slide1 l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
How to reform a terrain into a pyramid PowerPoint Presentation
Download Presentation
How to reform a terrain into a pyramid

Loading in 2 Seconds...

play fullscreen
1 / 27

How to reform a terrain into a pyramid - PowerPoint PPT Presentation


  • 132 Views
  • Uploaded on

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. Outline. Motivations and definitions

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

How to reform a terrain into a pyramid


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

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

slide2

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
slide3

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

slide4

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

slide6

Motivations

How to extract the feature of the following picture (or data distribution) ?

  • Image processing
  • Data mining
  • Statistics
slide8

Image segmentation

Partition the picture into an object and background

slide9

Extract as a pyramidic (or “fuzzy”) object

Looks like a sliced onion, but different from the “onion structure” in computational geometry.

slide10

Mathematical formulation

  • Input: A nonnegative function ρ on Rd and a familyF of regions in Rd . We assume .
  • Output: A “pyramid” function from (0,∞) to F. That is, a series of regions P(t) (0<t<∞) of F satisfying that P(t)  P(t’) if t > 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)

F is a set of squares for pyramids in Egypt.

slide11

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

slide12

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

slide13

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(|F|3) time

Unfortunately, |F| is often large.

slide14

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

A=R(t) and B=R(t’)

A ∪B=R(t)

A∩B=R(t’)

and

slide15

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)

slide16

p

p

Rectangle containing a given point p

Rectangles containing p

Union of rectangles stabbed at p

Corresponding pyramid

slide17

Optimal pyramid for the rectangle unions stabbed at p

Input: pixel grid with n pixels, positivematrices ρ and μ representing functions, and a grid point p.

Output: The optimal pyramid of ρ for U(p)

Theorem. The optimal pyramid for U(p) can be computed in O(n log n γ) time, where γ is the input precision.

slide18

Algorithm to compute the slide of the optimal pyramid at height t

(for p=(0,0))

Matrix(ρ- t μ)

Table of prefix sums

12

12

4

4

8

8

11

11

11

11

10

10

4

4

7

7

10

10

9

9

3

3

4

4

6

6

5

5

2

2

2

2

4

4

4

4

2

2

-

-

2

2

1

1

0

0

-

-

2

2

-

-

4

4

-

-

8

8

slide19

Computation of the region

12

12

4

4

8

8

11

11

11

11

10

10

4

4

7

7

10

10

9

9

3

3

4

4

6

6

5

5

2

2

2

2

4

4

4

4

2

2

-

-

2

2

1

1

0

0

-

-

2

2

-

-

4

4

-

-

8

8

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.

slide20

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 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.

slide21

t

4

4

4

4

3

3

1

1

-

-

1

1

3

3

3

3

0

0

-

-

1

1

-1

-2

3

3

2

2

-

-

1

1

-

-

3

3

2

2

2

2

0

0

-

-

2

2

-

-

4

4

1

1

-

-

1

1

-

-

2

2

-

-

2

2

-

-

4

4

s

The cut maximizing the sum of node weights of dominated vertices

slide22

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

slide23

1

-3

1

-2

1

3

2

-1

Construction of G’ (an example)

s

1

1

2

3

1

2

3

1

t

slide24

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

slide26

Output of SONAR data mining system

(

System for Optimized Numeric Association Rules)

Given a database that contains

3.54% of unreliable customers

(

Age, Balance)

(

CardLoanDelay

= yes)

R contains about 10% of customers

and maximizes the probability

(14.39%) of unreliable customers.

slide27

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