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Computer Vision Spring 2006 15 385 685 Instructor S Narasimhan Wean 5403 T R 3 00pm 4 20pm Announcements Homework 1 is due on Thursday in class Homework 2 will be out on Thursday Start homeworks early Post questions on bboard Binary Images Properties and Methods Lecture 4 Binary Images Images with only two values 0 or 1 Simple to process and analyze Very useful for industrial applications Binary Images Obtained from gray level or color image g x y by Thresholding Characteristic Function b x y 1 if g x y T 0 if g x y T Topics Discussed Geometric Properties Continuous and Discrete Binary Images Multiple Objects Connectivity Sequential iterative processing Selecting a Threshold Bimodal Histogram Threshold Geometric Properties of Binary Images Assume b x y b x y is continuous only one object Area Zeroth Moment A b x y dx dy y Position Center of Mass First Moment 1 x x b x y dx dy A 1 y y b x y dx dy A x Geometric Properties of Binary Images b x y Orientation Difficult to define Axis of least second moment For mass Axis of minimum inertia y y x y x r Minimize x E r b x y dx dy 2 Which equation of line to use y x y r y mx b x 0 m We use x sin y cos 0 are finite Minimizing Second Moment Find and that minimize E for a given b x y We can show that So r x sin y cos E x sin y cos b x y dx dy 2 dE Using 0 we get A x sin y cos 0 d Note Axis passes through the center So change co ordinates x x x x y y y y Minimizing Second Moment We get E a sin 2 b sin cos c cos 2 where a x 2 b x y dx dy b 2 x y b x y dx dy c y 2 b x y dx dy second moments w r t We are not done yet x y Minimizing Second Moment 2 2 E a sin b sin cos c cos b dE Using 0 we get tan 2 a c d sin 2 b b 2 a c 2 cos 2 a c b 2 a c 2 Solutions with ve sign must be used to minimize E Why Emin roundedness Emax Discrete Binary Images Assume b x y is discrete only one object b Area Zeroth Moment A ij j Position Center of Mass First Moment 1 x i bij A b i j i 1 y j bij A Second Moments a i bij 2 b 2 ij bij c j 2 bij Note a b c