Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)
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Transcript of Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)
![Page 1: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/1.jpg)
Hank Childs, University of OregonLecture #2
Fields, Meshes, and Interpolation (Part 1)
![Page 2: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/2.jpg)
I’m back
• Next travel: Oct 26th
– Will announce plan as we get closer
![Page 3: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/3.jpg)
Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 4: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/4.jpg)
Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 5: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/5.jpg)
Project #1
• Goal: write a specific image
• Due: “today”• % of grade: 2%
How is project #1 going?
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Project #2
• Will assign on Monday• Don’t want to have to rush through lecture …
will only make things confusing.
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Office Hours
• Who wants OH today?
![Page 8: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/8.jpg)
Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Fields– Meshes– Interpolation
![Page 9: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/9.jpg)
Scientific Visualization
• An interdisciplinary branch of science – primarily concerned with the visualization of three-dimensional
phenomena (architectural, meteorological, medical, biological, etc.)
– the emphasis is on realistic renderings of volumes, surfaces, illumination sources, and so forth, perhaps with a dynamic (time) component.
• It is also considered a branch of computer science that is a subset of computer graphics.
• The purpose of scientific visualization is to graphically illustrate scientific data to enable scientists to understand, illustrate, and glean insight from their data.
Source: wikipedia
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Information Visualization
• The study of (interactive) visual representations of abstract data to reinforce human cognition. – The abstract data include
both numerical and non-numerical data, such as text and geographic information.
Source: wikipedia
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SciVis vs InfoVis• “it’s infovis when the spatial representation is chosen, and
it’s scivis when the spatial representation is given”
(A)
(B)
(E)
(C)
(D)
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What sorts of data?
Of course, lots of other data too…
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What Is Visualization Used For?
• 3 Main Use Cases:– Communication– Confirmation– Exploration
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How Visualization Works
• Many visual metaphors for representing data– How to choose the right tool from the toolbox?
• This course:– Describe the tools– Describe the systems that support the tools
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Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 16: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/16.jpg)
Computer Graphics
• Defined: pictorial computer output produced, through the use of software, on a display screen, plotter, or printer.
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What is computer graphics good for?
• Ed Angel book:– Display of information– Design– Simulation and animation– User interfaces
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The Basics of Using a Graphics System
• You define:– Geometric primitives, typically a triangle mesh– Coloring for those geometry primitives– A model for a camera (where you are and what you are looking
at)• And then:
– Communicate this information using an interface (e.g., OpenGL)– And something (e.g., GPU) renders the scene
• VTK will do this for us.• Visualization: We will focus on transforming data to
geometric primitives.
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Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 20: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/20.jpg)
Elements of a VisualizationLegend
Provenance Information
Reference Cues
Display of Data
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Elements of a Visualization
What is the value at this location?How do you know?
What data went into making this picture?
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Where does temperature data come from?
• Iowa circa 1980s: people phoned in updates
6:00pm: Grandmacalls in 80F. 6:00pm: Grandma’s friend calls in 82F.
What is the temperature along the white line?
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What is the temperature at points between Ralston and Glidden?
D=0, Ralston, IA
Distance D=10 miles, Glidden, IA
~~
Tem
pera
ture
80F
82F
81F
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Ways Visualization Can Lie
Data Errors:Data collection is inaccurateData collected too sparsely
Visualization Program
Visualization Errors:Illusion of certainty Poor choices of parameters
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Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 26: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/26.jpg)
Fields & Spaces• Fields are defined over
“spaces”.– We will be considering
2D & 3D spaces.• Defined by an origin and
three vectors (or two vectors) that define orientation.
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Scalar Fields
• Defined: associate a scalar with every point in space.
• What is a scalar?– A: a real number
• Examples:– Temperature– Density– Pressure
The temperature at 41.2324° N, 98.4160° W is 66F.
Fields are defined at every location in a space (example space: USA)
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Vector Fields
• Defined: associate a vector with every point in space.
• What is a vector?– A: a direction and a
magnitude• Examples:– Velocity
The velocity at location (5, 6) is (-0.1, -1)
The velocity at location (10, 5) is (-0.2, 1.5)
Typically, 2D spaces have 2 components in their vector field, and 3D spaces have 3 components in their vector field.
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Vector Fields
Representing dense data is hard and requires special techniques.
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More fields (discussed later in course)
• Tensor fields• Functions• Volume fractions• Multi-variate data
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Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 32: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/32.jpg)
Mesh
What we want
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An example mesh
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An example mesh
Where is the data on this mesh?
(for today, it is at the vertices of the triangles)
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An example mesh
Why do you think the triangles change size?
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Anatomy of a computational mesh
• Meshes contain:– Cells– Points
• This mesh contains 3 cells and 13 vertices
• Pseudonyms:• Cell == Element ==
Zone• Point == Vertex ==
Node
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Types of Meshes
Curvilinear Adaptive Mesh Refinement
Unstructured
We will discuss all of these mesh types more later in the course.
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Rectilinear meshes
• Rectilinear meshes are easy and compact to specify:– Locations of X positions– Locations of Y positions– 3D: locations of Z positions
• Then: mesh vertices are at the cross product.• Example: – X={0,1,2,3}– Y={2,3,5,6}
Y=6
Y=5
Y=3
Y=2X=1X=0 X=2 X=3
![Page 39: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/39.jpg)
Rectilinear meshes aren’t just the easiest to deal with … they are also very common
![Page 40: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/40.jpg)
Quiz Time
• A 3D rectilinear mesh has:– X = {1, 3, 5, 7, 9}– Y = {2, 3, 5, 7, 11, 13, 17}– Z = {1, 2, 3, 5, 8, 13, 21, 34, 55}
• How many points?• How many cells?
= 5*7*9 = 315
= 4*6*8 = 192
Y=6
Y=5
Y=3
Y=2X=1X=0 X=2 X=3
![Page 41: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/41.jpg)
Definition: dimensions
• A 3D rectilinear mesh has:– X = {1, 3, 5, 7, 9}– Y = {2, 3, 5, 7, 11, 13, 17}– Z = {1, 2, 3, 5, 8, 13, 21, 34, 55}
• Then its dimensions are 5x7x9
![Page 42: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/42.jpg)
How to Index Points
• Motivation: many algorithms need to iterate over points.
for (int i = 0 ; i < numPoints ; i++){ double *pt = GetPoint(i); AnalyzePoint(pt);}
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Schemes for indexing points
Logical point indices Point indices
What would these indices be good for?
![Page 44: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/44.jpg)
How to Index Points
• Problem description: define a bijective function, F, between two sets: – Set 1: {(i,j,k): 0<=i<nX, 0<=j<nY, 0<=k<nZ}– Set 2: {0, 1, …, nPoints-1}
• Set 1 is called “logical indices”• Set 2 is called “point indices”
Note: for the rest of this presentation, we will focus on
2D rectilinear meshes.
![Page 45: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/45.jpg)
How to Index Points• Many possible conventions for indexing points and
cells.• Most common variants:– X-axis varies most quickly– X-axis varies most slowly
F
![Page 46: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/46.jpg)
Bijective function for rectilinear meshes for this course
int GetPoint(int i, int j, int nX, int nY){ return j*nX + i;}
F
![Page 47: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/47.jpg)
Bijective function for rectilinear meshes for this course
int *GetLogicalPointIndex(int point, int nX, int nY)
{ int rv[2]; rv[0] = point % nX; rv[1] = (point/nX);
return rv; // terrible code!!}
![Page 48: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/48.jpg)
int *GetLogicalPointIndex(int point, int nX, int nY){ int rv[2]; rv[0] = point % nX; rv[1] = (point/nX); return rv;}
F
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Quiz Time #2
• A mesh has dimensions 6x8.• What is the point index for (3,7)?• What are the logical indices for point 37?int GetPoint(int i, int j, int nX, int nY){ return j*nX + i;}
int *GetLogicalPointIndex(int point, int nX, int nY){ int rv[2]; rv[0] = point % nX; rv[1] = (point/nX); return rv; // terrible code!!}
= 45= (1,6)
![Page 50: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/50.jpg)
Quiz Time #3
• A vector field is defined on a mesh with dimensions 100x100
• The vector field is defined with double precision data.
• How many bytes to store this data?
= 100x100x2x8 = 160,000
![Page 51: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/51.jpg)
Bijective function for rectilinear meshes for this course
int GetCell(int i, int j int nX, int nY){ return j*(nX-1) + i;}
![Page 52: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/52.jpg)
Bijective function for rectilinear meshes for this course
int *GetLogicalCellIndex(int cell, int nX, int nY){ int rv[2]; rv[0] = cell % (nX-1); rv[1] = (cell/(nX-1)); return rv; // terrible code!!}
![Page 53: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/53.jpg)
Outline• Projects & OH• Intro– SciVis vs InfoVis– Very, very basics of computer graphics
• The Data We Will Study– Overview– Fields– Meshes– Interpolation
![Page 54: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/54.jpg)
Linear Interpolation for Scalar Field F
Goal: have data at some points & want to interpolate data to any location
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Linear Interpolation for Scalar Field F
A B
F(B)
F(A)
X
F(X)
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Linear Interpolation for Scalar Field F
• General equation to interpolate:– F(X) = F(A) + t*(F(B)-F(A))
• t is proportion of X between A and B– t = (X-A)/(B-A)
A B
F(B)
X
F(X)F(A)
![Page 57: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/57.jpg)
Quiz Time #4
• General equation to interpolate:– F(X) = F(A) + t*(F(B)-F(A))
• t is proportion of X between A and B– t = (X-A)/(B-A)
• F(3) = 5, F(6) = 11• What is F(4)? = 5 + (4-3)/(6-3)*(11-5) = 7
![Page 58: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/58.jpg)
Bilinear interpolation for Scalar Field F
F(0,0) = 10 F(1,0) = 5
F(1,1) = 6F(0,1) = 1
What is value of F(0.3, 0.4)?
F(0.3, 0) = 8.5
F(0.3, 1) = 2.5
= 6.1
• General equation to interpolate: F(X) = F(A) + t*(F(B)-F(A))
Idea: we know how to interpolate along lines. Let’s keep doing that
and work our way to the middle.
![Page 59: Hank Childs, University of Oregon Lecture #2 Fields, Meshes, and Interpolation (Part 1)](https://reader036.fdocuments.in/reader036/viewer/2022081506/5697bfa71a28abf838c98ff6/html5/thumbnails/59.jpg)
Rayleigh Taylor Instability