Contour Shape Analysis Using Crystalline Flow 2001.Nov.7.

Slides:

Advertisements

Similar presentations

CHAPTER 11 Vector-Valued Functions Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11.1VECTOR-VALUED FUNCTIONS.

Advertisements

Chapter 14 Section 14.5 Curvilinear Motion, Curvature.

Optimization. f(x) = 0 g i (x) = 0 h i (x)

Copyright © Cengage Learning. All rights reserved.

Curves and Polygons in the Plane

CONVEX AND CONCAVE LENSES OBJECTIVE: To find the focal point and focal length of convex and concave lens.

Polygons – Concave and Convex Turning Point Quiz Copyright © 2010 Kelly Mott.

Convex Mirrors. 2 Convex Mirror F and C are behind mirror.

AP Physics B Mrs. Wallace. Reflection Reflection occurs when light bounces off a surface. There are two types of reflection Specular reflection Off a.

Each pixel is 0 or 1, background or foreground Image processing to

Skill – C O N T O U R S. What are they and why are they used? Contours are lines joining places of equal elevation above sea level. Contours lines provide.

Physics 1502: Lecture 30 Today’s Agenda Announcements: –Midterm 2: Monday Nov. 16 … –Homework 08: due Friday Optics –Mirrors –Lenses –Eye.

USING GEOMETRIC MODELING FOR FEATURE RECOGNITION IN ARCHAEOLOGICAL VESSELS DEZHI LIU FEATURE GROUP PRISM/ASU 3DK – 3DK – September 15, 2000.

Chapter 11 Representation and Description. Preview Representing a region involves two choices: In terms of its external characteristics (its boundary)

Chapter 14 Section 14.3 Curves. x y z To get the equation of the line we need to know two things, a direction vector d and a point on the line P. To find.

Curved Mirrors The most common type of curved mirror is a spherical mirror A spherical mirror has the shape of a section from the surface of a sphere.

Please put your box number on your homework from now on. Box numbers are written in orange on the homework I am handing back. They are also posted in the.

Math Vocabulary Unit 1: Geometry What am I? ©Mrs. DeAmicis

Geometric Analysis of Packings Gady Frenkel, M. Blunt, P. King & R. Blumenfeld.

Ray Diagrams for spherical mirrors. Finding the focal point Center of Curvature (C)- if the mirror actually was a sphere, this is the center of that sphere.

Polygons Polygons. Polygon Any shape where every segment intersects exactly two others at its endpoints.

Measures: Perimeter & Area Mathematics and Millennials – 6th.

1 Mesh Parametrization and Its Applications 동의대학교 멀티미디어공학과 김형석 포항공과대학교 ( 이윤진, 이승용 )

Part 6: Graphics Output Primitives (4) 1.  Another useful construct,besides points, straight line segments, and curves for describing components of a.

Session 31 Check your homework with a neighbor. Which property justifies the statement? 1.If  A   B and  B   C, then  A   C. 2.If  M   N, then.

1 2 Curved mirrors have the capability to create images that are larger or smaller than the object placed in front of them. They can also create images.

Concave Mirrors Reflection, Image Height, and Distance.

Concave and Convex Mirrors

Spaces 1D, 2D, and 3D Point Line Plan Points/Vectors A point p = (x,y,z) is also the vector p Vector Operation The Length of the vector v Normalization.

Classify the triangle by its sides. The diagram is not to scale.

Vision. Normal Vision light is focused directly on the retina - can see clearly both near & far.

1.6 Classify Polygons. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that.

Urad_Budget_Part_I All variables shown are taken from 5-min data, averaged over a 1 h period F_AGRA = PGF + CenCorf Diff + Error = Diffusion + Calculation.

Polygon Definition Bounded by a closed circuit of straight-line segment. Term Edge : straight line segment Vertices : points.

Mirrors. Mirrors and Images (p 276) Light travels in straight lines, this is the reason shadows and images are produced (p 277) Real images are images.

Interior and Exterior Angles (6.1/6.2) Interior Angles in Convex Polygons Essential Question: How can you determine the number of degrees in any.

Reflection and Color Chapter Light modeled as a ray Light ray  a line in space that matches the direction of the flow of radiant energy (Imaginary.

Arc Length and Curvature

11 Chapter Introductory Geometry

Ch 1.6 CLASSIFY POLYGONS. In this section we will… Look at what makes something a polygon. Identify polygons by their sides and angles Identify polygons.

PHY 102: Lecture Wave Fronts and Rays 9.2 Reflection of Light

Quadrilaterals Jacob, Allie, and Jesenia. Definition ★ Quadrilateral means “4 sides ★ “Quad” means 4 and “lateral” means sides.

Sheng-Fang Huang Chapter 11 part I.  After the image is segmented into regions, how to represent and describe these regions? ◦ In terms of its external.

Enlargement Simple scale factors. Find the scale factor and the missing length ?

Tangential Wind vs Radial Wind Local Changes All variables shown are taken from 5-min data, averaged over a 1 h period Vtan = tangential wind component,

PGF and Flow Kinematics for z=12,14,16 km All variables shown are taken from 5- min data, averaged over a 1 h period.

Copyright © Cengage Learning. All rights reserved.

Constructing Objects in Computer Graphics

Lesson 1: Polygons, Triangles, Transversals and Proportional Segments

RAY DIAGRAMS FOR MIRRORS

Vector-Valued Functions and Motion in Space

Lenses and Ray Diagrams

Concept. Skills Check 1-5 (I will scan it in at the end of class) Going over the quiz.

Polygons with four sides

Copyright © Cengage Learning. All rights reserved.

Polygon Definition: A closed figure.

Copyright © Cengage Learning. All rights reserved.

CURVED MIRRORS.

11 Chapter Introductory Geometry

Characteristics of Lenses

4.4 Concave and Convex Mirrors

Computing Vertex Normals from Arbitrary Meshes

Ray Diagrams for spherical mirrors

Representation and Description

Curved / Spherical Mirrors

PLASTIC DEFORMATION & DISLOCATIONS

1.6 Classify Polygons.

Presentation transcript:

Contour Shape Analysis Using Crystalline Flow 2001.Nov.7

Crystalline Flow Evolution of a polygon

Crystalline Flow Outward Normal Velocity depends on Nonlocal Weighted Curvature

Curvature Inscribing Circle Wulff Shape

A Convex m-Polygon

Wulff Shape A Set of Unit Vectors:

Admissible Crystal A simple Polygon All outward normals belong to The normal of adjacent facet is parallel to the normal adjacent in the Wulff Shape Admissible Wulff Shape Not Admissible Jump!

Crystalline Flow : Nondecreasing in 2nd variable: Transition Number 0+1 : Length of facet of Wulff Shape: Length of i-th facet Nonlocal Curvature

Crystalline Flow Facet Disappearing at t = T* Case A: The polygon becomes convex near T* and all facet disappear at T*. All facets disappear at t = T*.

Crystalline Flow Facet Disappearing at t = T* Case B: Two parallel facets meet together.

Crystalline Flow is locally Lipschitz on is nondecreasing on for all Case B does not occur if

Crystalline Flow Facet Disappearing at t = T* Case C: At most two consecutive facets disappear.

Crystalline Flow

Crystalline Flow

Chain Coded Contour Make given polygon Admissible

Crystalline Flow

Scale Space Analysis Facet Number in Original Time White: Convex Black: Concave

Facet Extraction Trace concave(convex) facets back to t=0. Extraction Scale Extracted Facets

Facet Extraction As the extraction scale increases, more important facets are extracted.

Facet Extraction

Conclusions Crystalline for Contour Shape Analysis Wulff Shape Selection!