1. How to reform a terrain into a pyramid Takeshi Tokuyama (Tohoku U) Joint work with Jinhee Chun (Tohoku U) Naoki Katoh (Kyoto U) Danny Chen (U. Notre Dame) Pictures from web page of Institute of Egyptology, Waseda University, Japan

2. Outline Motivations and definitions One-dimensional problem General case, reducing to the longest path problem in a large DAG A two-dimensional problem reducing to the longest path problem in a small DAG A higher-dimensional problem reducing to the minimum s-t cut problem in a directed graph Applications and discussion

3. Re-shaping problem: Given a geometric object, transform it to a “well-shaped” object. The solution depends on the definition of “well-shaped”. Convex  convex hull, convex approximation Surface  surface reconstruction Smooth  smoothing Union of simple shapes  decomposition problem covering problem Mountain-like shape  Pyramid problem

4. Intuitive (but non-mathematical) formulation Input: A terrain corresponding to a nonnegative function ρ Procedure: Move earth from higher to lower positions (smoothing operation losing potential) Output: A mountain with the maximum positional potential If d=1, “mountain” means “a region below a nonnegative unimodal curve”. If d=2, a monochromatic image can be an input, where ρ indicates the brightness

5. Examples of pyramid construction problem

6. Motivations How to extract the feature of the following picture (or data distribution) ? Image processing Data mining Statistics

7. Extract a dense rectangle

8. Image segmentation Partition the picture into an object and background

9. Extract as a pyramidic (or “fuzzy”) object Looks like a sliced onion, but different from the “onion structure” in computational geometry.

10. Input: A nonnegative function ρ on R d and a family F of regions in R d. We assume. Output: A “pyramid” function from (0,∞) to F. That is, a series of regions P(t) (0 t’. Objective: Maximize ρ （ R) ： Integral of ρ over a region R μ gives the measure of the space (e.g. if μ≡1, μ （ R) is the volume) Mathematical formulation F is a set of squares for pyramids in Egypt.

11. Equivalence of two formulations For a pyramid with condition (1), positional potential f(x)= max { t : x P(t)} :surface function of the pyramid. For the optimal pyramid, for any t, This means “move earth to lower position” mass of terrain ― (1)

12. One dimensional problem Discrete version: ρ is a nonnegative function on the interval [0, n], and F is the set of all integral intervals in [0,n]. (output is rectilinear) Continuous version: ρ and μ are piecewise- linear functions with n linear pieces, and F is the set of all intervals. Theorem. The optimal pyramid can be computed in O( n log n) time. Use convex hull tree

13. Higher dimensional cases, if |F| is small Corollary: The optimal pyramid can be computed in O(|F| 3 ) time G(F): directed graph on F, and a directed edge e(R, R’) exists if and only if R R’ Theorem: A pyramid gives a directed path (R 1,R 2,…,R s ) in G(F), and the optimal pyramid gives the maximum weight path. t(e(R,R’)): solution of Weight: w(e(R,R’))= Unfortunately, |F| is often large.

14. Closed family of regions A family F of regions is called a closed family if it is closed under union and intersection operations. That is, A ∪ B and A∩B are members of F if A and B are members of F Lemma. If F is closed, the horizontal slice P(t) of the optimal pyramid is the region R(t) maximizing and A=R(t) and B=R(t’) A ∪ B=R(t) A∩B=R(t’)

15. What is the region family for this pyramid ? U(p)= “closure” of the set of all rectangles containing a given point p (rectangle unions stabbed at p)

16. Rectangle containing a given point p p Rectangles containing p p Union of rectangles stabbed at p Corresponding pyramid

17. Input: pixel grid with n pixels, positive matrices ρ and μ representing functions, and a grid point p. Output: The optimal pyramid of ρ for U(p) Optimal pyramid for the rectangle unions stabbed at p Theorem. The optimal pyramid for U(p) can be computed in O(n log n γ) time, where γ is the input precision.

18. Algorithm to compute the slide of the optimal pyramid at height t Matrix （ ρ- t μ ） (for p=(0,0)) -8-4-201 -22442 25643 9 10 74 11 12 1184 -8-4-201 -22442 25643 9 10 74 11 12 1184 Table of prefix sums

19. -8-4-201 -22442 25643 9 10 74 11 12 1184 -8-4-201 -22442 25643 9 10 74 11 12 1184 Computation of the region Linear time for computing a flat P(t). (Longest path in a DAG.) Binary decomposition process attains O(n log nγ) time to compute the pyramid.

20. Higher dimensional case F d (p) = closure (under union) of the family of d- dimensional axis-parallel orthogonal regions containing a grid point p Theorem The optimal pyramid of a d-dimensional terrain in a pixel grid with n pixels with respect to F d (p) can be computed in O(t(n,n) log nγ) time, where t(n,n) is the time to compute a minimum s-t cut in a directed graph with O(n) nodes and O(n) edges.

21. -4-2-2-11 -4-2022 -3-12 -2 3 -1033 -11344 -4-2-2-11 -4-2022 -3-123 -1033 -11344 s t The cut maximizing the sum of node weights of dominated vertices

22. The cut in G minimizing the sum of node weights of dominated vertices = minimum s-t cut in a modified directed graph G’ (Hochbaum(01)) G Positive weighted nodes Negative weighted nodes s t

23. Construction of G’ (an example) 23 1-21 -31 1 11 1 2 3 3 2 s t

24. Other closed region families Connected lower half region of a grid curve Closure of L-shape paths The optimal pyramid for these region families can be efficiently computed We can also handle its higher dimensional analogue

25. Data mining application of segmentation

26. Output of SONAR data mining system (System for Optimized Numeric Association Rules) Given a database that contains 3.54% of unreliable customers (Age, Balance) ∈ Ｒ ⇒ (CardLoanDelay = yes) R contains about 10% of customers and maximizes the probability (14.39%) of unreliable customers.

27. Conclusion Pyramid construction A new geometric optimization problem Application to fuzzy segmentation Application to data mining Polynomial time algorithms for special cases Open problems Is the problem NP hard for the families of rectilinear convex regions (or convex regions)? Give an efficient algorithm for the family of (axis parallel) rectangles