Chapter 8: Analysis of Oriented Patterns

Tuomas NeuvonenTommi Tykkälä

Outline

Oriented patterns in medical images Metrics Directional filtering Gabor filters Directional analysis & multiscale edge

detection Hough-Radon transform Example usage of Gabor filter

Oriented patterns

In natural materials, strength and functionality is derived from highly coherent structures and fibers

Bones, muscles, ligaments, blood vessels, brain white matter etc.

Patterns may contain meaningful information about the pathology

Example:Mammogram

Example: ligament healing, 3 weeks

Example: ligament healing, 6 weeks

Example: ligament healing, 14 weeks

Measures of Directional Distribution

Usually no need to separate α and (180º - α) → analysis limited to [0,180 º]

Analysis methods: The rose diagram The principal axis Angular moments Distance measures Entropy

The rose diagram

The rose diagram is a circular histogram of directional elements.

360 º divided into n sectors The radius is usually set proportional to the

area of corresponding dir. elements Linear proportionality can be achieved by

taking square root of the area as radius.

The principal axis

Corresponds the dominant axis of directional elements

Energy function for angle m= ∫x∫y[ xsin – ycos ]2f(x,y)dxdy Can be written with moments:

m= m20*sin2-2m11sincos+m02cos2 Minima of mis calculated by setting derivative to

zero → tan(2= 2m11/(m20-m02) →

Angular moments

Angular moments analogous to normalized moments

Mk=∑1N k(n)*p(n)

p(n) is normalized directional distribution vector (=circular histogram)

* (n) is the center of nth angle band in degrees

Distance measures

Directional distributions can be compared Useful for example when having an ideal

result and testing which method works the best

Euclidian distance is calculated between two directional distribution vectors

Entropy

Measures the scatter of directional elements H = - ∑1

N p(n)*log[p(n)] p(n) is the directional distribution vector as

before

Directional Filtering

Linear segments form a sinc function in Fourier domain:– Line in Fourier domain:

– Fan filter example, fig 8.2

y ax b

F u , v1 a Y

Y

Y

Yexp 2 j u

y ba

v y dy dy

F u , v2Ya

exp j 2 bua

sincua

v Y

Fourier transform of a line

Fan filter (fig 8.2)

Fourier domain techniques

Good:– select lines by their orientation

Bad:– junctions and occlusions smeared– truncation and spectral leaking (filter design

important)– Fourier domain filters not analytic, generalization

difficult– Difficulty in solving directional information at DC

(near origin of Fourier domain)

Gabor filters

Complex, sinusoidally modulated Gaussian functions

Optimal localization in freq and time domains Limited in time domain -> unlimited in spectral

domain (and vice versa)

Gabor filters

Uncertainty principle: In 2D: Gabor functions: (fig 8.7)

Essentially low-pass filters with directional selectivity

t f1

4

x y u v1

16 2

h x , y g x ' , y ' exp j 2 U x V y

x ' , y ' x cos y sin , x sin y cos

g x , y1

2 2exp

x 2 y 2

2 2

Gabor function (fig 8.7)

Division of the frequency domain by Gabor filters

Gabor filters

σ = spatial extent of the filter λ = aspect ratio orientation Proposed usage by Rolston and Rangayyan:

– Convolve band-limited and decimated versions of the image with the same wavelet

Gabor filters

Reconstruction of filter output:– Filter responses at different angles– Vector summation of responses (magnitude and

phase) – Figs. 8.10, 8.11.

Gabor filter responses (fig 8.10)

Gabor filtering (fig 8.11)

Directional analysis via Multi-scale Edge Detection

The goal is to get directional metrics from an image containing a big number of oriented collagen fibers

Problem: How to get the area of directional elements associated to a certain direction?

When solved, metrics can be calculated

Edge/Region detection (fig 8.12)

Step 1: Calculate stability map containing edge information from many scales

Step 2: Generate relative stability index map from stability map

Step 3: Extract lines from rsim Step 4: Extract regions from lines Step 5: Calculate areas for regions Step 6: Compute orientational distribution Step 7: Compute metrics (entropy, ang. moments…)

Directional analysis via Multi-scale Edge Detection

Hough-Radon Transform Analysis

With Hough-transform it is possible to detect lines from an image easily

Drawback: applicable to only binary images! Radon-transform similar but defined for grayscales

and has different coordinate system Hough-Radon is defined for grayscales and adds

gray levels in parameter space rather than increments by one

The basic idea

Map each (xn,yn,c) from grayscale image to a sine curve in parametric space (xicos(a)+yisin(a) more specifically)

Filter incremented sine curves using a peak detecting filter

Integrate columns of parametric image and normalize to get directional distribution

Hough-Radon dir. analysis (fig 8.18)

Problems

“Crosstalk”: several parallel lines cause false peaks in parametric space

False peaks are in 90deg angle compared to real lines in original image

Quantization errors: quantization levels of data in original and parametric space affect the accuracy of the results

Application: Bilateral Asymmetry in Mammograms

Asymmetry between left and right mammograms important for diagnosis

Problems:– Natural asymmetry– Alignment difficult– Distortions due to imaging conditions

Use Gabor wavelets to detect possible global disturbance

Application: Bilateral asymmetry...

Segmentation of fibroglandular disc– Gaussian mixture model of breast density, at least 4

tissue types– Model selection and expectation maximization (EM)

algorithm Delimitation of fibroglandular disc

– Apply constraints to EM algorithm

Segmentation

Application: Bilateral asymmetry...

Directional analysis with Gabor filters (Ferrari et al):

Basis functions Lack of orthogonality affects reconstruction Use even symmetric part of Gabor filter Choice of parameters: λ, σ, frequencies of

interest, number of scales, number of directions

x , y1

2 x y

e x p1

2 x 2

x

2

y 2

y

2j 2 W x

Examples of Gabor wavelets

Gabor in frequency domain

Application: Bilateral asymmetry...

Results– After Gabor filter analysis, construct rose diagrams– Use entropy, first and second moments of rose

diagram in objective assessment

Principal components

Rose diagram

Summary

Analysis of oriented patterns is an active field of study

Different metrics can be used to classify the level of directionality

Fourier based methods detect orientations, but perform poorly on junctions

Filter design important Gabor filters offer flexibility Hough-Radon transform is a general tool for

directionality analysis, but suffers from problems such as crosstalk and quantization errors