image foresting transform

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Image Foresting Transformfor Image Segmentation

Presented by:Michael FangWeilong Yang

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A Few Things to RecallImage Segmentation

◦Finding homogeneous regionsGraph-based Methods

◦Treating images as graphsImage Foresting Transform

◦Unification◦Efficiency◦Simplicity

Graph-Based Methods

G ={V, E}

V: graph nodesE: edges connection the nodes

PixelsPixel Similarity

Segmentation = Graph Partition

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Directed GraphsA directed graph is a pair (I, A), where I is a set of nodes and A is a set of ordered pairs of nodes.

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PathsA path is a sequence t1, t2, …, tk of

distinct nodes in the graph, such that (ti, ti+1) A for 1 i k – 1.

A path is trivial if k = 1;Path denotes the

concatenation of two paths, and , where ends at t and begins at t.

Path = s, t denotes theconcatenation of the longest prefix of and the last arc (s, t).

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Path CostsA path-cost function is a mapping

that assigns to each path a cost (), in some ordered set of cost values.

A function is said to be monotonic-incremental (MI) when

(t) = h(t),( s, t) = () (s, t),

where h(t) is a handicap cost value and satisfies: x’ x x’ (s, t) x’ (s, t) and x (s, t) x, for x, x’ and (s, t) A.

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Examples of MI Cost FunctionsAdditive cost function

sum(t) = h(t),sum( s, t) = sum() + w(s,

t),where w(s, t) is a fixed non-negative arc weight.

Max-arc cost function max(t) = h(t),

max( s, t) = max{max(), w(s, t)}, where w(s, t) is a fixed arc weight.

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Predecessor Map and Spanning ForestA predecessor map is a function P that

assigns to each node t I either some other node in I, or a distinctive marker nil I – in which case t is the root of the map.

A spanning forest is a predecessor map which takes every node to nil in a finite number of iterations (i.e., it contains no cycles).

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Paths of the Forest PFor any node t I, there is a

path P*(t) which is obtained in backward by following the predecessor nodes along the path.

P*(c) = a, b, c, where P(c) = b, P(b) = a, P(a) = nilP*(i) = i, where P(i) = nil

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Optimum-path ForestAn optimum-path forest is a spanning forest P, where (P*(t)) is minimum for all nodes t I. Consider cost function sum in the example below.

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An Image as a Directed GraphA grayscale image I is a pair (I, I),

where I is a finite set of pixels (points in Z2) and I assigns to each pixel t I a value I(t) in some arbitrary value space.

An adjacency relation A is a binary relation between pixels of I, which is usually translation-invariant.

Once A has been fixed, image I can be interpreted as a directed graph, whose nodes are the image pixels in I and whose arcs are defined by A.

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Seed PixelsIn some applications, we would like to use a predefined path-cost function but constrain the search to paths that start in a given set S I of seed pixels. This constraint can be modeled by defining

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IFT algorithm for Image Segmentation1. Path Cost

2. Four-Connected Adjacency

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IFT algorithm with FIFO policy(1)

Initialization

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IFT algorithm with FIFO policy(2)

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Growing Process

IFT algorithm with FIFO policy(3)

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IFT algorithm with FIFO policy(4)

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IFT algorithm with FIFO policy(4)

Another Example

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Input Image

Gradient Image

Seeds Labeling IFT

Framework of Image segmentation by IFT

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Experiment Results (1)

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Experiment Results (2)

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Experiment Results (3)

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Experiment Results (4)

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SummaryBasic concept of the Image

Foresting TransformIFT for image segmentationExperiment results