Lecture 1: Images and image filtering - Cornell University

CS4670: Computer Vision Noah Snavely Lecture 2: Edge detection From Sandlot Science Edge detection Convert a 2D image into a set of curves

Extracts salient features of the scene More compact than pixels Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity Edges are caused by a variety of factors

Images as functions Edges look like steep cliffs Characterizing edges An edge is a place of rapid change in the image intensity function image

Source: L. Lazebnik intensity function (along horizontal scanline) first derivative edges correspond to extrema of derivative

Image derivatives How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image, f, then compute the derivative Option 2: take discrete derivative (finite difference) How would you implement this as a linear filter?

: 1 -1 : -1 1

Source: S. Seitz Image gradient The gradient of an image: The gradient points in the direction of most rapid increase in intensity The edge strength is given by the gradient magnitude: The gradient direction is given by:

how does this relate to the direction of the edge? Source: Steve Seitz Image gradient Source: L. Lazebnik Effects of noise

Noisy input image Where is the edge? Source: S. Seitz Solution: smooth first f

h f*h To find edges, look for peaks in Source: S. Seitz Associative property of convolution

Differentiation is convolution, and convolution is associative: This saves us one operation: f Source: S. Seitz 2D edge detection filters Gaussian

derivative of Gaussian (x) Derivative of Gaussian filter x-direction y-direction The Sobel operator

Common approximation of derivative of Gaussian -1 0 1 1 2

1 -2 0 2 0

0 0 -1 0 1

-1 -2 -1 The standard defn. of the Sobel operator omits the 1/8 term doesnt make a difference for edge detection the 1/8 term is needed to get the right gradient magnitude Sobel operator: example Source: Wikipedia

Example original image (Lena) Finding edges gradient magnitude Finding edges

where is the edge? thresholding Questions?