How to reform a terrain into a pyramid

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
? Procedure Move earth from higher to lower
positions (smoothing operation losing
potential) Output A mountain with the maximum
positional potential
If d1, mountain means a region below a
nonnegative unimodal curve. If d2, a
monochromatic image can be an input, where ?
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
Mathematical formulation

• Input A nonnegative function ? on Rd and a
  family F of regions in Rd . We assume
  .
• Output A pyramid function from (0,8) to F.
  That is, a series of regions P(t) (0lttlt8) of F
  satisfying that P(t) P(t) if t gt t.
• Objective Maximize
• ?(R) Integral of ? over a region R
• µ gives the measure of the space
• (e.g. if µ1, µ(R) is the volume)

F is a set of squares for pyramids in Egypt.
11
Equivalence of two formulations
f(x) max t x P(t) surface function of
the pyramid. For the optimal pyramid, for any t,
? (1)
mass of terrain
This means move earth to lower position
For a pyramid with condition (1),
positional potential
12
One dimensional problem

• Discrete version ? 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 ? 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
G(F) directed graph on F , and a directed edge
e(R, R) exists if and only if R R
t(e(R,R)) solution of
Weight w(e(R,R))
Theorem A pyramid gives a directed path
(R1,R2,,Rs) in G(F), and the optimal pyramid
gives the maximum weight path.
Corollary The optimal pyramid can be computed in
O(F3) time
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 AnB 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
AR(t) and BR(t)
A ?BR(t)
AnBR(t)
and
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
p
Rectangle containing a given point p
Rectangles containing p
Union of rectangles stabbed at p
Corresponding pyramid
17
Optimal pyramid for the rectangle unions stabbed
at p
Input pixel grid with n pixels,
positive matrices ? and µ representing functions,
and a grid point p. Output The optimal pyramid
of ? for U(p)
Theorem. The optimal pyramid for U(p) can be
computed in O(n log n ?) time, where ? is the
input precision.
18
Algorithm to compute the slide of the optimal
pyramid at height t
(for p(0,0))
Matrix(?- t µ)
Table of prefix sums
11
12
11
8
4
11
12
11
8
4
9
10
10
7
4
9
10
10
7
4
2
5
6
4
3
2
5
6
4
3
-
2
2
4
4
2
-
2
2
4
4
2
-
8
-
4
-
2
0
1
-
8
-
4
-
2
0
1
19
Computation of the region
12
12
11
11
8
4
11
11
8
4
9
10
10
7
4
9
10
10
7
4
2
5
6
4
3
2
5
6
4
3
-
2
2
4
4
2
-
2
2
4
4
2
-
8
-
4
-
2
0
1
-
8
-
4
-
2
0
1
Linear time for computing a flat P(t). (Longest
path in a DAG.) Binary decomposition process
attains O(n log n?) time to compute
the pyramid.
20
Higher dimensional case
Fd(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 Fd(p) can be computed in O(t(n,n) log
n?) 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
t
-
1
1
3
4
4
-
1
1
3
4
4
-
1
0
3
3
-
1
0
3
3
-1
-2
-
3
-
1
2
3
-
3
-
1
2
3
-
4
-
2
0
2
2
-
4
-
2
0
2
2
-
4
-
2
-
2
-
1
1
-
4
-
2
-
2
-
1
1
s
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))
Positive weighted nodes
s
Negative weighted nodes
G
t
23
Construction of G (an example)
s
1
1
2
3
1
2
3
1
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