Introduction to Image Processing Edge Detection and Sharpening Grass Sky Tree ?

Edge Detection and Sharpening The Basic Theory Properties of 1st and 2nd Derivatives Edge Detection gradient operators based on 1st derivatives Laplacian as the 2nd derivative operator Edge Sharpening composite Laplacian unsharp masking

Origin of Edges Edges are caused by a variety of factors depth discontinuity surface color discontinuity illumination discontinuity surface normal discontinuity Edges are caused by a variety of factors Source: Steve Seitz

Edge Detection Edge is marked by points at which image properties change sharply To detect changes in a digital 2D image intensity function, f(x, y), a continuous image can be reconstructed, followed by the differentiation operator OR We take the discrete derivative (finite difference) shown below:

The Basic Theory An intensity jump (edge) generates a peak in the 1st derivative, and the peak generates a zero-crossing in the 2nd derivative (when the first derivative is at a maximum, the second derivative is zero) Gradient methods detect edges by looking for the maximum and minimum in the first derivative of the image Laplacian methods search for zero crossings in the second derivative of the image to find edges

Alternative 1st and 2nd Derivatives Sharpening filters are based on computing spatial derivatives of an image. The first-order derivative of a one-dimensional function f(x) is The second-order derivative of a one-dimensional function f(x) is

Properties of the 1st and 2nd Derivatives

Profile for 1st Derivative 5 4 3 2 1 6 7 -1 6 -6 1 2 -2 7 8

Profile for 2nd Derivative 5 4 3 2 1 6 7 -1 1 6 -12 -4 7 -7 9

Computing 1st Derivative 0 1 2 x -1 1 -2 2 Derivative of function I at 1: I’(1) = (I(2) - I(0))/2 Rearranging it we have: 2*I’(1) = -1*I(0) + 0*I(1) + 1*I(2) Equivalent to local filter operation using -1 -2 1 2 1 2 -1 -2 The standard definition of the Sobel operator omits the 1/8 term doesn’t make a difference for edge detection the 1/8 term is needed to get the right gradient value, however -1 1

1st Derivative Gradient Operators Roberts cross-gradient operators Prewitt operators Sobel operators

1st Derivative Gradient Operators Prewitt masks for detecting diagonal edges Sobel masks for

Properties of Image Gradient First-order derivatives: The gradient of an image Gxy at location (x,y) is defined as the vector: The gradient points in the direction of most rapid increase in intensity The edge strength is given by the gradient magnituge O OR approx. The gradient direction is given by: How does this relate to the edge direction?

Gradients in 2D For an image function, I(x,y), the gradient direction, (x,y), gives the direction of steepest image gradient: (x,y)  atan(Gy/Gx) This gives the direction of a line perpendicular to the edge Gxy Gy  Gx

Sobel Operator Original Sobel

Gradient Operators: Examples

Gradient Operators: Examples

Gradient Operators: Examples

Important Observation Note that Prewitt operator is a box filter convolved with a derivative operator Also note a Sobel operator is a [1 2 1] filter convolved with a derivative operator

Computing 2nd Derivatives 0 1 2 x The theory can be carried over to 2D as long as there is a way to approximate the derivative of a 2D image I’’(1) = (I’(1.5) - I’(0.5))/1 I’(0.5) = (I(1) - I(0))/1 and I’(1.5) = (I(2) - I(1))/1 I’’(1) = 1*I(0) – 2*I(1) + 1*I(2) Equivalent to local filter operation with 1 -2

Laplacian Operator Development of the Laplacian method The two dimensional Laplacian operator for continuous functions: The Laplacian is a linear operator.

Laplacian Operator

Laplacian Operator Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities Original Image Laplacian Filtered Image Laplacian Filtered Image Scaled for Display 23

But That Is Not Very Enhanced! Laplacian Filtered Image Scaled for Display The result of a Laplacian filtering is not an enhanced image We have to do more work in order to get our final image Subtract the Laplacian result from the original image to generate our final sharpened enhanced image 24

Laplacian Image Enhancement - = Original Image Laplacian Filtered Image Sharpened Image In the final sharpened image edges and fine detail are much more obvious 25

Simplified Image Enhancement The entire enhancement can be combined into a single filtering operation 26

Composite Laplacian Mask This gives us a new filter which does the whole job for us in one step -1 5 27

Composite Laplacian Mask

Unsharp Masking & Highboost Filtering to generate the mask: Subtract a blurred version of the image from itself add the mask to the original Highboost: k>1 gmask(x,y) = f(x,y) - f(x,y) g(x,y) = f(x,y) + k*gmask(x,y)

1st & 2nd Derivatives Comparisons Observations: 1st order derivatives generally produce thicker edges 2nd order derivatives have a stronger response to fine detail e.g. thin lines 2nd order derivatives produce a double response at step changes in grey level The 2nd derivative is more useful for image enhancement than the 1st derivative Stronger response to fine detail Simpler implementation Because these kernels approximate a second derivative measurement on the image, they are very sensitive to noise. To counter this, the image is often Gaussian smoothed before applying the Laplacian filter 30

Acknowlegements Slides are modified based on the original slide set from Dr Li Bai, The University of Nottingham, Jubilee Campus plus the following sources: Digital Image Processing, by Gonzalez and Woods http://www.comp.dit.ie/bmacnamee/materials/dip/lectures/ImageProcessing6-SpatialFiltering2.ppt