Chapter 10, Part IV

 Definitions: ◦ Let M 1, M 2 ….M R be sets denoting the coordinates of the points in the regional. ◦ Let C(M i ) be a set denoting the coordinates of the points in the catchment basin associated with regional minimum M i. M1M1 M2M2 M3M3 C(M1)C(M1) C(M2)C(M2) C(M3)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 C n (M i ) denote the set of coordinates of points in the catchment basin associated with M i that are flooded at stage n. ◦ Let C[n] denote the union of the flooded catchment basins portion at stage n as: C1(M1)C1(M1) C1(M2)C1(M2) C1(M3)C1(M3) C[1]={C 1 (M 1 ), C 1 (M 2 ), C 1 (M 3 )}

◦ Let Q[n] denote the set of connected components in T[n]. Q[1] = {Q 1, Q 2, Q 3 } Q1Q1 Q2Q2 Q3Q3

 Obtain C[n] from C[n-1] as follows: ◦ For each q  Q[n], consider p = q  C[n-1]. If (a) p is empty : a new minimum is encountered. Add q into C[n-1] to form C[n]. Q[1] = {Q 1, Q 2, Q 3 } Q1Q1 Q2Q2 Q3Q3 Q 4 is a new catchment basin. Add Q 4 to C[2]. Q1Q1 Q2Q2 Q3Q3 Q4Q4 Q[2] = {Q 1, Q 2, Q 3, Q 4 }

(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] = {Q 1, Q 2, Q 3 } Q1Q1 Q2Q2 Q3Q3 Q 1, Q 2, Q 3 belong to existing catchment basins. Add Q 1, Q 2, Q 3 to C[2]. Q1Q1 Q2Q2 Q3Q3 Q4Q4 Q[2] = {Q 1, Q 2, Q 3, Q 4 }

(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]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 dilation First dilation

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

Illustration of Dam Construction

Illustration 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, t i ) and f(x, y, t j ) taken at time t i and t j ◦ Form a difference image d ij (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.

 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, t 1 )… f(x, y, t n ). ◦ Let f(x, y, t 1 ) 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, t 1 ) and let f(x, y, k)=f(x, y, t k ). For any time k>1 three ADIs are defined as follows:

Motion 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 10.20