Image Foresting Transform

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# Image Foresting Transform - PowerPoint PPT Presentation

Image Foresting Transform. for Image Segmentation. Presented by: Michael Fang Weilong Yang. A Few Things to Recall. Image Segmentation Finding homogeneous regions Graph-based Methods Treating images as graphs Image Foresting Transform Unification Efficiency Simplicity.

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### Image Foresting Transform

for Image Segmentation

• Presented by:
• Michael Fang
• Weilong Yang
A Few Things to Recall
• Image Segmentation
• Finding homogeneous regions
• Graph-based Methods
• Treating images as graphs
• Image Foresting Transform
• Unification
• Efficiency
• Simplicity
Directed Graphs

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

Paths
• A 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).
Path Costs
• A 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.
Examples of MI Cost Functions
• Additive 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.
Predecessor Map and Spanning Forest
• A 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).
Paths of the Forest P
• For 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) = nil

P*(i) = i, where P(i) = nil

Optimum-path Forest

An 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.

An Image as a Directed Graph
• A 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.
Seed Pixels

In 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

IFT Algorithm for Image Segmentation
• Path Cost
IFT algorithm with FIFO policy(2)

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IFT algorithm with FIFO policy(4)

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IFT algorithm with FIFO policy(4)

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IFT algorithm with FIFO policy(4)

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IFT algorithm with FIFO policy(4)

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IFT algorithm with FIFO policy(4)

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Framework of Image segmentation by IFT

Input Image

Seeds Labeling

IFT

Summary
• Basic concept of the Image Foresting Transform
• IFT for image segmentation
• Experiment results
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