This document discusses spatial operations in image processing, including mathematical operations, spatial transformations, and intensity assignment. It covers array and matrix operations, linear and nonlinear operations, and basic arithmetic operations between images like addition, subtraction, multiplication and division. Spatial transformations include affine transforms that map pixel coordinates from one image to another. Intensity assignment is done through forward mapping, which can result in holes, or backward/inverse mapping, which interpolates intensities more efficiently.

1. SPACIAL DOMAIN
image processing

2. Contents
• Mathematical Operations in DIP
• Spatial Operations
• Geometric Spatial Transformations
• Intensity Assignment

3. Weeks 1 & 2 3
Introduction to Mathematical Operations in DIP
• Array vs. Matrix Operation
11 12
21 22
b b
B
b b
 
  
 
11 12
21 22
a a
A
a a
 
  
 
11 11 12 21 11 12 12 22
21 11 22 21 21 12 22 22
*
a b a b a b a b
A B
a b a b a b a b
 
 
  
 
 
11 11 12 12
21 21 22 22
.*
a b a b
A B
a b a b
 
  
 
Array product
Matrix product
Array
product
operator
Matrix
product
operator

4. Weeks 1 & 2 4
Introduction to Mathematical Operations in DIP
• Linear vs. Nonlinear Operation:
H is operator, f(x,y) is input image, g(x,y) is output image.
H is said to be a linear operator;
H is said to be a nonlinear operator if it does not meet the above
qualification.
 
( , ) ( , )
H f x y g x y

Additivity
Homogeneity
 
 
( , ) ( , )
( , ) ( , )
( , ) ( , )
( , ) ( , )
i i j j
i i j j
i i j j
i i j j
H a f x y a f x y
H a f x y H a f x y
a H f x y a H f x y
a g x y a g x y
 

 
 
   
 
   
 

5. Weeks 1 & 2 5
Arithmetic Operations
• Arithmetic operations between images are array operations.
The four arithmetic operations are denoted as
s(x,y) = f(x,y) + g(x,y)
d(x,y) = f(x,y) – g(x,y)
p(x,y) = f(x,y) × g(x,y)
v(x,y) = f(x,y) ÷ g(x,y)

6. 6
Example: Addition of Noisy Images for Noise Reduction
Noiseless image: f(x,y)
Noise: n(x,y) (at every pair of coordinates (x,y), the noise is uncorrelated and has zero
average value)
Corrupted image: g(x,y)
g(x,y) = f(x,y) + n(x,y)
Reducing the noise by adding a K set of noisy images, {gi(x,y)}
1
1
( , ) ( , )
K
i
i
g x y g x y
K 
 

7. Image averaging
• Image averaging is a digital image processing technique that is often
employed to enhance video images that have been corrupted by
random noise.
• The algorithm operates by computing an average or arithmetic mean
of the intensity values for each pixel position in a set of
captured images from the same scene or viewfield.

8. 8
Image Averaging

9. Weeks 1 & 2 9
An Example of Image Subtraction: Mask Mode Radiography
Mask h(x,y): an X-ray image of a region of a patient’s body
Live images f(x,y): X-ray images captured at TV rates after injection of the contrast medium
Enhanced detail g(x,y)
g(x,y) = f(x,y) - h(x,y)
The procedure gives a movie showing how the contrast medium propagates through the various
arteries in the area being observed.

10. 10
(b): X-ray images captured after injection of the contrast medium into
the bloodstream (iodine medium: Halogen – dark blue colored)
Image Subtraction

11. Weeks 1 & 2 11
An Example of Image Multiplication

12. Weeks 1 & 2 12
Set and Logical Operations

13. Weeks 1 & 2 13
Spatial Operations (P 107)
1) Single-pixel operations
Alter the values of an image’s pixels based on the intensity.
e.g.,
( )
s T z


14. Weeks 1 & 2 14
Spatial Operations
1) Single-pixel operations
Alter the values of an image’s pixels based on the intensity.
e.g.,
( )
s T z

Coding implementation
example
h
w

15. Weeks 1 & 2 15
Spatial Operations
1) Single-pixel operations
- Inverse imaging program --- running result
Input “Lenna” image Output inversed
“Lenna” image

16. Weeks 1 & 2 16
Geometric Spatial Transformations
► Geometric transformation (rubber-sheet transformation)
— A spatial transformation of coordinates
— intensity interpolation that assigns intensity values to the spatially transformed
pixels.
► Affine transform
( , ) {( , )}
x y T v w

   
11 12
21 22
31 32
0
1 1 0
1
t t
x y v w t t
t t
 
 
  
 
 

17. Weeks 1 & 2 17
( , ) {( , )}
x y T v w

   
11 12
21 22
31 32
0
1 1 0
1
t t
x y v w t t
t t
 
 
  
 
 
 Affine transform
 Geometric transformation

18. Weeks 1 & 2 18
Intensity Assignment
► Forward Mapping
It scans the pixels of the input image and, at each location, (v,w), computes the location (x,y) of
the corresponding pixel in the output image.
- Problem is that it’s possible that two or more pixels can be transformed to the same location
in the output image.
(v,w) is coordinate of input image pixel and (x,y) is coordinate of output image pixel.
► Inverse Mapping
Inverse mapping scans the output pixel location, and then, at each location (x,y) computes the
corresponding location in the input image using inverse mapping equation. It then
interpolates the nearest input pixels to determine the intensity of the output pixel value.
- Inverse mappings are more efficient to implement than forward mappings.
( , ) {( , )}
x y T v w

1
( , ) {( , )}
v w T x y



19. Forward Mapping / Backward Mapping
• Forward Mapping • Backward Mapping
(Inverse Mapping)

20. 20
Forward Mapping / Backward Mapping
• Forward Mapping
- problem: holes
• Backward Mapping