06 spatial filtering DIP

Spatial Filtering : 1

Spatial FilteringSpatial Filtering


Basics of Spatial FilteringBasics of Spatial Filtering

Image enhancement operations can also work with the values of the image pixels in the neighborhood and the corresponding values of a subimage that has the same dimensions as the neighborhood.

The subimage is called a filter, mask, kernel, template, or window.

The values in a filter subimage are referred to as coefficients, rather than pixels.

The process consists simply of moving the filter mask from point to point in an image.

For linear spatial filtering, the response is given by a sum of products of the filter coefficients and the corresponding image pixels in the area spanned by the filter mask


Linear Spatial FilterLinear Spatial Filter


Linear Spatial Filter (Cont.)Linear Spatial Filter (Cont.)

The result (or response), R, of linear filtering with the filter mask at a point (x, y) in the image is:

In general, linear filtering of an image f of size M X N with a filter mask of size m X n is given by the expression:

( 1, 1) ( 1, 1) ( 1,0) ( 1, )

(0,0) ( , ) (1,0) ( 1, ) (1,1) ( 1, 1)

R w f x y w f x y

w f x y w f x y w f x y

( 1)/2 ( 1)/2

( 1)/2 ( 1)/2

( , ) ( , ) ( , )m n

s m t n

g x y w s t f x s y t



When interest lies on the response, R, of an m X n mask at any point (x, y), and not on the mechanics of implementing mask convolution

1 1 2 2

1

mn mn

mn

i ii

R w z w z w z

w z



1 1 2 2 9 9

9

1

i ii

R w z w z w z

w z


Noise reduction by Spatial FilteringNoise reduction by Spatial Filtering

Image averaging takes advantage of information redundancy from the individual images to reduce noise

Not always possible to acquire so many images!

Alternative option: Take advantage of information redundancy from different pixels within the same image to reduce noise


Averaging FilterAveraging Filter

Instead of averaging between images, we can average neighboring pixels

9

1

1

9 ii

R z


Averaging FilterAveraging Filter


Weighted Average FilterWeighted Average Filter

Problem: Simple averaging of neighboring pixels lead to over-smoothing

Possible solution: Instead of weighting all neighboring pixels equally, assign higher weights to pixels that are closer to the pixel being convolved


Weighted Average FilterWeighted Average Filter


Weighted Averaging Filter:Weighted Averaging Filter:ExampleExample


Order-Statistic FiltersOrder-Statistic Filters

Nonlinear spatial filters

Steps

Order pixels within an area

Replace value of center pixel with value determined by ordering

Best known example: median filters

( ) ( ) ( )H af bg aH f bH g


Median FilterMedian Filter

Provides good noise reduction for certain types of noise such as impulse noise

Considerably less blurring than weighted averaging filter

Forces a pixel to be like its neighbors

Steps

Order pixels within an area

Replace value of center pixel with median value (half of all pixels have intensities greater than or equal to the median value)


Median Filter: ExampleMedian Filter: Example


Spatial Filtering: SharpeningSpatial Filtering: Sharpening

Goal: highlight or enhance details in images

Some applications:

Photo enhancement

Medical image visualization

Industrial defect detection


Sharpening Spatial FilteringSharpening Spatial Filtering

Goal: highlight or enhance details in images

Some applications:

Photo enhancement

Medical image visualization

Industrial defect detection

Basic principle:

Averaging (blurring) is analogous to integration

Therefore, logically, sharpening accomplished by differentiation


Derivatives of digital functionDerivatives of digital function

First-order derivative

Second-order derivative

( 1) ( )f

f x f xx

2

2( 1) ( 1) 2 ( )

ff x f x f x

x


Derivatives of digital functionDerivatives of digital function

First-order derivatives generally produce thicker edges

Second-order derivatives have stronger response to fine detail (e.g., thin lines and points)

First-order derivatives have stronger response to step changes

Second-order derivatives produce double response at step changes


Derivatives of digital function: ExampleDerivatives of digital function: Example


Sharpening using LaplacianSharpening using Laplacian

Second-order derivatives is better suited for most applications for sharpening

How?

By constructing a filter based on discrete formulation of second-order derivatives

Simplest isotropic derivative operator: Laplacian

2 22

2 2

f ff

x y


How to create Laplacian filterHow to create Laplacian filter

2 22

2 2

f ff

x y

2

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

ff x y f x y f x y

x

2

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

ff x y f x y f x y

y

2 ( 1, ) ( 1, ) ( , 1) ( , 1) 4 ( , )f f x y f x y f x y f x y f x y


Laplacian FilterLaplacian Filter


Sharpening using LaplacianSharpening using Laplacian

Problem: While applying Laplacian highlights fine detail, it de-emphasizes smooth regions (e.g., background features)

Results in featureless background with grayish fine details

Solution: Add original image to recover background features

2

2

( , ) ( , ), negative filter center( , )

( , ) ( , ), positive filter center

f x y f x yg x y

f x y f x y


ExampleExample


Sharpening using Unsharp MaskingSharpening using Unsharp Masking

Subtract blurred version of image from the

image itself to produce sharp image

( , ) ( , ) ( , )g x y f x y f x y


ExampleExample


Sharpening using High-boost FilteringSharpening using High-boost Filtering

Generalization of unsharp masking

As A increases, contribution of sharpening decreases

( , ) ( , ) ( , )g x y Af x y f x y


Sharpening using High-boost FilteringSharpening using High-boost Filtering


Example


First Derivatives for EnhancementFirst Derivatives for Enhancement

Implemented as magnitude of gradient in image processing

f

xf

f

y

1/222f f

fx y


ImplementationImplementation

Problem: Expensive to implement directly

Solution: Approximate using absolute values

Can be implemented using two 2x2 filters

Even filters difficult to implement, so 3x3 filters are used for approximation

9 5 8 6f z z z z

7 8 9 1 2 3

3 6 9 1 4 7

( 2 ) ( 2 )

( 2 ) ( 2 )

f z z z z z z

z z z z z z


Gradient MasksGradient Masks


Example: Defect DetectionExample: Defect Detection

06 spatial filtering DIP

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