Chapter 8: Analysis of Oriented Patterns Tuomas Neuvonen Tommi 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  ] 2 f(x,y)dxdy Can be written with moments: m  = m 20 *sin 2  -2m 11 sin  cos  +m 02 cos 2  Minima of m  is calculated by setting derivative to zero → tan(2  = 2m 11 /(m 20 -m 02 ) → 

Angular moments Angular moments analogous to normalized moments M k =∑ 1 N  k (n)*p(n) p(n) is normalized directional distribution vector (=circular histogram) *  (n) is the center of n th 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

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

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 (x n,y n,c) from grayscale image to a sine curve in parametric space (x i cos(a)+y i sin(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

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