An efficient image segmentation algorithm using bidirectional Mahalanobis

An efficient image segmentation algorithm using bidirectional Mahalanobis distance MASTERS THESIS By: Rahul Suresh COMMITTEE MEMBERS Dr. Stan Birchfield Dr. Adam Hoover Dr. Brian Dean

Thesis overview � Introduction � Related work � Background theory: ◦ Image as a graph ◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

What is segmentation � Dividing an image to disjoint regions such that similar pixels are grouped together Image Courtesy: [3]

What is segmentation � Image Segmentation involves division of image I into K regions: R 1, R 2, R 3, … RK such that: Every pixel must be assigned to a region Regions must be disjoint Size: 1 pixel to the entire image itself

What is segmentation � Pixels within a region share certain characteristics that is not found with pixels in another region. f is a function that returns TRUE if the region under consideration is homogenous

Applications of segmentation � Biomedical applications ◦ Used as a preprocessing step to identify anatomical regions for medical diagnosis/analysis. Brain Tissue MRI Segmentation [1] CT Jaw segmentation [2]

Applications of segmentation � 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 Face segmentation Iris Segmentation

Applications of segmentation � Astronomy: analysis Preprocessing step before further Segmentation of Nebula [4]

What is good segmentation? (Manual segmentations from BSDS) Which segmentation is “correct”?

What is good segmentation? � “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

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 � Related work � Background theory: ◦ Image as a graph ◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Related work 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 Image Courtesy: [5]

Related work: Mean shift and k means � 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 clusters 3. 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: Spectral theory and Normalized cuts � Represent � Using image as a 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: MST based approach � 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(Nlog. N) time � Discussed in detail in the next section

Thesis overview � Related work � Background theory: ◦ Image as a graph ◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Background: Image as a graph � 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(N 2) ◦ 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

Thesis overview � Related work � Background theory: ◦ Image as a graph ◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Background: Kruskal’s MST � 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 • Sorting: O(mlog(m)) time • Find. Set and Merge: O(mα(N)) time [very slow growing] OVERALL TIME: O(m log(m))

Thesis overview � Related work � Background theory: ◦ Image as a graph ◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Background: MST Segmentation � Use � In Minimum Spanning Tree to segment image. 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 weights 3. For every edge ei in E, 1. If Find. Set(ui) ≠ Find. Set(vi) AND Is. Similar(ui , vi)=TRUE Merge(Find. Set(ui) , Find. Set(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 : • 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!!

Background: MST Segmentation � Drawback 1: LEAK Felzenszwalb and Huttenlocher 2004

Background: MST Segmentation � Drawback 2: SENSITIVITY TO PARAMETER k ◦ Notice how granularity changes by varying k • k is arbitrary • k is affected by the size of the image

Thesis overview � Related work � Background theory: ◦ Image as a graph ◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Our algorithm: overview � Objective � Constructing � Sort � For ◦ image grid edges in ascending order every edge If Merge criterion is satisfied � Merge

Our algorithm: Objective � 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.

How to compare two regions in a graph? MST APPROACH Int(Ru) Int(Rv) D(u, v) u v 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 Ø Represent each region as a Gaussian Ø Check if the Gaussians are similar: Ø Mahalanobis distance is less than 2. 5

Our algorithm: overview � Objective � Constructing � Sort � For ◦ image grid edges in ascending order every edge If Merge criterion is satisfied � Merge

Our algorithm: Building image grid � 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 vi

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: overview � Objective � Constructing � Sort � For ◦ image grid edges in ascending order every edge If Merge criterion is satisfied � Merge

Our algorithm- Naïve approach

Our algorithm: overview � Objective � Constructing � Sort � For ◦ image grid edges in ascending order every edge If Merge criterion is satisfied � Merge

Our algorithm: Merge Criterion � While adding edge ei to the MST, regions Ru and Rv are merged if the following criterion is satisfied: Forces small regions to merge Around 2. 5 is a good threshold Bidirectional Mahalanobis

Our algorithm: overview � Objective � Constructing � Sort � For ◦ image grid edges in ascending order every edge If Merge criterion is satisfied � Merge

Our algorithm: Merge � Merging regions Ru and Rv ◦ Update information at the root node of the disjoint set datastructure (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(N 2).

Our algorithm: Speed-up � 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(log. N) amortized time. � Although the asymptotic running time is O(Nplog. N), it is still slower than MST

Our algorithm: Approximation � Do not update weights or re-sort edges after every iteration. � This runs in O(Nlog. N) time ◦ Speed comparable to MST � Our experiments show that the approximated algorithm still improves upon the drawback of MST: ◦ Leak ◦ Sensitivity to parameter k

Thesis overview � Related work � Background theory: ◦ Image as a graph ◦ Kruskals’ Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Results � Tested the algorithm on: ◦ Synthetic images ◦ Berkeley Segmentation Dataset � Compared its performance with MST based segmentation algorithm

Results � Our algorithm overcomes leak! Gradient ramp MST merges the entire image into 1 region Our algorithm has 2 stable regions

Results � Effect MST based Segmentation of parameter tau on granularity k is arbitrary! Varies for different image sizes Our algorithm • τ represents the distance between Gaussians! • Best results when 2< τ<2. 5

Results

Results � We ran the segmentation Mona Lisa exhaustively for multiple values of τ � Studied the effect of τ on the number of regions formed � Notice that curve flattens for Man τ in the range 2 -2. 5. ◦ Represents “stable” regions ◦ Segmentation unaffected by parameter change cf. Yu 2007 Notice the flat regions

Results: BSDS dataset � We ran the algorithm exhaustively on Berkeley Segmentation dataset. � Our algorithm produced more “correct” segmentations than MST segmentations. � Segmentation � Some was sharper in our algorithm. specific example illustrated in the subsequent slides

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

Thesis overview � Related work � Background theory: ◦ Image as a graph ◦ Kruskal’s Minimum Spanning Tree ◦ MST based segmentation � Our algorithm � Results � Conclusion and future work

Conclusion � 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(N 2) time. ◦ Using skip list improves to O(Nlog. N) but still not as fast as MST � An approximated algorithm is proposed: ◦ Runs in O(Nlog. N) time and speed comparable to MST ◦ Still overcomes the two drawbacks of MST based segmentation

Future work � Pre-processing: � Efficiency: Homographic filtering 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

References [1] http: //picsl. upenn. edu/Project/Brain. Tissue. Segmentation [2] http: //etidweb. tamu. edu/faculty/beasley/publications/Beasley_Mediviz_Visualize. Edge. Weights_2011. pdf [3] http: //www. eecs. berkeley. edu/Research/Projects/CS/vision/bsds/ [4] http: //atc. udg. edu/~llado/CVpapers/icip 11 b. pdf [5] Stan Birchfield. Image Segmentation. Lecture Notes