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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.
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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 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(1) Initialization C(t) I t
IFT algorithm with FIFO policy(2) ∝ ∝ ∝ ∝ 5 5 5 5 1 2 3 4 ∝ ∝ ∝ ∝ 5 5 5 5 5 6 7 8 ∝ ∝ ∝ ∝ 5 5 5 5 9 10 11 12 ∝ ∝ 0 0 0 5 5 0 0 0 13 14 15 16 13 0 0 16
IFT algorithm with FIFO policy(3) Growing Process
IFT algorithm with FIFO policy(4) ∝ ∝ ∝ ∝ 5 5 5 5 1 2 3 4 ∝ ∝ ∝ ∝ 5 5 5 5 5 6 7 8 5 ∝ ∝ ∝ 5 5 5 5 9 10 11 12 ∝ 0 5 0 0 5 5 0 13 14 15 16 5 5 0 5 5 0 16 9 14
IFT algorithm with FIFO policy(4) ∝ ∝ ∝ ∝ 5 5 5 5 1 2 3 4 ∝ ∝ ∝ ∝ 5 5 5 5 5 6 7 8 5 ∝ ∝ 5 5 5 5 5 9 10 11 12 5 0 5 0 0 5 5 0 13 14 15 16 5 5 5 5 5 5 5 5 14 15 9 12
IFT algorithm with FIFO policy(4) ∝ ∝ ∝ ∝ 5 5 5 5 1 2 3 4 5 ∝ ∝ ∝ 5 5 5 5 5 6 7 8 5 5 ∝ 5 5 5 5 5 9 10 11 12 5 0 5 0 0 5 5 0 13 14 15 16 5 5 5 5 5 5 5 5 5 5 12 15 10 14 5
IFT algorithm with FIFO policy(4) ∝ ∝ ∝ ∝ 5 5 5 5 1 2 3 4 5 ∝ ∝ ∝ 5 5 5 5 5 6 7 8 5 5 ∝ 5 5 5 5 5 9 10 11 12 5 0 5 0 0 5 5 0 13 14 15 16 5 5 5 5 5 5 5 5 12 5 15 10
IFT algorithm with FIFO policy(4) ∝ ∝ ∝ ∝ 5 5 5 5 1 2 3 4 5 ∝ ∝ 5 5 5 5 5 5 6 7 8 5 5 5 5 5 5 5 5 9 10 11 12 5 0 5 0 0 5 5 0 13 14 15 16 5 5 5 5 5 5 5 5 5 5 5 10 11 15 8
IFT algorithm with FIFO policy(4) 5 5 5 5 5 5 5 5 1 2 3 4 5 5 5 5 5 5 5 5 5 6 7 8 5 5 5 5 5 5 5 5 9 10 11 12 5 0 5 0 0 5 5 0 13 14 15 16
Framework of Image segmentation by IFT Input Image Gradient Image Seeds Labeling IFT
Summary • Basic concept of the Image Foresting Transform • IFT for image segmentation • Experiment results
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