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## PowerPoint Slideshow about 'Image Foresting Transform' - dorian-phelps

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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 SI of seed pixels. This constraint can be modeled by defining

IFT Algorithm for Image Segmentation

- Path Cost
- Four-Connected Adjacency

IFT algorithm with FIFO policy(3)

Growing Process

IFT algorithm with FIFO policy(4)

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Summary

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

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