1. What is an image?
f(x,y):2 
Image is a 2D rectilinear array of pixels (picture element)
N=M=256
N=M=30

2. What is an image? No continuous values - Quantization L=1 (1 bit)
L=255 (8 bits)
L=3 (2 bits)
L=15(4 bits) 8
170 15
255

3. An image is just 2D? No! – It can be in any dimension Example 3D:
Voxel-Volume Element

4. An image is just 2D? No! – It can be in any dimension
An image is a n-dimensional rectilinear array of elements

5. Does an image just map to scalars?

6. Roy van Pelt, PhD & Anna Vilanova, PhD TU/e Biomedical Image Analysis Group, 2012

7. Sampling and Quantization
Sampling is digitizing the coordinate values of our function, e.g., f(x,y).
Quantization is digitizing the amplitude values.
In practice the sampling and quantization depend on the sensor arrangement that does the measurements.

8. A B

9. Digital vs Continuous Imagey x mm
Is the distance in mm between samples in x direction
is the distance in mm between samples in y direction y x
Spatial resolution defines the smallest spatial change that we will be able to distinguish, in spatial units!
Measures for that are dots per unit distance dpi, e.g., (dots per inch).

10. Contrast
Dynamic range is lowest and highest intensity level that an image shows
Contrast is the difference in intensity between the highest and the lowest level.
High Dynamic range implies high contrast
Intensity resolution smallest discernible change in intensity level.
Usually integer power of 2, measured by number of bits.
Whether you can distinguish all levels or not depends on human perception.

11. False Contouring

12. We use the data we know to estimate the values in unknown positions.
Image Interpolation
We use the data we know to estimate the values in unknown positions. x ?

13. Image Interpolation–Nearest Neighbour
We use the data we know to estimate the values in unknown positions. x ?

14. Example How does it work in 2D?

15. Image Interpolation – Linear Interpolation
We use the data we know to estimate the values in unknown positions.
Explain the rounding under... x ?

16. Example How does it work in 2D?

17. Interpolation
There are other methods for interpolation of higher order. Meaning more neighbors are involved and more complex curves are fitted.

18. Transformations

19. Motivation How can we transform images?
Apply transformation to all pixels
First do translation, then rotation, then scaling

20. Motivation Transformation in 2D
Transformation using homogenous coordinates

21. Homogenous coordinates
Allow to manipulate n-dim vectors in a n+1-dim space
A point p can be written as vector
In homogenous coordinates we add a scaling factor
To transform the homogenous coordinates in normal coordinate, divide by the n+1 coordinate.

22. Homogenous coordinates
we note
Proof:

23. Translation
Classic
Homogenous coordinates

24. Rotation (clockwise)
Classic
Homogenous coordinates

25. Translation and rotation
Classic
Homogenous coordinates

26. Translation, rotation and scaling
Classic
Homogenous coordinates

27. Affine Transformation
A transformation that preserves lines and parallelism (maps parallel lines to parallel lines) is an affine transformation.

28. Demonstration

29. Rigid Transformations in 3d
Around x-axis (counter-clockwise)
Around y-axis
Around z-axis
General

30. Image transformation Tsd Destination image Source image
For each position Pd in the destination image we search
the pixel color I(Pd).

31. Image transformation Tsd Destination image Source image
First we compute a position Ps in the source image.

32. Image transformation P is not integer. How do we compute I(Pd)=I(Ps)?
Tsd
P is not integer.
How do we compute I(Pd)=I(Ps)?
Answer: by interpolation

33. Demonstration