Targil 2 Image enhancement and edge detection. For both we will use image derivatives.

Image enhancement Histogram enhancement (histogram equalization…) Reducing noise (smoothing, median) Sharpening Emphasize the details Make the edges stronger Problem: we magnify the noise

Sharpening: Subtracting The Laplacian F(x) F’(x) F’’(x) F(x)-F’’(x)

Reminder : Convolution Kernel, Convolver image For example: means that

Image derivatives (Convolve with [1 -1]) (Convolve with [1 -1] T ) A better kernel: (Convolve with ½*[1 0 -1])

Image derivatives (cont’) Problem: the image is not continuous. A better approximation: Locally approximate the image with a smooth surface. Compute the derivatives of this surface. Popular kernels:

The second derivative Check that:

The Laplacian Equation: The matrix: Subtracting the Laplacian:

Sharpening Example

Edge Detection Why do we need it ? A compact representation of the image More robust to light changes. Easier to follow (tracking and computations of camera motion) Segmentation: usually, edges are located at transitions between objects Used for texture analysis

Edge Detection What are “edges” ? How to find the edges ? How to compute the exact location of an edge ? T-junction Transition between objects Texture Noise Wide edge

The gradient The vector of derivatives Edge Size Edge Direction Derivative in Direction 

The gradient OriginalGradient

* = Example: Derivatives 0 1 * = I x = I y =

= Gradient I x 2 + I y 2

Edge Localization-Zero Crossing Where exactly is the edge ? Zero crossing of f’’ f f’’ Problem: f’’ is very noisy Smooth first !

A smoothing with a 2D Gaussian (We usually use the binomial coefficients instead.)

Canny Edge Detection Computing the image derivatives Gx, Gy –Smoothing with a Gaussian. –Using simple derivative kernels. Compute the edge direction: Take only the local maxima in that direction (to get an edge with width 1) Hysteresis: Edge linking with two thresholds Q.: What will be the width of the Gaussian?

Example Original Canny