Introduction to Mathematical tools in used in DIP

Array vs Matrix operations Linear vs Non-Linear operations Digital image processing use various mathematical tools Array vs Matrix operations Linear vs Non-Linear operations Arithmetic operations Set and Logic operations Spatial operations Probabilistic methods

Array vs. Matrix operations Let us consider two images as follows a11 a12 a21 a22 b11 b12 b21 b22 and Array product is given by a11 a12 a21 a22 b11 b12 b21 b22 = a11b11 a12b12 a21b21 a22b22 Now Matrix product is given by a11 a12 a21 a22 b11 b12 b21 b22 = a11b11+a12b21 a11b12+a12b22 a21b11+a22b21 a21b12+a22b22

Arithmetic operations S(x,y)=f(x,y)+g(x,y) S(x,y)=f(x,y)-g(x,y) S(x,y)=f(x,y)*g(x,y) S(x,y)=f(x,y)/g(x,y)

Addition operation is used frequently for Image Enhancement Addition operation is used frequently for Image Enhancement. The noise can be reduced by adding a no.of noisy images and taking the average

A frequent application of image subtraction is in the enhancement of differences between images which are not noticeable by human.

Common use of image multiplication is in masking also called as region of interest operations.

Set operations

Logic operations Logic operations are extensively used in Image morphology.

Spatial operations Single pixel operations Neighborhood operations There are 3 types of spatial operations Single pixel operations Neighborhood operations Geometric spatial transformations

Single pixel operations The simplest operation we perform on a digital image is to alter the values of its individual pixels based on their intensity. It can be expressed as s=T(z) z-intensity of original image s-intensity of transformed image

Neighborhood operations Let Sxy denote the set of neighborhood centered on an arbitrary point (x,y) in an image f. Neighborhood processing generates a corresponding pixel at the same coordinates in an output image g. 1 2 4 6 7 0 2 2 4 1 0 0 5 7 7 7 3 3 2 0 0 0 1 4 3 5 1 1 0 0 1 2 4 6 7 0 2 4 4 1 0 0 5 7 7 7 3 3 2 0 0 0 1 4 3 5 1 1 0 0 S=(1+2+4+2+2+4+5+7+7)/9=4

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Geometric spatial transformations Geometric transformations modify the spatial relationship between pixels in an image. The transformation of coordinates may be expressed as (x,y)=T{(v,w)}. Where (x,y) is coordinates of pixels in transformed image (v,w ) is coordinates of pixels in original image. one of the most used spatial coordinate transformations is the affine transform given by [x y 1]=[v w 1] T By using this we can scale, rotate, translate a set of coordinate system depending on T value

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Image transforms Sometimes it is better to perform image processing tasks in transform domain and applying inverse transform to return to spatial domain.

Example

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