Module 3 - Spatial Domain, Slides of Digital Image Processing

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Digital Image Processing

Image Enhancement (Spatial Filtering 2)

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Contents

In this lecture we will look at more spatial filtering techniques

• (^) Spatial filtering refresher
• (^) Sharpening filters
  • (^1) st derivative filters
  • (^2) nd derivative filters
• (^) Combining filtering techniques

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Sharpening Spatial Filters

Previously we have looked at smoothing filters which remove fine detail Sharpening spatial filters seek to highlight fine detail

• (^) Remove blurring from images
• (^) Highlight edges Sharpening filters are based on spatial differentiation

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Spatial Differentiation

Differentiation measures the rate of change of a function Let’s consider a simple 1 dimensional example Images taken from Gonzalez & Woods, Digital Image Processing (2002)

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st

Derivative

The formula for the 1 st derivative of a function is as follows: It’s just the difference between subsequent values and measures the rate of change of the function f ( x 1 ) f ( x ) x f    

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st

Derivative (cont…)

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

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nd

Derivative (cont…)

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

of 39 Using Second Derivatives For Image Enhancement The 2 nd derivative is more useful for image enhancement than the 1 st derivative

• (^) Stronger response to fine detail
• (^) Simpler implementation
• (^) We will come back to the 1 st order derivative later on The first sharpening filter we will look at is the Laplacian
• (^) Isotropic
• (^) One of the simplest sharpening filters
• (^) We will look at a digital implementation

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The Laplacian (cont…)

So, the Laplacian can be given as follows: We can easily build a filter based on this

[ ( 1 , ) ( 1 , )

2  f  f x  y  f x  y  f ( x , y  1 )  f ( x , y  1 )]  4 f ( x , y ) 0 1 0 1 -4 1 0 1 0

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The Laplacian (cont…)

Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities Images taken from Gonzalez & Woods, Digital Image Processing (2002) Original Image Laplacian Filtered Image Laplacian Filtered Image Scaled for Display

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Laplacian Image Enhancement

In the final sharpened image edges and fine detail are much more obvious Images taken from Gonzalez & Woods, Digital Image Processing (2002)

Original Image Laplacian Filtered Image Sharpened Image

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Laplacian Image Enhancement

Images taken from Gonzalez & Woods, Digital Image Processing (2002)

of 39 Simplified Image Enhancement (cont…) This gives us a new filter which does the whole job for us in one step 0 -1 0 -1 5 - 0 -1 0 Images taken from Gonzalez & Woods, Digital Image Processing (2002)

of 39 Simplified Image Enhancement (cont…) Images taken from Gonzalez & Woods, Digital Image Processing (2002)