spatial filtering

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USING MATLAB IMAGEPROCESSING TOOL

SPATIAL FILTERING TECHNIQUESThe techniques used here include: Average Laplacian Gaussian Unsharp Prewitt Sobel Median

What is Spatial Filtering?

It is an image operation where each pixel value is changed by a function of the intensities of

pixels in a neighbourhood.

Why do we need Spatial Filters?It is necessary to highlight or suppress features

in an image based on spatial frequency such as suppressing noise or highlighting specific image characteristics. This is what spatial filters do.

h(x,y)

How does it work?

Spatial filtering involves selecting a specific mask or kernel of a particular size and correlating or convolving that mask on an input image to receive a new image; the output image which is less noise and more smooth.

The mask chosen is based upon the operation you wish to carry out on the image.

F(x,y) g(x,y) = f(x,y)**h(x,y)

Original image 2D Convolution Product

Output Image

.

CORRELATION CONVOLUTIONA process of moving the

filter mask over the image and computing the sum of products at each location

Correlation of a filter w(x,y) of size m*n with an image f(x,y)is:

A process of moving a previous 180 degrees rotated filter mask over the image and computing the sum of products at each location.

Convolution of a filter w(x,y) of size m*n with an image f(x,y) is:

w(x,y) f(x,y)= ∑∑w(s,t)f(x+s,y+t)

w(x,y) * F(x,y)= ∑∑ w(s,t)f(x-s,y-t)a

b .

LAPLACIAN FILTERINGDefined as the sum of the second derivatives

laplace operator and calculated as the sum of differenced over the nearest neighbours of the central pixel.

It uses highlights intensity discontinuities in the image and deemphasize regions with slow varying intensity levels.

It tends to produce images having greyish edge lines and other discontinuities and dark featureless background

One application operator is to restore fine detail to an image which has been smoothed to remove noise

The Laplacian L(x,y) of an image with pixel intensity values I(x,y) is given by: ∆^2f=d^2f/dx^2+d^2f/dy^2

The chosen kernels approximates the second derivative measurement on the image.

Original Image

Laplacian filter

Original image

Image resulted from convolving a laplacian mask

Image resulted after converting the image to type double and then

convolving with a laplacian mask

Image resulted from subtracting the laplacian

image from original Image

Code c=imread('ultra.jpg'); >> size(c)

ans =

1345 1741 3

>> c1=rgb2gray(c);%converting image to grayscale, working with a 2D image now >> size(c1)

ans =

1345 1741 >> % applying a laplacian filter >> w=fspecial('laplacian',0) % generating the laplacian mask

w =

0 1 0 1 -4 1 0 1 0

>> g1=imfilter(a,w,'replicate');% filtering image with the laplacian mask and the replicate is padding the image in the borders

>> figure,imshow(g1,[]) >> a22=im2double(a); >> g2=imfilter(a22,w,'replicate'); >> figure,imshow(g2,[]) >> g3=a22-g2; >> figure,imshow(g3,[]) >> figure,subplot(2,2,1),imshow(a,[]),subplot(2,2,2),imshow(g1,[]),subplot(2,2,3),imshow(g2,[]),subplot(2,2,4),imshow(g3,[])

Gaussian Smoothing

Gaussian filtering is done by convolving each point in the input array with a gaussian kernel and then summing them to produce the output.

It is the result of blurring an image by a gaussian function typically to reduce image noise and reduce detail. Formulae- G(x,y)=1/2пσ^2*е^-x^2+y^2/(2σ^2).

Where x is the distance from the original image in the horizontal axis.

Where y is the distance from the origin in the vertical axisWhere σ is the standard deviation of the gaussian

distributionThis formula produces a surface whose contours are

concentric circles with a gaussian distribution from the centre point. Values from this distribution are used to build a convolution matrix which is applied to the original image.

Choosing a sigma Value for Gaussian Smoothing

The sigma value can be thought of as an approximation of just how much you want the image to spread or blur in pixels.

It is also referred to as the population standard deviation.

Usually try to choose a sigma value which is 1/3 the filter mask you are using. Here the filter decays to nearly zero at edges and you avoid discontinuities in the filtered image.

The choice of sigma all depends on what you want to do.

Laplacian Filter after GaussianSmoothing

CODE gaussian+laplacian

myfilter=fspecial('gaussian',[3,3],0.5);%generating gaussian mask figure,imshow(imfilter(a22,myfilter,'replicate'));% filtering with gaussian mask >> w=fspecial('laplacian',0)

w =

0 1 0 1 -4 1 0 1 0

>> a22=im2uint8(mat2gray(z)); % increasing the dynamic range of the image >> g5=a22-imfilter(a22,w,'replicate'); >> figure,imshow(g5) >> figure,subplot(3,3,1),imshow(a),title('Original Image'),subplot(3,3,2),imshow(z,

[]),title('Gaussian Smoothing'),subplot(3,3,3),imshow(g5,[]),title('Laplacian Filtering After Gaussian Smoothing')

>> figure,subplot(1,3,1),imshow(a),title('Original Image'),subplot(1,3,2),imshow(z,[]),title('Gaussian Smoothing'),subplot(1,3,3),imshow(g5,[]),title('Laplacian Filtering After Gaussian Smoothing')

Gaussian smoothing with different sigma values + Laplacian Filter

CODE>> myfilter=fspecial('gaussian',[r,c],0.1);

%using a sigma value of 0.1 to generate gaussian kernel

>> z=imfilter(a22,myfilter,'replicate');>> g5=a22-imfilter(a22,w,'replicate');>> myfilter=fspecial('gaussian',[r,c],0.9);>> z=imfilter(a22,myfilter,'replicate');

Laplacian Filter + Gaussian smoothing with different mask size

3 by 3 Mask

5 by 5 Mask

7 by 7 Mask 9 by 9 Mask

CODE

%using different filter mask sizefor r=3:2:9;for c=3:2:9;myfilter=fspecial('gaussian',[r,c]);z=imfilter(a22,myfilter,'replicate');w=fspecial('laplacian',0); a22=im2uint8(mat2gray(z));g5=a22-imfilter(a22,w,'replicate');for n=1:4subplot(2,2,n),imshow(g5,[]);endendend

AVERAGE FILTERReplace each pixel by the average of pixels in a

square window surrounding this pixel.This has the effect of eliminating pixel values

which are unrepresentative of their surroundingsIt is used for reducing noise in an imageIt flattens local differences and reduces

sharpnessHowever, it blurs details and edgesIt is simple and easy to implement method of

smoothing images that is reducing the amount of intensity variations between one pixel and the next.

3 by 3 Average FilterOriginal image

Code AVERAGE FILTER >> w=fspecial('average',[3 3])

w =

0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111

>> gs=imfilter(a,w,'replicate'); >> figure,imshow(gs),title('Image After An Average Filter Is Applied')

>> s=fspecial('average',[5 5])

s =

0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 0.0400 >> gs=imfilter(a,s,'replicate'); >> figure,imshow(gs),title('Image After An Average 5 by 5 Filter Is Applied') >> s=fspecial('average',[11,11]); >> gs=imfilter(a,s,'replicate'); >> figure,imshow(gs),title('Image After An Average 11 by 11 Filter Is Applied')

Mask SizeMasks are usually chosen to have odd

dimensions to provide a centre pixel location. The output is written to that pixel.

Larger filters do more smoothing but also provides more blurring.

The low frequency response becomes more pronounced as the filter is increased

Larger filters removes noise but it also removes detailed information from edged and other image features.

Median FilterIs a non-linear filtering technique often used to

remove noise with laplacian distribution.The main idea of the median filter is to run through

entry by entry replacing each entry with the median of neighbouring entries.

It is used in reducing impulse or salt and pepper noise

It is useful in preserving edges in an image while reducing noise

Can smooth pixels whose values differ significantly from their surroundings without affecting other pixels.

3 by 3 For loop Median FilterOriginal image

Code >> a=imread('ultra.jpg'); size(a)

ans =

1345 1741 3

>> a=rgb2gray(a); >> %pad the matrix with zeros on all sides >> newA=zeros(size(a)+2); >> b=zeros(size(a)); >> %copy the original image matrix to padded matrix >> for x=1:size(a,1) for y=1:size(a,2) newA(x+1,y+1)=a(x,y); end end >> for i=1:size(newA,1)-2 for j=1:size(newA,2)-2 window=zeros(9,1); inc=1; for x=1:3 for y=1:3 window(inc)=newA(i+x-1,j+y-1);% first window selected inc=inc+1; end end med=sort(window); %place the median element in the output matrix b(i,j)=med(5);% the fifth element will be the middle element place that element in the output matrix end end >> %convert the output matrix to 0-255 range image type >> b=uint8(b); >> figure,imshow(b),title('Image After Median Filtering'); >> figure,imshow(a),title('Original Image')

3 by 3 filter with Matlab FunctionOriginal image

5 by 5 Median FilterOriginal image

Code% applying median filtering with matlab built in

functiongm=medfilt2(a,'symmetric');% symmetric is similar

to replicate, defining how you want to pad the imagefigure,imshow(gm),title('Image after Median filter

with Matlab built in function')gm=medfilt2(a,[5 5],'symmetric');figure,imshow(gm,[]),title('Image after 5*5 Matlab

Median Filter')gm=medfilt2(a,[7 7],'symmetric');figure,imshow(gm,[])gm=medfilt2(a,[9 9],'symmetric');figure,imshow(gm,[])

Unsharp FilterIt is a simple sharpening operator which

derives its name from the fact that it enhances edges via a procedure which subtracts an unsharp or smoothed version of the image from the original image.

Unsharp masking produces an edge image g(x,y) from an input image f(x,y).

g(x,y)=f(x,y)-fsmooth(x,y).Sharpening can help to emphasize texture and

detail and is critical when post-processing most digital images.

Sharpening the ImageOriginal image

Code%using a Sharpen Filter Maskce=fspecial('unsharp')cf=imfilter(a,ce,'corr','symmetric');% corr is

correlatingfigure,imshow(cf,[]),title('Image After using

Unsharp Mask')

Smoothing before Sharpening

Code% trying smoothing before sharpening the image by using a

gaussian filtercx=fspecial('gaussian',[3 3],0.5);cff=imfilter(a,cx,'conv','replicate');cx2=fspecial('unsharp');cff2=imfilter(cff,cx2,'conv','replicate');subplot(2,2,4),imshow(cff2,[]),title('Smoothing Image

Before Sharpening')sharpmore2=imfilter(cff2,cx2,'replicate');figure,imshow(sharpmore2,[]),title('Excessive Sharpening

after Smoothing')cff3=imfilter(cff2,cx,'replicate');sharpmore3=imfilter(cff3,cx2,'replicate');figure,subplot(1,2,1),imshow(sharpmore2,

[]),title('Excessive Sharpening after Smoothing is Applied Before ist Sharp Mask'),subplot(1,2,2),imshow(sharpmore3),title('Excessive Sharpening After Smoothing After Ist Sharp Mask Applied')

Smoothing before Sharpening by choosing a kernel independent of

Matlab

Code %Alternative Way of Sharpening the Image subplot(2,2,1),imshow(a),title('Original Ultrasound'); gaussianfilter=[1 4 7 4 1;4 20 33 20 4; 7 33 55 33 7;4 20 33 20 4;1 4 7

4 1];% choosing own weighted kernel gaussianfilter=gaussianfilter/sum(sum(guassianfilter));%normalizing gaussianfilter=gaussianfilter/sum(sum(gaussianfilter)); gaussianUltra=imfilter(a,gaussianfilter); subplot(2,2,2),imshow(gaussianUltra), title('Gaussian Ultra sound'); edgefilter=[-1,-1,-1;8,-1,-1;-1,-1,-1] edgeultra=imfilter(gaussianUltra,edgefilter); subplot(2,2,3),imshow(edgeultra); title('Edges from Blurred Ultrasound'); sharpultra=gaussianUltra+edgeultra; subplot(2,2,4),imshow(sharpultra); title('Sharp Ultrasound(sharper than original');

Disk FilterIs simple a circular averaging filterIt does the same as the average filter

mentioned earlier.The radius controls how big an area the

operator should locate when spreading pixels.

Original image

Code % using a disk filter on the imagecc=fspecial('disk')c3=imfilter(a,cc,'conv','replicate');figure,imshow(c3,[]),title('Image After using a

Disk Mask')

Changing Radius values in Disk Filtering

Radius of 2

Radius of 4

Radius of 6

Radius of 6

Radius of 18

Radius of 16

Radius of 14

Radius of 12

Code for r=2:2:8cs=fspecial('disk',r); fg=imfilter(a,cs,'conv','replicate');% conv refers to convolution for ss=1:4subplot(2,2,ss),imshow(fg,[])endend

for r=12:2:18cs=fspecial('disk',r); fg=imfilter(a,cs,'conv','replicate'); for ss=1:4subplot(2,2,ss),imshow(fg,[])endend

Sobel FilterThe sobel operator is a discrete

differentiation operator, computing an approximation of the gradient of the image intensity function.

It is based on convolving the image with a small, separable and integer valued filter in the horizontal and vertical direction.

The final step involves approximating the gradient magnitude.

Vertical and Horizontal Gradient Sobel Images

Vertical and Horizontal Sobel Image resulted from Subtracting sobel normalized images from original

images

Code % now using a sobel Mask cr=fspecial('sobel') % generate a vertical gradient mask cr=

1 2 1 0 0 0 -1 -2 -1

cr2=imfilter(a,cr,'conv','replicate'); figure,imshow(cr2,[]) crr=cr‘ % will give a horizontal gradient mask cr3=imfilter(a,crr,'conv','replicate'); figure,subplot(1,2,1),imshow(cr2,[]),title('Normalized Vertical gradient

from Sobel operator'),subplot(1,2,2),imshow(cr3,[]),title('Normalized Horizontal gradient from Sobel operator')

cr4=im2uint8(mat2gray(a)); cr5=imfilter(cr4,cr,'conv','replicate'); cr6=cr4-cr5; % to get the final image cr5=imfilter(cr4,crr,'conv','replicate'); cr7=cr4-cr5; figure,subplot(1,2,1),imshow(cr6,[]),title('Vertical Sobel mask Applied') subplot(1,2,2),imshow(cr7,[]),title('Horizontal Sobel Mask Applied')

Sobel Gradient Magnitude & Direction using Gmag Function

Code%Finding the image gradient[gmag,gdir]=imgradient(a,'sobel');figure,imshowpair(gmag,gdir,'montage');title('Gradient Magnitude,Gmag(left), and

Gradient Direction,Gdir(right),using Sobel Method')

imtool(gmag)cd=im2uint8(gmag);

Magnitude of Sobel Gradient directly from the Vertical and Horizontal components

code a=imread('ultra.jpg'); >> a=rgb2gray(a); >> hy=fspecial('sobel'); >> hx=hy'; >> a=im2uint8(mat2gray(a)); >> gx=imfilter(a,hx); %x component of sobel gradient >> gy=imfilter(a,hy); %y component of sobel gradient >> igx2=im2double(gx); >> igy2=im2double(gy); >> gmag=hypot(igx2,igy2); %HYPOT Robust computation of the

square root of the sum of squares

>> figure,imshow(gmag) >> a=im2double(a); >> finalimage=a-gmag; %retrieving final image >> figure,imshow(finalimage);

Prewitt FilterThe prewitt operator is a discrete

differentiation operator, computing an approximation of the gradient of the image intensity function.

At each point in the image, the result of the prewitt operator is either the corresponding gradient vector or the norm of this vector.

The prewitt operator is based on convolving the image with small separable and integer valued filter in horizontal and vertical direction.

Used to detect edges.

Vertical and Horizontal Gradient Prewitt Images

Prewitt gradient Magnitude

code >> hy=fspecial('prewitt') hy =

1 1 1 0 0 0 -1 -1 -1 >> hx=hy'; >> a=im2uint8(mat2gray(a)); >> gx=imfilter(a,hx); %x component prewitt gradient >> gy=imfilter(a,hy); %y component of prewitt gradient >> igx2=im2double(gx); igy2=im2double(gy); gmag=hypot(igx2,igy2); figure,imshow(gmag) a=im2double(a); finalimage=a-gmag; figure,imshow(finalimage);

spatial filtering

Documents

laplacian image

image fx

filtering image

d image

image noise

y original image

input image

image operation