Image Foresting Transform
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Image Foresting Transform for Image Segmentation Presented by: Michael Fang Weilong Yang 1
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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. - PowerPoint PPT Presentation
Transcript of 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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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
