Image Processing and Analysis (ImagePandA)

9 – Shape

Christoph Lampert / Chris Wojtan

Outline

Representation Boundary following Signatures

R(theta) Convex deficiency Skeletons

Shape Descriptors Boundary

Fourier descriptors Statistical Moments

Regional Simple: area, permieter, ratio, … Topology (Euler characteristic) E = C-H Statistical Moments

Moment Invariants

Morphing Blending Morphing

Level set methods

2

Overview

Want to be able to compare and describe shapes and images Need meaningful geometric representations Need metrics which are similar for similar images

Boundary description Shape of boundary/border

Region description Color/texture/stuff within shape

3

Simple metrics

Area Perimeter Ratio between Area & Perimeter

CS 484, Fall 2012 ©2012, Selim Aksoy 4

Boundary following1. Let be the uppermost, leftmost point on the shape

Denote the west neighbor of . Examine the 8-neighbors of , starting with and proceeding

clockwise. Let be the neighbor which is a part of the shape Let be the neighbor immediately preceding Store and

2. Let and 3. Starting with , find the first clockwise neighbor of

which is on the shape4. Let be that neighbor and be the neighbor before it.5. Repeat steps 3 and 4 until and the next boundary

point found is . The ordered sequence is the boundary.

5

Boundary following

6

Boundary following

7

Distance vs. Angle signature

8

Invariant to translation, scale, and rotation Assuming: start from centroid, consistently pick

starting point

Distance vs. Angle signature

9

Distance vs. Angle signature

10

Invariant to translation, scale, and rotation, assuming: start from centroid

How would you compute the centroid? consistently pick starting point

eg farthest from centroid Parameterize by angle

so # of pixels is irrelevant

Convex defficiency

11

Convex shape All points within shape can be connected by

straight lines without leaving the shape Convex hull of an arbitrary shape

Connect all points in with lines to produce a new shape

is the convex hull of

Convex defficiency

12

The ratio in area between and can give us a measure of a shape’s convexity

Invariant to scale, rotation, translation You can also use this to segment a boundary

Skeletons

13

The difference/ratio in area between and can give us a measure of a shape’s convexity

You can also use this to segment a boundary

Moment Invariants

2-D moment of order of a digital image of size is defined as

where and are integers.

14

Moment Invariants

2-D central moment of order is defined as

where and

The normalized central moments are:

where

15

Moment Invariants

Invariant moments

)

16

Moment Invariants

are invariant to: Translation Scale change Mirroring (within a minus sign) Rotation

17

Moment Invariants

18