Topographic analysis: change detection algorithm

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PATC Algorithm and Implementation

Version 0.1 Date: 22/10/2011

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Permutation analysis of topographic change

(PATC)

Implementation details

Authors: Neil O’ Leary, Balwantray Chauhan, Paul Artes

Dalhousie University

Purpose

This document provides the algorithmic details for the PATC to enable implementation on

any platform.

List of variables for algorithm

w: width (in pixels) of mean topographic/RNFLT image

T: mean topographic/RNFLT image (w x w)

(x, y): pixel location in image T

w a: width (in super-pixels) of topographic/RNFLT image

(xa, y a ): super-pixel location in image T

s: number of T in longitudinal series

nP: number of permutations = max( n! , 1000 ) (optimal maximum number t.b.c.)

i: index of each examination in series

j: index of reference image to which series is registered

Tji: ith T in series aligned to image j

P- , P+: overall significance for negative and positive change in topographic series




Version 0.1 Date: 22/10/2011

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Algorithm

1. Generate 4 dimensional array M of size s x s x w x w containing image series registered

using each image as a reference image

2. * Pre-process all images in array M (smoothing, super-pixel averaging, etc.) giving array Ma

of size s x s x w a x w a

3. Generate np random reorderings in nP x s matrix (or read from look-up table)

4. Examine observed sequence:

a. Perform univariate test (OLSLR or other candidate)

b.

Compute one-sided test-statistics at each pixel (testing independently for negativeand positive change): t-

x,y and t+x,y (unlike [1]: examines negative change only)

c. Obtain pixel-wise p-values(p-x,y and p+

x,y ) from t-x,y and t+x,y compared to parametric

t-distribution (unlike [1]: uses permutation distribution)

d. † Combine one sided p-values using Fisher equation = S-obs and S+

obs [2]

e. * Use weighting/censoring to combine p-values using (e.g. using Truncated

Product Method [3], clustering [4], effect-size and spatial weighting)

5. Examine each permuted sequence (k = 1 to np )

a. Read k th sequence from permutation list

b. Use series of Tji , (i = 1 to s) to match baseline in permuted sequence

c. Repeat steps 4 (a)-(d) to obtain S-k and S+

k

6. Compare S-obs and S+

obs with distributions of S-k and S+

k to obtain P- and P+ respectively

7. Output observed change maps, P- and P+

* Optional steps: to be optimised




Version 0.1 Date: 22/10/2011

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References

[1] Patterson et al. (2005) “ A New Statistical Approach for Quantifying Change in Series of

Retinal and Optic Nerve Head Topography Images”: Invest Ophthalmol Vis Sci; 46:5,1659-1667

[2] Fisher (1925) "Statistical methods for research workers": Oliver and Boyd

[3] Zaykin et al., (2002) "Truncated product method for combining P-values": Genet

Epidemiol; 22:170-185.

[4] Anselin (1995) "Local Indicators of Spatial Association - LISA": Geographical Analysis;

27:93-115

Total run-times in MATLAB (using matrix-wise linear

regressions) for 8 mean topographies (1000 permutations)

Super-pixel size (pixels) Time (mins) Notes

1 x 1 <9Peripheral 20 pixels not

examined

2 x 2 <4 Registration performed onpre-averaged images

4 x 4 <1Registration performed on

pre-averaged images

Potential speed-ups

Storing data from registration (arrays M and Ma )

Working in single precision vs. double ( Useful for developmental stage only ) Storing “cube” of p-values and slopes from OLS at

each location and each permutation sequence can reduce re-calculation to test differing

combination functions

Topographic analysis: change detection algorithm

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