Post on 25-Feb-2016
description
An efficient image segmentation algorithm using bidirectional
Mahalanobis distance
MASTERS THESISBy: Rahul Suresh
COMMITTEE MEMBERSDr.Stan BirchfieldDr.Adam Hoover
Dr.Brian Dean
Introduction Related work Background theory:
◦ Image as a graph◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Dividing an image to disjoint regions such that similar pixels are grouped together
What is segmentation
Image Courtesy: [3]
Image Segmentation involves division of image I into K regions: R1, R2, R3, … RK such that:
What is segmentation
Every pixel must be assigned to a region
Regions must be disjoint
Size: 1 pixel to the entire image itself
Pixels within a region share certain characteristics that is not found with pixels in another region.
What is segmentation
f is a function that returns TRUE if the region under consideration is homogenous
Biomedical applications◦ Used as a preprocessing step to identify
anatomical regions for medical diagnosis/analysis.
Applications of segmentation
Brain Tissue MRI Segmentation [1]
CT Jaw segmentation [2]
Object recognition systems:◦ Lower level features such as color and texture are
used to segment the image◦ Only relevant segments (subset of pixels) are fed
to the object recognition system. Saves computational cost, especially for
large scale recognition systems
Applications of segmentation
As a preprocessing step in face and iris recognition
Applications of segmentation
Face segmentation Iris Segmentation
Astronomy: Preprocessing step before further analysis
Applications of segmentation
Segmentation of Nebula [4]
What is good segmentation?
Which segmentation is “correct”?
(Manual segmentations from BSDS)
“Correctness”- Are similar pixels grouped together and dissimilar pixels grouped seperately?
Granularity- Extent of resolution of segmentation◦ Consider example in the previous image
What is good segmentation?
There is ambiguity in defining “good”/ “optimal” segmentation.
An image can have multiple segmentations. “correct”◦ Make evaluation /benchmarking of segmentation
algorithm hard
What is good segmentation?
Related work Background theory:
◦ Image as a graph◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Related work Background theory:
◦ Image as a graph◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Some of the popular image segmentation approaches are:
Split and Merge approaches Mean Shift and k-means Spectral theory and normalized cuts Minimum spanning tree
Related work
Split and Merge approaches◦ Iteratively split
If evidence of a boundary exists
◦ Iteratively merge Based on similarity
Quad-tree used
Related work
Image Courtesy: [5]
Mean shift and k-means are related. Mean-shift:
◦ Represent each pixel as a vector [color, texture, space]
◦ Define a window around every point. 1. Update the point to the mean of all the points
within the window.2. Repeat until convergence.
Related work: Mean shift and k means
K-means:◦ Represent each pixel as vector [color, texture, space]◦ Choose K initial cluster centers
1. Assign every pixel to its closet cluster center.2. Recompute the means of all the clusters3. Repeat 1-2 until convergence.
Difference between K means and mean-shift:◦ In K-means, K has to be known beforehand◦ K-means sensitive to initial choice of cluster centers
Related work: Mean shift and k means
Represent image as a graph. Using graph cuts, partitions the image into
regions. In Spectral theory and normalized cuts,
Eigenvalues/vectors of the Laplacian matrix is used to determine the cut
Related work: Spectral theory and Normalized cuts
Use Minimum Spanning Tree to segment image.◦ Proposed by Felzenszwalb & Huttenlocher in 2004.◦ Uses a variant of Kruskals MST to segment images◦ Very efficient- O(NlogN) time
Discussed in detail in the next section
Related work: MST based approach
Related work Background theory:
◦ Image as a graph◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Graph G=(V,E) is an abstract data type containing a set of vertices V and edges E.
Useful operations using a graph:◦ See if path exists between any 2 vertices◦ Find connected components ◦ Check for cycles◦ Find the shortest point between any 2 vertices◦ Compute minimum spanning ◦ Graph partition based on cuts
Graph algorithms are useful in image processing
Background: Image as a graph
Image graph:◦ Pixels/group of pixels form vertices.◦ Vertices connected to form edges◦ Edge weight represents dissimilarity between vertices
Types of image graph:◦ Image grid◦ Complete graph◦ Nearest neighbor graph
Background: Image as a graph
Image grid:◦ Edges: every vertex (pixel) is connected with its
4 (or 8) x-y neighbors.◦ No of edges m= O(N) [Graph operations are
quick]◦ Fails to capture global properties
Background: Image as a graph
Complete graph:◦ Edges: Connect every vertex (pixel) with every
other vertex◦ No of edges m= O(N2)◦ Captures global properties◦ Graph operations are very expensive
Background: Image as a graph
Nearest neighbor graph:◦ Compromise between grid (fails to capture global
properties) and complete graph (too many edges).◦ Represent every vertex as a combination of color
and x-y features. [e.g. (R, G, B, x, y)]◦ Find the K=O(1) neighbors for each pixel using
Approximate nearest neighbor (ANN)◦ Edges: Connect every pixel to K nearest neighbors
Background: Image as a graph
Related work Background theory:
◦ Image as a graph◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Tree is a graph which is:◦ Connected◦ Has no cycles
Spanning tree: contains all the vertices of graph G◦ A graph can have multiple spanning trees
Minimum spanning tree is a spanning tree which has the least sum of weights of edges
Background: Kruskal’s MST
Background: Kruskal’s MST
• Sorting: O(mlog(m)) time• FindSet and Merge: O(mα(N)) time [very slow growing]
OVERALL TIME: O(m log(m))
Related work Background theory:
◦ Image as a graph◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Use Minimum Spanning Tree to segment image. In Kruskal’s MST algorithm,
◦ Edges are sorted in ascending order of weights◦ Edges are added in order to the spanning tree as long as a
cycle is not formed.◦ All vertices added to ONE spanning tree
If Kruskal’s is applied directly to image segmentation:◦ We will end up with ONE segment (entire image)
Background: MST Segmentation
Variant of Kruskal’s used in image segmentation.1. Create an image grid graph.2. Sort edges in the increasing order of weights3. For every edge ei in E,
1. If FindSet(ui) ≠ FindSet(vi) AND IsSimilar(ui ,vi)=TRUE
Merge(FindSet(ui) ,FindSet(vi) )
Instead of one MST, we end up with a forest of K trees
◦ Each tree represents a region
Background: MST Segmentation
We add an edge ei connecting regions Ru and Rv to a tree only if :
Background: MST Segmentation
• D(Ru Rv): edge weight connecting vertices u and v• Int(Ri): maximum edge weight in region Ri
WE MERGE IF THE EDGE WEIGHT IS LOWER THAN THE MAXIMUM EDGE WEIGHT IN
EITHER REGIONS!!
Drawback 1: LEAK
Background: MST Segmentation
Felzenszwalb and Huttenlocher 2004
Drawback 2: SENSITIVITY TO PARAMETER k◦ Notice how granularity changes by varying k
Background: MST Segmentation
• k is arbitrary• k is affected by the size of the image
Related work Background theory:
◦ Image as a graph◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Objective Constructing image grid Sort edges in ascending order For every edge
◦ If Merge criterion is satisfied Merge
Our algorithm: overview
Improve upon the drawbacks of MST ALGORITHM:◦ Addressing Leak:
Represent regions as a Gaussian distribution. Use Bidirectional Mahalanobis distance to compare Gaussians.
◦ Overcome sensitivity to parameter k: Propose parameter τ that is
independent of image size Works well for 2-2.5
Provide a mathematical intuition for it. Propose an approximation that enables real-time
implementation.
Our algorithm: Objective
How to compare two regions in a graph?
MST APPROACH
D(u,v)u v
Int(Ru) Int(Rv)
Check if D(u,v) < Int(Ru) && D(u,v) < Int(Rv)• Leak can happen
How to compare two regions in a graph?
OUR APPROACH
D(u,v)u v
Represent each region as a Gaussian Check if the Gaussians are similar:
Mahalanobis distance is less than 2.5
Objective Constructing image grid Sort edges in ascending order For every edge
◦ If Merge criterion is satisfied Merge
Our algorithm: overview
Initialize Vertices: ◦ Every pixel is mapped to a vertex◦ Information about vertex vi is stored at the ‘i’th entry of
the disjoint set data structure D. The ‘i’th entry in D contains following information:
◦ Root node◦ Zeroth, first and second order moments◦ List of all the edges connected to vertex v i
Our algorithm: Building image grid
Initialize Edges:◦ Between neighboring pixels in x-y space◦ Number of edges m= O(N)◦ Use List to maintain edges
Edge weight: ◦ Euclidean distance between pixels to begin with◦ Mahalanobis distance between Gaussians as region grows
Note that Euclidean distance is a special instance of Mahalanobis distance
Our algorithm: Building image grid
Objective Constructing image grid Sort edges in ascending order For every edge
◦ If Merge criterion is satisfied Merge
Our algorithm: overview
Our algorithm- Naïve approach
Objective Constructing image grid Sort edges in ascending order For every edge
◦ If Merge criterion is satisfied Merge
Our algorithm: overview
While adding edge ei to the MST, regions Ru and Rv are merged if the following criterion is satisfied:
Our algorithm: Merge Criterion
Forces small regions to merge
Bidirectional Mahalanobis
Around 2.5 is a good threshold
Objective Constructing image grid Sort edges in ascending order For every edge
◦ If Merge criterion is satisfied Merge
Our algorithm: overview
Merging regions Ru and Rv
◦ Update information at the root node of the disjoint set data-structure (Similar to MST)
◦ Updating information about root node, zeroth, first and second order moment is easy
However, after merging Ru and Rv
◦ All edges connected to either Ru or Rv have to be updated w.r.t. (Ru ∪
Rv )
◦ The edges have to be re-sorted.
◦ The above operations will slow down the overall running time to O(N2).
Our algorithm: Merge
To speed up weight update that needs to be performed after every iteration,◦ For every region in the DSDS, we store the pointers to
all the edges connected to it.◦ When 2 regions are merged, we merge their neighbor
lists also◦ Assuming that the number of neighbors for every
region is constant, every iteration of merging neighbors can also be accomplished in O(1) time
Our algorithm: Speed-up
Re-sorting the edges after every merge is an expensive operation. ◦ We use skip lists to maintain edges. ◦ Skip list is a data structure that helps maintain sorted
items. Every insert, delete and search operation takes O(logN)
amortized time.
Although the asymptotic running time is O(NplogN), it is still slower than MST
Our algorithm: Speed-up
Do not update weights or re-sort edges after every iteration.
This runs in O(NlogN) time◦ Speed comparable to MST
Our experiments show that the approximated algorithm still improves upon the drawback of MST:◦ Leak◦ Sensitivity to parameter k
Our algorithm: Approximation
Related work Background theory:
◦ Image as a graph◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Tested the algorithm on:◦ Synthetic images◦ Berkeley Segmentation Dataset
Compared its performance with MST based segmentation algorithm
Results
Our algorithm overcomes leak!
Results
Gradient ramp MST merges the entire image into 1 region
Our algorithm has 2 stable
regions
Effect of parameter tau on granularity
Results
MST based Segmentatio
n
Our algorithm
k is arbitrary! Varies for different image
sizes
• τ represents the distance between Gaussians!• Best results when 2< τ<2.5
Results
We ran the segmentation exhaustively for multiple values of τ
Studied the effect of τ on the number of regions formed
Notice that curve flattens for τ in the range 2-2.5.◦ Represents “stable” regions◦ Segmentation unaffected by
parameter change
ResultsMona Lisa
Man Notice the flat regions
cf. Yu 2007
We ran the algorithm exhaustively on Berkeley Segmentation dataset.
Our algorithm produced more “correct” segmentations than MST segmentations.
Segmentation was sharper in our algorithm. Some specific example illustrated in the
subsequent slides
Results: BSDS dataset
Results
Notice the Man on the Hill
Results
Notice the face of the man kneeling down!
Results
Notice how a small leak has merged grass with part of
bison’s body
Results: Our results are much sharper
Related work Background theory:
◦ Image as a graph◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation
Our algorithm Results Conclusion and future work
Thesis overview
Proposed a new segmentation algorithm that improves upon the drawbacks of MST:◦ Leak
Represent regions as Gaussians Use bidirectional Mahalanobis distance to compare
regions◦ Sensitivity to parameter k
τ = 2.5 works well for all images, represents normalized distance between Gaussian distributions
Shown experimentally to be “stable”
Conclusion
In the worst case scenerio, ◦ naïve version of our algorithm runs in O(N2) time. ◦ Using skip list improves to O(NlogN) but still not as
fast as MST An approximated algorithm is proposed:
◦ Runs in O(NlogN) time and speed comparable to MST◦ Still overcomes the two drawbacks of MST based
segmentation
Conclusion
Pre-processing: Homographic filtering Efficiency: Speed up the original version of
the algorithm using more sophisticated priority queues.
Benchmarking: Mathematical study of the accuracy of our segmentation algorithm on BSDS dataset
Future work
[1] http://picsl.upenn.edu/Project/BrainTissueSegmentation[2] http://etidweb.tamu.edu/faculty/beasley/publications/Beasley_Mediviz_VisualizeEdgeWeights_2011.pdf[3] http://www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/[4] http://atc.udg.edu/~llado/CVpapers/icip11b.pdf[5] Stan Birchfield. Image Segmentation. Lecture Notes
References