Introduction to Image Processing

Grass

Sky

TreeTree

? ?

Sharpening Spatial Filters

Edge Detection

• The goal is to mark points at which image intensity changes sharply

• Sharp changes in image properties reflect important events

• To detect such changes (edges) find peaks in the 1st derivative of intensity or zero-crossings in the 2nd derivative

The 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

• 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

)()1( xfxfx

f

)(2)1()1(2

2

xfxfxfx

f

Alternative 1st and 2nd Derivatives

Properties of the 1st and 2nd Derivatives

Profile for 1st Derivative

5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7

-1 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0

Profile for 2nd Derivative

5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7

-1 0 0 0 0 1 0 6 -12 6 0 0 1 1 -4 1 1 0 0 7 -7 0 0

1st Derivative in One DimensionI

0 1 2 x

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 0 1

-1 0 1

-2 0 2

-1 0 1

-1 -2 -1

0 0 0

1 2 1

0 1 2

-1 0 1

-2 -1 0

• 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

Computing 2nd Derivatives

I

0 1 2 x

I’’(1) = (I’(1.5) - I’(0.5))/1I’(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 1

• The theory can be carried over to 2D as long as there is a way to approximate the derivative of a 2D image

Roberts cross-gradient operators

Prewitt operators

Sobel operators

1st Derivative Gradient Operators

Prewitt masks for detecting diagonal edges

Sobel masks for detecting diagonal edges

1st Derivative Gradient Operators

• First-order derivatives:– The gradient of an image Gxy at location (x,y) is defined

as the vector:

– The magnitude of this vector:

– The direction of this vector:

yfxf

y

x

G

Gf

Properties of Image Gradient

yx GGf

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

Gx

Gy

Gxy

Sobel Operator

Original Sobel

yx GGf

Gradient Operators: Examples

Gradient Operators: Examples

Gradient Operators: Examples

• 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

Important Observation

• Development of the Laplacian method– The two dimensional Laplacian operator for continuous

functions:

– The Laplacian is a linear operator.

2

2

2

22

y

f

x

ff

),(2),1(),1(2

2

yxfyxfyxfx

f

),(2)1,()1,(2

2

yxfyxfyxfy

f

),(4)]1,()1,(),1(),1([2 yxfyxfyxfyxfyxff

Laplacian Operator

Laplacian Operator

• Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities

OriginalImage

LaplacianFiltered Image

LaplacianFiltered Image

Scaled for Display

Laplacian Operator

But That Is Not Very Enhanced!

• 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

LaplacianFiltered Image

Scaled for Display

fyxfyxg 2),(),(

Laplacian Image Enhancement

In the final sharpened image edges and fine detail are much more obvious

- =

OriginalImage

LaplacianFiltered Image

SharpenedImage

positive. ismask Laplacian theoft coefficiencenter theif ),(

negative. ismask Laplacian theoft coefficiencenter theif ),(),(

2

2

fyxf

fyxfyxg

Simplified Image Enhancement

The entire enhancement can be combined into a single filtering operation

),1(),1([),( yxfyxfyxf )1,()1,( yxfyxf

)],(4 yxf

fyxfyxg 2),(),(

),1(),1(),(5 yxfyxfyxf )1,()1,( yxfyxf

Composite Laplacian Mask

• This gives us a new filter which does the whole job for us in one step

0 -1 0

-1 5 -1

0 -1 0

f2

Composite Laplacian Mask

1st & 2nd DerivativesComparisons

• 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

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