Deducing Local Influence Neighbourhoods in Images Using Graph Cuts

1
Deducing Local Influence Neighbourhoods in Images
Using Graph Cuts

• Ashish Raj, Karl Young and Kailash Thakur
• Assistant Professor of Radiology
• University of California at San Francisco, AND
• Center for Imaging of Neurodegenerative
• Diseases (CIND)
• San Francisco VA Medical Center
• email ashish.raj_at_ucsf.edu
• Webpage http//www.cs.cornell.edu/rdz/SENSE.htm
• http//www.vacind.org/faculty

2
San Francisco, CA
3
Overview

• We propose a new image structure called local
  influence neighbourhoods (LINs)
• LINs are basically locally adaptive
  neighbourhoods around every voxel in image
• Like superpixels
• Idea of LIN not new, but first principled cost
  minimization approach
• Thus LINs allow us to probe the intermediate
  structure of local features at various scales
• LINs were developed initially to address image
  processing tasks like denoising and interpolation
• But as local image features they have wide
  applications

4
Local neighbourhoods as intermediate image
structures
Low level
High level
Too cumbersome Computationally expensive Not
suited for pattern recognition
Prone to error propagation Great for graph
theoretic and pattern recognition
Good intermediaries between low and high levels?
5
Outline

• Intro to Local Influence Neighbourhoods
• How to compute LINs?
• Use GRAPH CUT energy minimzation
• Some examples of LINs in image filtering and
  denoising
• Other Applications
• Segmentation
• Using LINs for Fractal Dimension estimation
• Use as features for tracking, registration

6
Local Influence Neighbourhoods

• A local neighbourhood around a voxel (x0, y0) is
  the set of voxels close to it
• closeness in geometric space
• closeness in intensity
• First attempt use a space-intensity box

• Definition of e, d arbitrary
• Produces disjoint, non-contiguous, holey, noisy
  neighbourhoods!
• Need to introduce prior expectations about
  contiguity
• We develop a principled probabilistic approach,
  using likelihood and prior distributions

7
Example Binary image denoising

• Suppose we receive a noisy fax
• Some black pixels in the original image were
  flipped to white pixels, and some white pixels
  were flipped to black
• We want to recover the original

8
Problem Constraints
likelihood

• Our Constraints
• If a pixel is black (white) in the original
  image, it is more likely to get the black (white)
  label
• Black labeled pixels tend to group together, and
  white labeled pixels tend to group together

prior
original image
9
Example of box vs. smoothness
10
Example of box vs. smoothness
11
A Better neighbourhood criterion

• Incorporate closeness, contiguity and smoothness
  assumptions
• Set up as a minimization problem
• Solve using everyones favourite minimization
  algorithm
• Simulated Annealing
• (just kidding) - Graph Cuts!
• A) Closeness lets assume neighbourhoods follow
  Gaussian shapes around a voxel

12
A) Closeness criterion in action
13
B) Contiguity and smoothness

• This is encoded via penalty terms between all
  neighbouring voxel pairs

G(x) Sp,q V(xp, xq) V(xp, xq) distance metric
Define a binary field Fp around voxel p s.t. 0
means not in LIN, 1 means in LIN

• Closeness

B) Contiguity/smoothness
Bayesian interpretation this is the log-prior
for LINs
14
MAP can be written as energy minimization

• E.g. consider linear system y Hx n
• Pr(yx) (likelihood function) exp(- y-Hx2)
• Pr(x) (prior PDF) exp(-G(x))
• MAP can be converted to energy minimization by
  taking logarithm
• xest arg min y-Hx2 G(x)

15
Markov Random Field Priors

• Imposes spatial coherence (neighbouring pixels
  are similar)
• G(x) Sp,q V(xp, xq)
• V(xp, xq) distance metric

• Potential function is discontinuous, non-convex
• Potts metric is GOOD but very hard to minimize

16
Bottomline

• Maximizing LIN prior corresponds to the
  minimization of
• E(x) Ecloseness(x) Esmoothness(x)
• MRF priors encode general spatial coherence
  properties of images
• E(x) can be minimized using ANY available
  minimization algorithm
• Graph Cuts can speedily solve cost functions
  involving MRFs, sometimes with guaranteed global
  optimum.

17
Graph Cut based Energy Minimization
18
How to minimize E?

• Graph cuts have proven to be a very powerful tool
  for minimizing energy functions like this one
• First developed for stereo matching
• Most of the top-performing algorithms for stereo
  rely on graph cuts
• Builds a graph whose nodes are image pixels, and
  whose edges have weights obtained from the energy
  terms in E(x)
• Minimization of E(x) is reduced to finding the
  minimum cut of this graph

19
Minimum cut problem

• Mincut/maxflow problem
• Find the cheapest way to cut the edges so that
  the source is separated from the sink
• Cut edges going from source side to sink side
• Edge weights now represent cutting costs

20
Graph construction

• Links correspond to terms in energy function
• Single-pixel terms are called t-links
• Pixel-pair terms are called n-links
• A Mincut is equivalent to a binary segmentation
• I.e. mincut minimizes a binary energy function

21
Table1 Edge costs of induced graph
22
Graph Algorithm

• Repeat graph mincut for each voxel p

23
Examples of Detected LINs
24
Results Most Popular LINs
25
Filtering with LINs

• Use LINs to restrict effect of filter
• Convolutional filters

• Rank order filter

26
Maximum filter using LINs
27
Median filter using LINs
28
EM-style Denoising algorithm
Noise model O I n
Image prior

• Likelihood for i.i.d. Gaussian noise

Maximize the posterior
29
Bayesian (Maximum a Posteriori) Estimate
likelihood
posterior
prior

• Here x is LIN, y is observed image
• Bayesian methods maximize the posterior
  probability
• Pr(xy) ? Pr(yx) . Pr(x)

30
EM-style image denoising
Joint maximization is challenging We propose
EM-style approach Start with Iterate
We show that
31
Results LIN-based Image Denoising
32
Results Bike image
33
Table1 Denoising Results
34
Other Applications of LINs

• LINs can be used to probe scale-space of image
  data
• By varying scale parameters sx and sn
• Measuring fractal dimensions of brain images
• Hierarchical segmentation superpixel concept
• Use LINs as feature vectors for
• image registration
• Object recognition
• Tracking

35
Hierarchical segmentation

• Begin with LINs at fine scale
• Hierarchically fuse finer LINs to obtain coarser
  LINS ? segmentation

36
How to measure Fractal Dimension using LINs?

• How LINs vary with changing sx and sn depends on
  local image complexity
• Fractal dimension is a stable measure of
  complexity of multidimensional structures
• Thus LINs can be used to probe the multi-scale
  structure of image data

37
FD using LINs

• For each voxel p, for each value of sx, sn
• count the number N of voxels included in Bp

phase transition
.

• Slope of each segment local fractal dimension

extend to (sx , sn) plane
38
Possible advantages of LIN over current techniques

• LINs provide FD for each voxel
• Captures the FD of local regions as well as
  global
• Ideal for directional structures and oriented
  features at various scales
• Far less susceptible to noise
• (due to explicit intensity scale sn which can be
  tuned to the noise level)
• Enables the probing of phase transitions

39
Possible Discriminators of Neurodegeneration

• Fractal measures may provide better
  discriminators of neurodegeneration (Alzheimers
  Disease, Frontotemporal Dementia, Mild Cognitive
  Disorder, Normal Aging, etc)
• Possibilities
• Mean (overall) FD -- D(0)
• Critical points, phase transitions in (sx, sn)
  plane
• More general Renyi dimensions D(q) for q 1
• Summary image feature f(a) ?? D(q)
• Phase transitions in f(a)
• Fractal structures can be characterized by
  dimensions D(q), summary f(a) and various
  associated critical points
• These quantities may be efficiently probed by the
  Graph Cut based local influence neighbourhoods
• These fractal quantities may provide greater
  discriminability between normal, AD, FTD, etc.

40
Summary

• We proposed a general method of estimating local
  influence neighbourhoods
• Based on an optimal energy minimization
  approach
• LINs are intermediaries between purely
  pixel-based and region-based methods
• Applications include segmentation, denoising,
  filtering, recognition, fractal dimension
  estimation,
• in other words, Best Thing Since Sliced Bread

41
Deducing Local Influence Neighbourhoods in Images
Using Graph Cuts

• Ashish Raj
• CIND, UCSF
• email ashish.raj_at_ucsf.edu
• Webpage http//www.cs.cornell.edu/rdz/SENSE.htm
• http//www.vacind.org/faculty