Overview: Interactive Computer Graphics

� Interactive Graphics System Model

� Graphics Pipeline

� Coordinate Systems

� Modeling Transforms

� Cameras and Viewing Transform

� Lighting and Shading

� Color

� Rendering

� Visible Surface Algorithms

� Rasterization

� Hardware

ITCS 6010/8010 1 Review: Computer Graphics

The Viewing Pipeline

3D Model 3D World Viewing Transform

CameraSpecification

3DClipping

Projectionto 2D

RenderingDisplay

LightingSpecification

Model coords World Coords�

Camera coords

Clippingcoords

Screen coords

ITCS 6010/8010 2 Review: Computer Graphics

The Viewing Pipeline

ITCS 6010/8010 3 Review: Computer Graphics

Coordinate Systems

� Model Coordinates

� World Coordinates

� Viewing (Camera, Eye) Coordinates

� Perspective/Image Coordinates

� Screen/Display Coordinates

ITCS 6010/8010 4 Review: Computer Graphics

Graphics Primitives/Attributes

Primitives

� Point, Line, Polyline, Polygon, Rectangle, Circle

� Cubic Curves and Surfaces (Bezier, B-Spline, NURBS)

� Marker, Polymarker, Text

� Texture (2D, 3D)

Attributes

� Lines/Markers: style, width

� Color (R,G,B or index into lookup table)

� Filled Primitives: style, pattern

� Text: style (font), size, origin

ITCS 6010/8010 5 Review: Computer Graphics

Affine Transforms

� Most common in computer graphics.

� Permits a simple matrix representation

� Can be composed

Definition

[Qx Qy Qz

]=

a d gb e hc f k

Px

Py

Pz

+

Tx

Ty

Tz

~Q = ~M ~P + ~T

Primitive Affine Transforms:Scale, Rotate, Translate

ITCS 6010/8010 6 Review: Computer Graphics

Homogeneous Coordinates

� Allows all 3 primitive transforms to use a matrix representation.

� Add a fourth coordinate, w; additional column and row to matrix.

� To convert PH to P2D, divide each coordinate by the w and discardthe 3rd coordinate.

� w = 0 represents points at infinity.

PH = (Px, Py, Pz, Pw)

P3d = (Px/Pw, Py/Pw, Pz/Pw)

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Modeling Transformations

Translation

T (tx, ty, tz) =

1 0 0 tx

0 1 0 ty

0 0 1 tz

0 0 0 1

Scale

S(sx, sy, sz) =

sx 0 0 00 sy 0 00 0 sz 00 0 0 1

ITCS 6010/8010 8 Review: Computer Graphics

Modeling Transformations

Rotation

Rz(θ) =

cosθ −sinθ 0 0sinθ cosθ 0 0

0 0 1 00 0 0 1

Rx(θ) =

1 0 0 00 cosθ −sinθ 00 sinθ cosθ 00 0 0 1

Ry(θ) =

cosθ 0 sinθ 00 1 0 0

−sinθ 0 cosθ 00 0 0 1

ITCS 6010/8010 9 Review: Computer Graphics

Cameras and 3D Viewing

� A mechanism for specifying the the viewer location, orientation andvolume of the world that is of interest.

� A pin-hole camera model is assumed.

X�

Y�

Z�

VIEWER�

(CAMERA)

U

V�

N

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Camera Parameters

� View Reference Point, ~V RP .

� Look At vector, ~N .

� Up Vector, ~V .

� Vector ~U = ~N × ~V

� Window

� Camera (viewer) position.

� Near and Far clipping planes.

These parameters specify a view volume (frustum) that is a truncatedpyramid .

ITCS 6010/8010 11 Review: Computer Graphics

View Volume

ProjectionPlanet

Back

Frontt

Viewer�

N

� The volume of 3D space that is projected and displayed.

� Volume defined by a 2D window on the projection plane and frontand back clipping planes (hither/yon, near/far).

� View volume is specified in camera coordinates.

� View volume is a truncated frustum .

ITCS 6010/8010 12 Review: Computer Graphics

Viewing Transform

X

Y�

Z�

U

V�

N(0, 0, 0)

P (x, y, z)

Q (a, b, c)

World Coordinates Camera Coordinates(Right Handed) (Left Handed)

(rx, ry, rz)

� Object points in world coordinates are transformed into the cameracoordinate system with

� The viewer at the origin.

� Look-at vector along the negative Z axis.

� The Up vector along the Y axis.

ITCS 6010/8010 13 Review: Computer Graphics

Viewing Transform

The viewing transform is an affine transform, consisting of a rotation andtranslation, as follows:

Mwv =

ux uy uz rx

′

vx vy vz ry′

nx ny nz rz′

0 0 0 1

with ~r′ = (−~r.~U,−~r.~V ,−~r. ~N)

ITCS 6010/8010 14 Review: Computer Graphics

Geometric Projections

Projection

“An operation that transforms points in N-dimensional space to anotherof dimensionality less than n”

Geometric Projection

Application to geometric coordinate systems (Cartesian, Polar, Sphericaland Cylindric systems)

Types of Projections

� Planar

� Non-Planar

Application to Computer Graphics

1. Viewing 3D environments on a 2D screen.

2. Visualization of multi-variate data from scientific applications.ITCS 6010/8010 15 Review: Computer Graphics

Planar Projections

PLANAR

PARALLEL PERSPECTIVE

OBLIQUE ORTHOGRAPHIC

AXONOMETRIC� MULTI−VIEW

1−pt 2−pt�

3−pt�

ISOMETRIC DIMETRIC TRIMETRIC�

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Planar Projections

Parallel Projection

◦ Projectors are parallel to each other.

Perspective

◦ Projectors converge to a point.

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Perspective Projection

� Projectors converge to a point.

� All points on the same projector project to the same point, destroyingdepth information.

� Depth information must be carried along the pipeline for use in visi-ble surface calculations.

� Perspective transformation effectively involves a division of the 3Dpoint coordinates by its depth (from the view point).

Projection Plane at n = 0

t’P1

P2

�

P’

P(pu, pv, pn)

E(eu, ev, en)

ITCS 6010/8010 18 Review: Computer Graphics

Perspective Transformation

Projection Plane at n = 0

t’P1

P2

�

P’

P(pu, pv, pn)

E(eu, ev, en)

If ~P = (pu, pv, pn) is an object point, ~E = (eu, ev, en) is the cameraposition and ~P ′ = (p′u, p

′v, p

′n) is the projection point, assuming the

projection plane contains the origin of the camera coordinate system, itcan be shown that

p′u =pu

1− pn/en

p′v =pv

1− pn/en

ITCS 6010/8010 19 Review: Computer Graphics

Assuming p′n = pn

1−pn/enas a pseudo-depth, we can write

P ′ = (pu, pv, pn, 1− pn/en)

= [pu, pv, pn, 1] ~Mp

with

Mp =

1 0 0 00 1 0 00 0 1 −1/en

0 0 0 1

ITCS 6010/8010 20 Review: Computer Graphics

Lighting and Shading

The Rendering Equation

I(x, x′) = g(x, x′)

[ε(x, x′) +

∫S

ρ(x, x′, x′′)I(x′, x′′)dx′′]

whereI(x, x′) = Intensity of light passsing from x′ to x.g(x, x′) = “Geometry term”ε(x, x′) = Emitted light intensity from x′ to x.ρ(x, x′, x′′) = Light intensity scattered from x′′ to x by

a patch of surface at x′.

ITCS 6010/8010 21 Review: Computer Graphics

Local Lighting Model

� Consists of 3 components, ambient, diffuse and specular

Ambient Reflection

Ia = kaIa

� Ia is the ambient intensity, constant for all objects, and ka, the ambi-ent reflection coefficient.

� Use to model illumination that arrive at object points through indi-rect sources, such as nearby objects through multiple reflections orscattering.

ITCS 6010/8010 22 Review: Computer Graphics

Diffuse Reflection

Id = Ipkdcosθ

= Ipkd( ~N • ~L)

where ~N is the unit normal to the surface and ~L, the vector (of unitlength) to the light source, Ip the light source intensity.

� We approximate diffuse reflection using Lambert’s law.

� Intensity of light reflected is a constant, and independent of viewerposition.

ITCS 6010/8010 23 Review: Computer Graphics

Specular Reflection (Phong Model)

θ

L N

θ α

R

V Cos ( )�

α

α α α90 90 90

α4

64

Cos αCos

�

Is = kscosnα

= ks(~R • ~V )n

ITCS 6010/8010 24 Review: Computer Graphics

Local Illumination Model

Including ambient and diffuse reflections, the local lighting model be-comes

I = Ia + Id + Is

I = kaIa + Ip[kd( ~N • ~L) + ks(~R • ~V )n]

Assuming a wavelength based distribution of light and l light sources,

Iλ = Iaλka +

l∑1

fattiIpλi[kdλcos( ~N • ~Li) + ksλ( ~Ri • ~V )n]

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Rendering Algorithms: Gouraud Shading

N1

N2

N3

N4

N v

I

I

I

1

2

3

I a I bI p

y

y

y

1

s

2

� Face Normals are averaged to obtain vertex normals.

� Vertex Intensities (I1, I2, I3) are obtained by application of a lightingmodel.

ITCS 6010/8010 26 Review: Computer Graphics

Rendering Algorithms: Phong Shading

A�

B

C

N

NN c

b

a

� Compute Normals at vertices of polygon.

� Interpolate each component of normal at any interior point of poly-gon

� Apply illumination model.

� Can easily be incorporated into a scanline algorithm.

ITCS 6010/8010 27 Review: Computer Graphics

Visible Surface Algorithms

Z (Depth) Buffer Algorithm (Catmull ’74)

� A buffer of Z or depth values is maintained, one for each pixel.

� During scan conversion, if primitive’s depth is less (closer) than theexisting value in the Z buffer, replace it and corresponding color (in-tensity) in the frame buffer.

� Each primitive is scanconverted once, independent of any otherprimitive and in any order.

� Z (depth) can be calculated from the polygon plane equation, orinterpolated from depth at polygon vertices

ITCS 6010/8010 28 Review: Computer Graphics

Z-Buffer Algorithm

For each object primitive{

For each pixel in projection{

Z = primitive’s depth value at pixel(x,y).if Z < Z BUF[x][y]{

Write ZBUF (x, y, Z)Write Pixel (x, y, pixel color)

}}

}

ITCS 6010/8010 29 Review: Computer Graphics

Rasterization

Scanline Polygon Fill Algorithm

With some modifications and additional parameters, is used to scancon-vert polygons along with the Z buffer algorithm

Approach

� Exploit the geometry of the polygon.

� Use scanline and edge coherence for efficient filling.

I0� I1 I2

�I3

�

A�

B

C

D

E

j�

ITCS 6010/8010 30 Review: Computer Graphics

Color Models

RGB Color Model

Red

Green

Blue

Yellow�

Cyan

Magenta White�

(1, 1, 1)

Black(0,0,0)

Application

– Color CRT Monitors.ITCS 6010/8010 31 Review: Computer Graphics

CMY Color Model

� Complement of the RGB model.

� Uses subtractive primaries, Cyan, Magenta, Yellow. CMY

=

111

−

RGB

Application

� HardCopy Devices, such as ink jet color printers.

ITCS 6010/8010 32 Review: Computer Graphics

HSV Color Model

HS

V�

whitered

yellowgreen

cyan

blue magenta

black

� Model defined in a cylindrical coordinate system.

� Colors are defined in a hexcone or six-sided pyramid.

� Based on the artist’s model of tints, shades and tones.

⇒ Hue is measured around the vertical axis.

⇒ Saturation ranges from 0 on the centerline to 1 on the triangularsides.

⇒ Value ranges from 0.0 (black) at the apex of the pyramid to 1.0(white) at the base.

ITCS 6010/8010 33 Review: Computer Graphics

Display Architecture

PeripheralDevicesCPU

System Memory

VideoController

Display

System Bus

Display Processor

FrameBuffer

� Video Controller reads the frame buffer through a special port.

� A single address space system.

� Memory contention is still a problem, but the flexibility of a singleaddress space (for raster ops, for instance) outweighs this factor.

ITCS 6010/8010 34 Review: Computer Graphics

Graphics Processing Units (GPU)

� Modern graphics cards (Nvidia GEForce, ATI Radeon etc) are ex-tremely powerful computing platforms for processing/shading largeamounts of geometry

� Pipelined architecture

� Support for 3D textures (multitexturing, dependent texturing)

� May be used for general purpose computing

� Examples: Accelerated ray tracing, solving linear systems, fluid flow.

ITCS 6010/8010 35 Review: Computer Graphics