1. Introduction to Mathematical tools in used in DIP

2. 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

3. 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 =
a11b a12b12
a21b a22b22
Now Matrix product is given by
a11 a12
a21 a22
b11 b12
b21 b22 =
a11b11+a12b a11b12+a12b22
a21b11+a22b a21b12+a22b22

4. 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)

5. 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

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

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

8. Set operations

9. Logic operations
Logic operations are extensively used in Image morphology.

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

11. 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

12. 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.
S=( )/9=4

13. Contd…

14. 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

15. Contd…

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

17. Example

18. THANK YOU