Computer Graphics Inf4/MSc 1 Computer Graphics Lecture 4 View Projection Taku Komura.

Computer Graphics Inf4/MSc

1

Computer Graphics

Lecture 4

View Projection

Taku Komura


Measuring the BRDF

• Measured using a device called gonioreflectometer– Casting light from various directions to the object,

and capturing the light reflected back


Problems with Measured BRDF

• Includes a lot of error

• Huge amount of time to capture

• The data size is enormous – 18 hours acquisition time, 30GB raw data according

to [Ngan et al. EGSR ’05]

-> Fitting the acquired data into analytical models


Analytical models

• Empirical models

– Gouraud, Phong models or more complex models

• Microfacet models

– Assuming the surface is composed of a large number of micro mirrors

– Each reflect light back to the specular direction


Microfacet Theory

• [Torrance & Sparrow 1967]– Surface modeled by tiny mirrors – Value of BRDF at

• # of mirrors oriented halfway between and where is the incoming direction, is the out

going directionAlso considering the statistics of the shadowing/masking

• Modulated by Fresnel, shadowing/masking

[Shirley 97]


Examples : Satin


Examples : velvet


8

Implementation of viewing.

• Transform into camera coordinates.

• Perform projection into view volume or screen coordinates.

• Clip geometry outside the view volume.

• Remove hidden surfaces (next week)


Transformations

lwlcwproj vMMMv

11/04/23

Local to world matrix

World to cameramatrix

Projection matrix

Local coordinates

Screen coordinates


View Transformation (from lecture 2)

02/10/09Lecture 4 10

We want to know the positions in the camera coordinate system

vw = Mc→w vc

vc = Mc → w vw

= Mw→c vw

-1

Point in theworld coordinate

Point in thecamera coordinate

Camera-to-world transformation


11

View ProjectionWe want to create a picture of the scene viewed from the camera

Two sorts of projection

Parallel projection

Perspective projection


12

Mathematics of Viewing

• We need to generate the transformation matrices for perspective and parallel projections.

• They should be 4x4 matrices to allow general concatenation.

• And there’s still 3D clipping and more viewing stuff to look at.


13

Parallel projections (Orthographic projection)

• Specified by a direction of projection, rather than a point.

• Objects of same size appear at the same size after the projection


14

Parallel projection.

1000

0000

0010

0001

orthM

Orthographic Projection onto a plane at z = 0.

xp = x , yp = y , z = 0.


15

Perspective Projection• Specified by a center of projection and the focal

distance (distance from the eye to the projection plane)

Objects far away appear smaller, closer objects appear bigger


Projection Matrix• Here we will follow the projection transform

method used in OpenGL • The camera facing the –z direction


17

Perspective projection – simplest case.

d

y

z

ProjectionPlane.

P(x,y,z)

Pp(xp,yp,-d)

Centre of projection at the origin,Projection plane at z=-d.d: focal distance

x


18

Perspective projection – simplest case.

d

x

y

z

P(x,y,z)

Pp(xp,yp,-d)

z

P(x,y,z)

d

z

P(x,y,z)

d

y

x

xp

yp

dz

y

z

ydy

dz

x

z

xdx

z

y

d

y

z

x

d

x

pp

pp

/ ;

/

;

:trianglessimilarFrom


19

Perspective projection.

01/d-00

0100

0010

0001

:matrix 4x4 a as drepresente becan ation transformThe

perM

TTT

pp dzzyxdz

ydz

xddyx

1..1


20


TT

perp

Tp

dzzyxWZYX

z

y

x

PMP

WZYXP

/

101/d-00

0100

0010

0001

point projected general theRepresent


21


ddz

y

dz

x

W

Z

W

Y

W

X

dzzyxP Tp

,/

,/

,,

: 3D back to come W toDropping

/Trouble with this formulation :

Centre of projection fixed at the origin.


22

Alternative formulation.

z

P(x,y,z)

d

x

xp

z

P(x,y,z)

d

y

yp

Projection plane at z = 0Centre of projection at z = d

1)/( ;

1)/(

dby Multiply

;

:trianglessimilarFrom

dz

y

dz

ydy

dz

x

dz

xdx

dz

y

d

y

dz

x

d

x

pp

pp


23

Alternative formulation.

11/d-00

0000

0010

0001

perM

z

P(x,y,z)

d

x

xp

z

P(x,y,z)

d

y

yp

Projection plane at z = 0,Centre of projection at z = d

Now we can allow d


24

Problem

• After projection, the depth information is lost

• We need to preserve the depth information for hidden surface remove during rasterization


25

3D View Volume

• The volume in which the visible objects exist• For orthographic projection, view volume is a

box.• For perspective projection, view volume is a

frustum.• The surfaces outside the view volume must be

clipped Far clipping plane.

Near clipping plane

left

right

Need to calculate intersectionWith 6 planes.


26

Canonical View Volume• We can transform the frustum view volume into a

normalized canonical view volume using the idea of perspective transformation

• Much easier to clip surfaces and apply hidden surface removal


27

Transforming the View Frustum• Let us define parameters (l,r,b,t,n,f) that

determines the shape of the frustum

• The view frustum starts at z=-n and ends at z=-f, with 0<n<f

• The rectangle at z=-n has the minimum corner at (l,b,-n) and the maximum corner at (r,t,-n)


28

Transforming View Frustum into a Canonical view-volume

• The perspective canonical view-volume can be transformed to the parallel canonical view-volume with the following matrix:

0100

200

02

0

002

)0](,[

nf

fn

nf

nfbt

bt

bt

nlr

lr

lr

n

P

thenfnfnzIf

p


29

Final step.

• Divide by W to get the 3-D coordinates– Where the perspective projection actually gets done

• Now we have a ‘canonical view volume’.– Don’t flatten z due to hidden surface calculations.

• 3D Clipping– The Canonical view volume is defined by:

-1x 1, -1 y 1 , -1 z 1– Simply need to check the (x,y,z) coordinates and see if they are

within the canonical view volume


30

Reading for View Transformation

• Foley et al. Chapter 6 – all of it,– Particularly section 6.5

• Introductory text, Chapter 6 – all of it,– Particularly section 6.6

• Akenine-Moller, Real-time Rendering Chapter 3.5

Computer Graphics Inf4/MSc 1 Computer Graphics Lecture 4 View Projection Taku Komura.

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