ADIs.

Let R(x,y)= f(x, y, t1) and let f(x, y, k)=f(x, y, tk). For any time k>1 three ADIs are defined as follows:

otherwiseyxA

TkyxfyxRifyxAyxA

k

kk ),(

),,(),(1),(),(

1

1

otherwiseyxP

TkyxfyxRifyxPyxP

k

kk ),(

),,(),(1),(),(

1

1

otherwiseyxN

TkyxfyxRifyxNyxN

k

kk ),(

),,(),(1),(),(

1

1

Motion Detection Using ADIsMotion Detection Using ADIs

1. The location and size of the moving object can be determined in the positive ADI.

2. The direction and speed of the moving object can be determined in the absolute and negative ADIs.

A key to the success of these techniques is to select a proper reference image.

The difference of two images cancels the stationary parts in both images.

Extracting the background as a reference image makes it easy to inspect moving objects in motion pictures.

Let the first image as initial image. Use the positive ADI to outline the size and

location of the moving object.

This region becomes entirely black

When the moving object has moved completely out of the position in the reference image, the background in the present frame can be duplicated in the location previously occupied by the object.

When all moving objects are completely out of their original positions, the reference image with the background left can be generated.

Example 10.20Example 10.20