Math in image processing

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1 Math in image processing

2 Math in image processing Nyquist theorem

3 Math in image processing Discrete Fourier Transformation

4 Math in image processing Image enhancement: scaling

5 Math in image processing Image enhancement: histogram equalization cumulative histogram improved image

6 Math in image processing Image enhancement: filtering (low- or high-pass) Reducing the amplitudes of low-freq peak we can avoid some of the artefacts.

7 Math in image processing: segmentation An idea is to find the clusters or subsets of the image which can be considered (or its characteristics) as homogeneous. skull CSF (cerebrospinal fluid) White matter or Grey matter

8 Math in image processing: segmentation First and simple way to do it manually (frequently is applied, for example, in the case of tumour segmentation).

9 Math in image processing: segmentation Thresholding

10 Edge-based segmentation Edges MRA Direction High threshold edges Low threshold edges Gradient Math in image processing: segmentation

11 Math in image processing: segmentation FSL: BET Fslview allows one to visualise the results in three projections

12 Math in image processing: segmentation FSL: BET Artefacts from bad parametrisation

13 Math in image processing: segmentation FSL: BET Extracted/rendered brain Extracted/rendered skull

14 Math in image processing: segmentation FSL: FAST WM CSF GM Rendered T1 raw data

15 Math in image processing: segmentation FreeSurfer How to segment properly the GM

16 Math in image processing: segmentation FreeSurfer Convert it into computer model

17 Math in image processing: registration Cerebral cortex can be segmented in special regions (Broadmann, 1909). The question: how can we compare the same regions for different subjects?

18 Math in image processing: registration Idea is to find a transformation T which allows one to align two images. One image is fixed, when another is moving. where S is a similarity, P is a penalty Different variants of the similarity functions: Sum of squared differences Mutual information

19 Math in image processing: registration What kind of geometrical transformation we can use? Rotation (rigid body transformation)

20 Math in image processing: registration What kind of geometrical transformation we can use? Rotation Translation (rigid body transformation)

21 Math in image processing: registration What kind of geometrical transformation we can use? Rotation Translation Scaling (nonrigid)

22 Math in image processing: registration Simple 2D case The same case but generalized to 3D

23 Math in image processing: registration

24 Math in image processing: registration Affine transformation: wiki page example

25 Math in image processing: registration Non-linear transformations

26 Math in image processing: registration Non-linear transformations: often before it we do affine transformation Warp functions This transformation is reversible!

27 Math in image processing: fitting Y We need to fit the measured data to model function X

28 Math in image processing: fitting Y We need to fit the measured data to model function If function is linear it's more or less easy to do: y = ax + b X

29 Math in image processing: fitting Y We need to fit the measured data to model function If function is linear it's more or less easy to do: y = ax + b X Outlier

30 Math in image processing: fitting Robust estimators in regression methods

31 Problems 1. Install FSL ( and try different utilities from the package. 2. Install SPM ( and try different utilities from the package. 3. Install FreeSurfer ( and try different utilities form the package. 4. Install ITK-SNAP, try to segment the images from the given examples. 5. Extract the brain using BET utility with minimal artefacts 6. Extract WM, GM, and CSF tissues using FAST with minimal artefacts 7. Are the unit transformation in registration procedure commutative? 8. Perform a coregistration of 4D volumes of diffusion dataset

32 Literature Bankman, Handbook of medical imaging. Processing and analysis Smith, Digital signal processing Sonka and Fitzpattrick, Handbook of medical imaging, vol.2. Medical image processing and analysis

33 Next topic is Statistics

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