CS 636 Start Up Advanced Rendering Techniques · CS636_Pres4 Author: David Breen Created Date:...
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1 CS 636 Advanced Rendering Techniques Dr. David Breen Online Wednesday 6PM ® 8:50PM Presentation 4 4/29/20 1 Start Up n Any questions from last time? n Go over acceleration techniques? n Or intersection algorithms? 2 Slide Credits n Leonard McMillan, Seth Teller, Fredo Durand, Barb Cutler - MIT n G. Drew Kessler, Larry Hodges - Georgia Institute of Technology n John Hart - University of Illinois n Rick Parent - Ohio State University 3 Transformations & Hierarchical Models 28 n Add an extra dimension • in 2D, we use 3 x 3 matrices • In 3D, we use 4 x 4 matrices n Each point has an extra value, w Homogeneous Coordinates x' y' z' w' = x y z w a e i m b f j n c g k o d h l p p' = M p 29 n Most of the time w = 1, and we can ignore it Homogeneous Coordinates x' y' z' 1 = x y z 1 a e i 0 b f j 0 c g k 0 d h l 1 30
Transcript of CS 636 Start Up Advanced Rendering Techniques · CS636_Pres4 Author: David Breen Created Date:...
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CS 636 Advanced Rendering Techniques
Dr. David BreenOnlineWednesday 6PM ® 8:50PM
Presentation 44/29/20
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Start Upn Any questions from last time?n Go over acceleration techniques?n Or intersection algorithms?
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Slide Creditsn Leonard McMillan, Seth Teller,
Fredo Durand, Barb Cutler - MITn G. Drew Kessler, Larry Hodges - Georgia
Institute of Technologyn John Hart - University of Illinoisn Rick Parent - Ohio State University
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Transformations & Hierarchical Models
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n Add an extra dimension• in 2D, we use 3 x 3 matrices• In 3D, we use 4 x 4 matrices
n Each point has an extra value, w
Homogeneous Coordinates
x'y'z'w'
=
xyzw
aeim
bfjn
cgko
dhlp
p' = M p
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n Most of the time w = 1, and we can ignore it
Homogeneous Coordinates
x'y'z'1
=
xyz1
aei0
bfj0
cgk0
dhl1
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Translate (tx, ty, tz)n Why bother with the
extra dimension?Because now translations can be encoded in the matrix!
Translate(c,0,0)
x
y
p p'
c
x'y'z'1
=
xyz1
1000
0100
0010
tx
ty
tz
1
x'y'z'
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Scale (sx, sy, sz)n Isotropic (uniform)
scaling: sx = sy = sz
n You only have to implement uniform scaling
x'y'z'1
=
xyz1
sx
000
0sy
00
00sz
0
0001
Scale(s,s,s)
x
p
p'
qq'
y
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Rotationn About z axis
x'y'z'1
=
xyz1
cos fsin f
00
-sin fcos f
00
0010
0001
ZRotate(f)
x
y
z
p
p'
θ
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Rotation
n About x axis:
n About y axis:
x'y'z'1
=
xyz1
0cos asin a
0
0-sin acos a
0
1000
0001
x'y'z'1
=
xyz1
cos q0
-sin q0
sin q0
cos q0
0100
0001
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Rotationn About (kx, ky, kz), an
arbitrary unit vector (Rodrigues Formula)
x'y'z'1
=
xyz1
kxkx(1-c)+ckykx(1-c)+kzskzkx(1-c)-kys
0
0001
kykx(1-c)-kzskyky(1-c)+c
kzky(1-c)+kxs0
kxkz(1-c)+kyskykz(1-c)-kxskzkz(1-c)+c
0
where c = cos q & s = sin q
Rotate(k, q)
x
y
z
θ
k
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How are transforms combined?
(0,0)(1,1)
(2,2)
(0,0)
(5,3)
(3,1)Scale(2,2) Translate(3,1)
TS =20
02
00
10
01
31
20
02
31=
Scale then Translate
Use matrix multiplication: p' = T ( S p ) = ((TS) p)
Caution: matrix multiplication is NOT commutative!
0 0 1 0 0 1 0 0 1
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Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p
Translate then Scale: p' = S ( T p ) = ST p
(0,0)(1,1)
(4,2)(3,1)
(8,4)
(6,2)
(0,0)(1,1)
(2,2)
(0,0)
(5,3)
(3,1)Scale(2,2) Translate(3,1)
Translate(3,1) Scale(2,2)
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TS =200
020
001
100
010
311
ST =20
02
00
10
01
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Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p
200
020
311
20
02
62
=
=
Translate then Scale: p' = S ( T p ) = ST p
0 0 1 0 0 1 0 0 1
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Transformations in Ray Tracing
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Transformations in Modelingn Position objects in a scenen Change the shape of objectsn Create multiple copies of objectsn Projection for virtual camerasn Animations
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Scene Description
Scene
LightsCamera ObjectsMaterialsBackground
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Simple Scene Description File
Camera {center 0 0 10direction 0 0 -1up 0 1 0 }
Lights {numLights 1DirectionalLight {
direction -0.5 -0.5 -1color 1 1 1 } }
Background { color 0.2 0 0.6 }
Materials {numMaterials <n>
<MATERIALS> }
Group { numObjects <n><OBJECTS> }
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Hierarchical Models
n Logical organization of scene
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Group { numObjects 3Group {
numObjects 3Box { <BOX PARAMS> }Box { <BOX PARAMS> }Box { <BOX PARAMS> } }
Group {numObjects 2Group {
Box { <BOX PARAMS> }Box { <BOX PARAMS> }Box { <BOX PARAMS> } }
Group {Box { <BOX PARAMS> }Sphere { <SPHERE PARAMS> }Sphere { <SPHERE PARAMS> } } }
Plane { <PLANE PARAMS> } }
Simple Example with Groups
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Group { numObjects 3Group {
numObjects 3Box { <BOX PARAMS> }Box { <BOX PARAMS> }Box { <BOX PARAMS> } }
Group {numObjects 2Group {
Box { <BOX PARAMS> }Box { <BOX PARAMS> }Box { <BOX PARAMS> } }
Group {Box { <BOX PARAMS> }Sphere { <SPHERE PARAMS> }Sphere { <SPHERE PARAMS> } } }
Plane { <PLANE PARAMS> } }
Adding Materials
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Adding Transformations
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Using Transformations
n Position the logical groupings of objects within the scene
n Transformation in group
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Directed Acyclic Graph is more efficient and useful
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Processing Model Transformattions
n Goaln Get everything into world coordinates
n Traverse graph/tree in depth-first ordern Concatenate transformationsn Can store intermediate transformationsn Apply/associate final transformation to
primitive at leaf noden What about cylinders, superquadrics, etc.?
n Transform ray!
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Transform the Ray
n Map the ray from World Spaceto Object Space
World Space
r majorr minor
(x,y)
(0,0) Object Space
r = 1
pWS = M pOS
pOS = M-1 pWS
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Transform Rayn New origin:
n New direction:originOS = M-1 originWS
directionOS = M-1 (originWS + 1 * directionWS) - M-1 originWS
originOS
originWS
directionOS
directionWS
Object SpaceWorld Space
qWS = originWS + tWS * directionWS
qOS = originOS + tOS * directionOS
directionOS = M-1 directionWS
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Transforming Points & Directions
n Transform point
n Transform direction
n Map intersection point and normal back to world coordinates
Homogeneous Coordinates: (x,y,z,w)W = 0 is a point at infinity (direction)
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Transform Normalsn Why? They’re used for shading
object color only Diffuse Shading
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Transforming Normalsn A surface normal is a property, not a
geometric entity
n Correct normal transformation matrix:
See https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/geometry/transforming-normals
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Given overlapping shapes A and B:
Union Intersection Subtraction
Constructive Solid Geometry (CSG)
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How can we implement CSG?
Points on A, Outside of B
Points on B, Outside of A
Points on B, Inside of A
Points on A, Inside of B
Union Intersection Subtraction
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Collect all the intersections
Union Intersection Subtraction
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Implementing CSG
1. Test "inside" intersections:• Find intersections with A,
test if they are inside/outside B• Find intersections with B,
test if they are inside/outside A
2. Overlapping intervals:• Find the intervals of "inside"
along the ray for A and B• Compute
union/intersection/subtractionof the intervals
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Constructive Solid Geometry
Ray Tracing CSG Models
Rick ParentOhio State U.
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Seminal paper Ray tracing CSG models
nS.D. Roth, "Ray Casting for Modeling Solids", Computer Graphics and Image Processing 18 (2): 109–144, February 1982
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CSGn Form object as booleans of primitive
objectsn Primitives: sphere, cube, cylinder, conen Boolean operators: union, intersection,
differencen Tree structure used to manage operations
n Leaf nodes are primitive objectsn Intermediate nodes specify combination
operator
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Ray tracing CSG modelsn Intersect ray with primitivesnProduces “spans” along raynPerform Boolean operations on
spansnDetermines intersection of evaluated
modelnCalculate normal at intersection
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Union
Min (tCmin, tBmin )
B
C +
BC
Ray intersects union: at first intersection
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Possible ways for 2 spans to overlap
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Intersection
++
If ((tCmin< tBmin ) and (tCmax> tBmin ) ): tBmin Else If ((tBmin< tCmin ) and (tBmax> tCmin ) ): tCmin Else: none
BC
B
C
First time in B and in C
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Difference
-
B C
If ((tBmin< tCmin ): tBmin Else if (tCmax< tBmax ): tCmaxElse: none
B
C
First time in B not in C
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Difference-
BC
If ((tCmin< tBmin): tCmin Else if (tBmax< tCmax ): tBmaxElse: none
B
C
First time in C not in B
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PrimitivesAnything that can be intersected (easily) with a ray
Conics: solve analytically using R(t)Convex polyhedraA plane (a cutting plane is useful)
can be used as a modeling tool (boolean operations) surface model (e.g., polyhedron) computed from CGS
or Can be used as a model representation
keep tree structure and ray trace directly
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Controlling the Combinations
-+
+
?
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Tree Structure
circlerectangle rectangle
+ +
+ -
T1T2 T3
T4
T5- +
+T1
T2
T3
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Tree Structure #1
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Tree Structure
circlerectangle rectangle
+ +
- +
T1T2 T3
T4
T5- +
+T1
T2
T3
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Tree Structure #2
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Tree Structure• Intersect ray with leaf nodes (primitive objects)• Combine intersection spans according to intermediate nodes
• union• intersection• difference
• Might create multiple spans
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Union of Spans
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Intersection of Spans
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Difference of Spans
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Normals of CSG intersections
Normal of some surface (or its negation)
Union or intersection: positive normal of intersected surface
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Difference normals• Intersection is one of:
• tmin of positive object – normal of surface• tmax of negative object – negated normal
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Add transformations to tree
http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/model/csg.html
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Bounding Volumes
circlerectangle rectangle
+ +
+ -
T1T2 T3
T4
T5
Construction•Use bounding volumes at leaf nodes• Union bounding volumes at interior nodes
Traversal•Top-down•Test bounding volume at interior
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Example
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Example
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Example
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Example (Simon Chorley)
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Example (Simon Chorley)
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For example:
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Wrap Upn Discuss status/problems/issues with
this week’s programming assignment
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