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

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

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

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

Does an image just map to scalars?

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

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.

A B

Digital vs Continuous Image y 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).

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.

False Contouring

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 ?

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

Example How does it work in 2D?

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

Example How does it work in 2D?

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

Transformations

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

Motivation Transformation in 2D Transformation using homogenous coordinates

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.

Homogenous coordinates we note Proof:

Translation Classic Homogenous coordinates

Rotation (clockwise) Classic Homogenous coordinates

Translation and rotation Classic Homogenous coordinates

Translation, rotation and scaling Classic Homogenous coordinates

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

Demonstration

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

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

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

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

Demonstration