Graph-based Segmentation

Main Ideas Convert image into a graph Vertices for the pixels Vertices for the pixels Edges between the pixels Edges between the pixels Additional vertices and edges to encode other constraints Additional vertices and edges to encode other constraints Manipulate the graph to segment the image

Papers Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images Boykov and Jolly Boykov and Jolly Minimize an energy function Minimize an energy function Efficient Graph-based Segmentation Felzenszwalb and Huttenlocher Felzenszwalb and Huttenlocher Cluster the vertices based on edge weight Cluster the vertices based on edge weight

Boykov and Jolly Binary image segmentation Classify pixels as object or background Classify pixels as object or background Their contribution is adding interactivity Their contribution is adding interactivity Minimise an energy function E(A) = B(A) + λR(A) E(A) = B(A) + λR(A) A: Segmentation (assign each pixels to object or background) B(A): The cost of all the edges between object pixels and background pixels R(A): The cost of deciding a pixel to be object or background

Creating the Graph Each pixel has a corresponding vertex Additionally, a source (“object”) and a sink (“background”) Each pixel vertex has an edge to its neighbours (e.g. 8 adjacent neighbours in 2D), an edge to the source, an edge to the sink

Edge Weights between pixels Weight of edges between pixel vertices are determined by the B() function Low score when boundary is likely to pass between the vertices high score when vertices are probably part of the same element E.g. the difference in pixel intensities, the gradient

Edges to Source/Sink If pixel is known to be an object, use a high weight (K) to the source, zero weight to the sink K is chosen so that it will never be cut K is chosen so that it will never be cut Conversely, if pixel is background, use weight K to the sink, zero weight to the source Otherwise, weigh edges to source and sink appropriately using the R() function Note that the edge to the source is the “likelihood” for the pixel being the background – we break this edge when the pixel is assigned to the background Note that the edge to the source is the “likelihood” for the pixel being the background – we break this edge when the pixel is assigned to the background

Applications Handles arbitrary number of dimensions Finds global minimum energy Needs “good” user input to work effectively Need intelligent functions Need to select λ

Felzenszwalb and Huttenlocher Download the program from the webpage: Minimal documentation Short README file Short README file Paper Paper

The program Comes as a tar.gz archive and.zip archive Process Extract archive Extract archive ‘make’ (Makefile supplied) ‘make’ (Makefile supplied) Program consists of A.cpp “wrapper” file (only calls the functions) A.cpp “wrapper” file (only calls the functions) Actual algorithm functions are in.h files Actual algorithm functions are in.h files Basic portable C++ code Basic portable C++ code

Program Testing Built on Mac OS X, Linux, Windows cygwin Gcc toolchain, but any C++ compiler should work Gcc toolchain, but any C++ compiler should work Supplied basic Makefile Supplied basic Makefile Results were basically the same between platforms Colors are chosen randomly Colors are chosen randomly Results obtained are not the same as posted on the website Image files on the website may be modified (scaled/compressed/downsampled) Image files on the website may be modified (scaled/compressed/downsampled)

Algorithm Create a graph Each vertex corresponds to a vertex Each vertex corresponds to a vertex Edges are between “neighbouring” vertices Choose a small neighbourhood to reduce computation time (otherwise we have a complete graph) Choose a small neighbourhood to reduce computation time (otherwise we have a complete graph) Weight on the edge is the 5D distance between the points (for a 2D image) Weight on the edge is the 5D distance between the points (for a 2D image) 5D vector = x position, y position and 3 color components

Parameters σ: Use this value and do Gaussian smoothing (preprocessing the image to reduce noise) k: threshold value for doing the clustering min: “hack” parameter the smallest cluster size must contain at least this many vertices – clusters that are too small will be merged with other clusters until sufficiently large the smallest cluster size must contain at least this many vertices – clusters that are too small will be merged with other clusters until sufficiently large

Clustering Put each vertex in a component Sort edges by weight Take each edge in turn If the edge is between vertices in two different components A and B, we can merge if the edge weight is lower enough than the threshold If the edge is between vertices in two different components A and B, we can merge if the edge weight is lower enough than the threshold Threshold is the minimum of the following value, computed on A and B Threshold is the minimum of the following value, computed on A and B (Lowest weight edge in minimum spanning tree of the component) + (k / size of component)

Notes Low edge weights between vertices that are likely to be in the same cluster As a cluster gets larger, it becomes harder to add vertices to it Heuristic – not really minimising a particular energy function More similar to “region growing” More similar to “region growing” User has select “good” parameters to get good results

Effect of σ Increased smoothing results in removal of noise Can cause “bleeding” – the algorithm has difficulty separating background from the object if the boundaries are too smooth

Reference Images

Increasing σ Clouds are recognised as one object Palm tree gets confused with ocean

Grain Increasing σ introduces more blurring (reduces the edge weight between pixels)

Vertebrae MRI Gets rid of noise (bottom left, right hand side), but purple vertebrate piece bleeds out

Increasing Threshold Clusters more aggresively Palm tree is confused with ocean and clouds

Grain Non-grain pixels are almost all clustered together Measure of how “similar” all the pixels of an object should be

MRI Vertebrae and region next to vertebrae are very similar shade so are easily confused

Increasing ‘min’ value Segmentation is the same, but small components are merged with neighboring ones

Grain Easy to control change, gets rid of small artifacts

MRI Not much effect if regions are already large

Parameter Tweaking Need to manually tune parameters to get a good image Image of MRI after selecting σ = 0.6 σ = 0.6 k = 200 k = 200 min = 60 min = 60

Performance Need to tune parameters by hand Very fast Program usually takes a couple seconds on the test images provided Program usually takes a couple seconds on the test images provided Only takes ppm images in RAW data format Theoretically, algorithm generalises to arbitrary number of dimensions and arbitrary number of features per pixel

Command-line tools Working with images involves opening up the results in an image viewer – can get messy