Second-Order Connected Attribute Filters Using Max-Trees

Transcription

1 Second-Order Connected Attribute Filters Using Max-Trees Georgios K. Ouzounis and Michael H. F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands

2 Presentation Outline Connectivity and Connected Operators Connectivity Classes Connected Openings Attribute Filters Second-Order Connectivity Clustering Based Connectivity Partitioning Based Connectivity Computing Second-Order Attribute Filters The Max-Tree Algorithm The Dual Input Max-Tree Filtering Experiments Conclusions

3 Connectivity and Connected Filters A connectivity class C is the set of all connected subsets of some universal set E. Some limitations apply: C and {x} C if C n C with n = 1,...N and N n=1 C n then N n=1 C n C A binary image represented by a set X E can be partitioned into its connected components or grains C i. Connected components can be extracted by means of a connected opening Γ x : Γ x (X) = i {C n C x C n and C n X} (1) Because all C n are connected, their union is also connected. Furthermore x / X, Γ x (X) =.

4 Attribute filters The simplest filtering schemes for binary images work by preserving or removing connected components Given that connected filters work on image structures, filter selection criteria can be defined in terms of the properties or attributes of these structures. Criterion T : T (C) = (Attr(C) λ) (2) with Attr(C) some real-valued attribute of C, and λ the attribute threshold Γ T (X) = x X Γ T (Γ x (X)) (3) 1. compute attribute for each connected component 2. keep components of which attribute value exceeds some threshold λ

5 Second-Order Connectivity Many generalizations of the standard (4- or 8- way) connectivity have been proposed aiming to improve the robustness and increase the versatility of these filters. Connectivity generalizations yield an approximation as to when image objects can be considered connected or when connected regions can be treated as separate objects. An example is Second-Order Connectivity which relies on an underlying connectivity. Second-Order connected sets are given by a connected opening extracting the connected components found at the intersection between the original image X and the connectivity map ψ(x). ψ refered to as the generalizing operator is usually a structural opening, closing or dilation. C ψ denotes the connectivity class and Γ ψ x the corresponding connected opening.

6 Clustering-based Connectivity Let ψ be an increasing and extensive operator, i.e. X ψ(x) If x X, the connected opening Γ ψ x looks at connected components of ψ(x), and intersects the one returned with X. i.e. Γ ψ x(x) = { Γx (ψ(x)) X, x X, x / X (4) This clusters nearby connected components according to C into new, larger ones. X ψ(x) Γ ψ p (X) Γ ψ q (X) Note that p = (65, 85) and q = (200, 225).

7 Partitioning-based Connectivity Let ψ be an increasing, idempotent and anti-extensive operator: ψ(x) X If x ψ(x), the connected opening Γ ψ x looks at connected components of ψ(x). If x X \ ψ(x) a singleton set {x} is returned. i.e. Γ x (ψ(x)) if x ψ p (X) Γ ψ x(x) = {x} if x X \ ψ p (X) x / X This partitions connected components into smaller ones. p = (65, 85) and q = (200, 225). (5) X ψ(x) Γ ψ p (X) Γ ψ q (X)

8 Computing Second-Order Attribute Filters Algorithm based on Max-Tree - hierarchical image representation for attribute filtering Tree nodes correspond to Peak Components and leaves to Regional Maxima Each node points to its parent and root node corresponds to background Node attributes stored in the tree structure. Filtering involves comparing node attributes against a threshold and removing them if not satisfying the criterion (different filtering rules). Flooding done by recursion. Data stored in a set of FIFO hierarchical queues pseudo code

9 Max-Tree representation input signal P 0 3 P 0 2 P 1 2 P 0 1 P 0 0 peak components C 0 3 C 0 2 C 0 1 C 1 2 C 0 0 C0 1 C0 2 C0 3 C0 2 C0 1 C1 2 C0 1 C0 0 labelling C 0 0 Max-Tree

10 The Dual Input Max-Tree Two input images - X and ψ(x) The hierarchical queue is shaped from the connectivity map and the number of pixels for each level in the Max-Tree by the histogram of the original image. The flooding function proceeds with pixels retrieved from ψ(x). If the intensity of a pixel p is the same in both X and ψ(x), the algorithm proceeds as the normal Max-Tree. If not then we have two cases: pseudo code

11 Clustering C 0 3 C 0 3 P 0 3 P 0 2 P 1 2 P 0 1 P 0 0 P 0 3 P 0 2 P 0 1 P 0 0 C 0 2 C 0 1 C 0 0 C 1 2 X ψ(x): closing Max-Tree of X Dual Input Max-Tree The attributes of C 0 2 and C 1 2 are merged to C 0 2 since all pixels at level h = 2 are clustered to a single connected component. Assume we are flooding at level h and encounter a pixel p which corresponds to h in X such that h [p] < h[p]. Then p is connected to the current active node at h [p] through the connected component at level h[p]; i.e. it defines a peak component at level h [p] to which p in ψ(x) is connected. C 0 2 C 0 1 C 0 0

12 Partitioning P 0 3 P 0 2 P 1 2 P 0 1 P 0 0 P 0 2 P 1 2 P 0 1 P 0 0 C 0 3 C 0 2 C 0 1 C 0 0 C 1 2 C 0 3 C 1 3 C 0 2 C 0 1 C 0 0 C 1 2 X ψ(x): opening Max-Tree of X Dual Input Max-Tree Assuming the node C 0 3 of the original Max-Tree vanishes in the connectivity map ψ(x), then in the Dual Input representation is split to a number of singleton nodes. Assume we are flooding at level h and encounter a pixel p which corresponds to h in X such that h[p] < h [p]. Then p is a discarded component in ψ(x) and hence defines a singleton node at level h [p]. Singletons upon detection finalize their node status and return and an attribute value of 1.

13 Filtering 3 C C C 0 2 C C C 0 2 C C C C P 0 3 P 0 2 P 1 2 P 0 1 P 0 0 P 0 2 P 0 1 P attribute values original filtered (λ = 10)

14 Vessel-Enhancement Filtering Isosurface projection of the original MRA at level 50 and of the filtered at level 3. Shape filtering using I/V 5/3 > λ as 3D shape criterion where I the moments of inertia and V the volume. The result can be computed in 8 s on a Pentium 4 at 2.8 GHz for a volume.

15 Conclusion and Prospects Linear time complexity - fast algorithm. Minimal difference with conventional Max-Tree timing Supports gray-scale images Ideal for the wider class of attribute filters Supports both types of second-order connectivity generalization (clustering and partitioning) in the same implementation It handles connectivity maps generated by flat operators that do not strictly need to be extensive or anti-extensive Drawbacks: Limited to flat structural operators Partitioning Connectivity generates an oversegmentation problem

16 Max-Tree Pseudo Code /* flood(h, thisattribute) : Flooding function at level h */ attribute = thisattribute /* accounts for child attributes */ while (not HQueue-empty(h)) /* First step: propagation */ { p = HQueue-first(h) /* retrieve priority pixel */ STATUS[p] = NumberOfNodes[h] /* STATUS = the node index */ for (every neighbor q of p) /* process the neighbors */ { if (STATUS[q] == "NotAnalyzed") { HQueue-add(ORI[q],q) /* add in the queue */ STATUS[q] = "InTheQueue" NodeAtLevel[ORI[q]] = TRUE /* confirm node existance */ if (ORI[q] > ORI[p]) /* check for child nodes */ { m = ORI[q] child_attribute = 0 do{ /* recursive child flood */ m = flood(m,child_attribute) } while (m!= h) attribute += child_attribute }}}} NumberOfNodes = NumberOfNodes[h] + 1 /* update the node index */ m = h-1 /* 2nd step: defines father*/ while ((m >= 0) and (NodeAtLevel[m] = FALSE)) m = m-1 if (m >= 0){ i = NumberOfNodes[h] - 1; j = NumberOfNodes[m]; } else The node C_i at level h has no father, i.e. its the root node NodeAtLevel[h] = FALSE; node->attribute = attribute; node->status = Finalized; thisattribute = attribute; return (m) Back

17 Dual Input Max-Tree Pseudo Code /* flood(h, thisattribute) : Flooding function at level h */ attribute = thisattribute + node->attribute /* node->attribute is */ /* added to account for pixels found during other calls to flood */ while (not HQueue-empty(h)) /* First step: propagation */ { p = HQueue-first(h) /* retrieve priority pixel */ STATUS[p] = NumberOfNodes[h] /* STATUS = the node index */ if(ori[p]!=h){ /* Detect intensity mismatch */ NodeAtLevel[ORI[p]]=TRUE /* Same for both cases */ node = Tree + NodeOffsetAtLevel[ORI[p]] + NumberOfNodes[ORI[p]] node->attribute ++ if(ori[p]>h){ /* Anti-extensive case */ node->parent = NodeOffsetAtLevel[h] + NumberOfNodes[h] node->status = Finalized; node->level = ORI[p] NumberOfNodes[ORI[p]] += 1; NodeAtLevel[ORI[p]] = FALSE attribute++ } /* Finalizing the singleton node */ } else attribute++ /* If pixel intensity is the same in both images*/ /* The rest as in Figure 2... */ return (m) Back