chapter 8: analysis of oriented patterns tuomas neuvonen tommi tykkälä
TRANSCRIPT
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