EUVIP 2026

Transcript

1. VECTOR ORDERING DERIVED FROM BINARY PARTITION TREE LEAF REORDERING FOR MATHEMATICAL MORPHOLOGY Olivier Lézoray Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ. GREYC UMR 6072, Caen, France olivier.lezoray@unicaen.fr https://lezoray.users.greyc.fr
2. Why do we need a vector ordering? The problem Mathematical morphology is naturally defined on a complete lattice, and requires a total order. ▶ Grayscale values have a natural total order. ▶ Color, multispectral or learned feature vectors do not. ▶ Lexicographic orderings are possible, but become unsuitable as the dimension grows. Standard solution Standard solution: construct an h-ordering, i.e. a surjection h : T → L onto a totally ordered lattice L, v i ≤h v j ⇐⇒ h(vi ) ≤ h(vj ) Our proposition Reorder the leaves of a Binary Partition Tree representing and image using both hierarchy and feature similarity O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 2 / 13
3. The ordering defines a palette and an index image Let the reordered vectors vi be P = {v′1 , . . . , v′m }. The image is represented by an index image I and an ordered palette P: f(v) = P[I(v)]. f I P Morphological operations are applied to the index and reconstructed from the palette : g(f(v)) = P[g(I(v))] Vector image f Index image I O. Lézoray grayscale MM Ordered palette P Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 3 / 13
4. From a BPT to a complete lattice Binary Partition Tree A BPT is obtained by hierarchical agglomerative clustering of image regions. ▶ Leaves correspond to the original pixels/regions. ▶ Each merge creates an internal node. An image and its BPT. ▶ With efficient data structures: O(m log m) construction. 1 Key observation At every internal node, the two children can be swapped. Thus the same BPT topology admits many valid leaf permutations. O. Lézoray 2 3 4 With a tree topology fixed, the leaf order can change How to order the leaves? Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 4 / 13
5. Leaf reordering: local traversal vs global optimization Existing BPT ordering The BPT can simply be traversed from left to right: ▶ Determined by the local child ordering. ▶ No explicit optimization of the final leaf sequence. Our viewpoint Instead of accepting one traversal, optimize over all permutations compatible with the tree topology. ▶ Used by Veganzones et al. for vector ordering. OLO: the reference algorithm Optimal Leaf Ordering minimizes OLO(π) = m−1 X d(πi , πi+1 ). i=1 1 2 3 4 2| m−1 {z } compatible leaf permutations It explicitly favors smooth consecutive leaf vectors. O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 5 / 13
6. UltraMixed: combine hierarchy and feature similarity Proposed objective For a leaf order π = (π1 , . . . , πm ): UltraMixed(π) = m−1 X [h(LCA(πi , πi+1 )) + λ d(πi , πi+1 )] i=1 where d is the Euclidean feature distance and λ ≥ 0 controls the trade-off ▶ h(LCA) keeps consecutive leaves close in the hierarchy. ▶ λd keeps consecutive vectors similar. ▶ Produces a path that respects both the tree structure and the original data similarities. ▶ As OLO: can be solved by dynamic programming. O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 6 / 13
7. How do we evaluate an ordering? 1. Palette smoothness 2. Index irregularity The ordered palette should form a short path in feature space: The total order should preserve spatial neighborhoods: m−1 ∥P∥TV = 1 X ′ ∥vi − v′i+1 ∥. m−1 i=1 ▶ Lower = smoother ordering. ▶ Measures feature-space continuity. O. Lézoray ∥∇I∥TV = 1 X X |I(vi )−I(vj )|. |V||E| v v ∼v i j i ▶ Lower = better spatial neighborhood preservation and lower artifacts ▶ Computed directly from I. Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 7 / 13
8. A single CIFAR image: smoothness vs spatial regularity Lex BPT OLO UltraMixed GPL P ∥P∥T V 0.4937 0.9165 0.7529 0.7720 0.8519 I ∥∇I∥T V 0.0936 0.0978 0.0869 0.0795 0.0838 What matters? ▶ Lexicographic order is very smooth, but has high spatial irregularity. ▶ OLO improves irregularity through leaf reordering. ▶ UltraMixed gives the lowest irregularity while remaining smooth. ▶ Optimizing only feature smoothness is not enough: the spatial structure encoded by the BPT matters. O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 8 / 13
9. CIFAR: consistent behavior over 30,000 images Experimental setting ▶ CIFAR training set: 30,000 color 32 × 32 images. ▶ BPT: average linkage. Main observation UltraMixed provides the best trade-off: short palette path and low index irregularity. Lexicographic smoothness is lower, but irregular and cannot be used with high-dimensional vectors. O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 9 / 13
10. Any kind of feature! Dilation Opening Closing VGG16 Patches Colors Erosion Figure: Morphological operations with a 3 × 3 structuring element with vectors: colors, 5 × 5 patches, and VGG16 features O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 10 / 13
11. Comparison with Veganzones et al. Original image UltraMixed [1] Erosion Dilation Opening Closing Figure: Comparison with Veganzones [1] with a 11 × 11 structuring element. O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 11 / 13
12. Conclusion 1. BPT leaf reordering: a way to build vector orderings for multivariate mathematical morphology. 2. The proposed UltraMixed objective combines two complementary sources of information: hierarchical structure + feature similarity 3. UltraMixed gives a strong balance between palette smoothness and low irregularity, with coherent morphological results. 4. The approach is generic and can operate on high-dimensional feature representations. Future work Improve the scalability of leaf-reordering methods for large images and larger feature sets. O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 12 / 13
13. The End O. Lézoray Vector Ordering Derived from Binary Partition Tree Leaf Reordering for Mathematical Morphology 13 / 13