歐陽明 Professor Dept. of CSIE and GINMNTU Ming Ouhyoung Shading

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Shading

Ming Ouhyoung 歐陽明 Professor Dept. of CSIE and GINM

NTU

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Illumination model

1) Ambient light (漫射)(or 環境反射) I = la * ka * Obj(r, g, b)

la : intensity of ambient light

ka : 0.0 ~ 1.0, Obj(r, g, b): object color 2) Diffuse reflection (散射)(or 漫反射)

I = Ip (r, g, b) * Kd * Obj(r, g, b) * COS(θ ) Ip (r, g, b): light color

3) Light source attenuation

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Specular reflection

^{(似鏡面反射)}

• I = Ks * Ip(r, g, b) * COSⁿ(α), Ks = specular-reflection coef.

• Phong illumination model

color of object = Obj (R, G, B)

= (Or, Og, Ob) or (light frequency) where 0.0 =< Od =< 1.0

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Faster specular reflection calculation:

Halfway vector approximation

• halfway vector

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Polygon shading : linear interpolation

a. flat shading : constant surface shading.

b. Gouraud shading: color interpolation shading.

c. Phong shading: vertex normal interpolation shading

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Phong Shading

• Use a big triangle, light shot in the center, as an example!

• The function is really an approximation to Gaussian distribution

macroscopic

• The distribution of microfacets is Gaussian. [Torrance, 1967]

(Beckmann distribution func.)

• Given normal direction N_a and N_b, N_m = ?

• interpolation in world or screen coordinate?

• in practice

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Bi-linear interpolation

. Linear interpolation:

A (x1, y1, z1) with color (r1, g1, b1); B (x2, y2, z2) with color (r2, g2, b2)

What is the color of point C (x3, y3, z3) located on the line AB.

C = color of A + t * (color of B- color of A), where t is | (C-A) |/ |(B-A)|

Similarly, we can process Bi-linear interpolation

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Gouraud Shading with Bilinear Interpolation

• Gouraud shading (smooth shading): color interpolation, for example,

Triangle with three vertices (x1, y1), (x2, y2), (x3, y3), each with red components R1, R2, R3

color is represented as (Red, Green , Blue)

Assuming a plane (in 3D) with vertices (x1, y1, R1), (X2, Y2, R2), and (X3, y3, R3)

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Henri Gouraud

(at 2018 Siggraph)

English pronunciation: /ger’raud/

French pronunciation: /’gu:hu:/

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Gouraud shading

• Vector equation of the plane is

(x,y) = s (x2-x1, y2 – y1) + t(x3-x1, y3-y1) + (x1, y1) solved for (s, t) , then

s = A1x + B1y +C1, t = A2x + B2y +C2

So, given point (x,y) in this plane, what is its color?

Answer: color of (x,y) = R1+ s (R2-R1) + t (R3-R1), or color = Ax + By +C, where

A = A1 (R2-R1)+ A2 (R3-R1) B = B1(R2-R1) + B2 (R3-R1)

C = C1(R2-R1) + C2(R3-R1) + R1

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How is the color calculated?

Since, (x,y) = s (x2-x1, y2 – y1) + t(x3-x1, y3-y1) + (x1, y1)

Therefore

, (x,y, R) = s (x2-x1, y2 – y1, R2-R1) + t(x3-x1, y3- y1, R3-R1) + (x1, y1, R1)

or, color R = s (R2-R1) + t (R3-R1) + R1

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Complexity of shading

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Phong: under Ivan Sutherland

• Bùi Tường Phong (Vietnamese: Bùi Tường Phong, December 14, 1942–1975) was a Vietnamese-born computer graphics researcher and pioneer.

• He came to the University of Utah College of

Engineering in September 1971 as a research assistant in Computer Science and he received his Ph.D. from the University of Utah in 1973.

• Phong knew that he was terminally ill with leukemia while he was a student. In 1975, after his tenure at the University of Utah, Phong joined Stanford as a

professor. He died not long after finishing his dissertation

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What is the color of copper?

• Reflection of copper

– drastic change as a function of incidence angle [Cook, 82"]

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New method: BRDF:Bi-directional Reflectance Density Function

• Use a camera to get the reflection of materials from many angles

• Light is also from many angles

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The Rendering Equation: Jim Kajiya

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BRDF

• BRDF: a four-dimensional function that defines how light is reflected at an opaque surface.

• The BRDF was first defined by Edward Nicodemus around 1965^[1]. The modern definition is:

F(ω_i, ωô^{) = dL(}ωô^)/dE(ω_i ) = dL(ωô)/L(ω_i )cos θ_i dω_i

– where L is the radiance, E is the irradiance, and θ_i is the angle made between ω_i and the surface normal, n.

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BSSRDF

• The Bidirectional Surface Scattering Reflectance Distribution Function (BSSRDF), is a further

generalized 8-dimensional function S(X_i,ω_i,X_o,ω_o), in which light entering the surface may scatter

internally and exit at another location. X

describes a 2D location over an object's surface.

• non-local scattering effects like shadowing,

masking, interreflections or subsurface scattering.

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SSBRDF, BSDF (Bidirectional scattering

distribution function)

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3D face scan

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Statute of Obama, US president

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Statute of Obama (Plaster), US president

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Homework #1

• Input : a file of polygons (triangles)

• test image : a teapot, a tube

• input format :

Triangle fr, fg, fb, br, bg, bb x y z nx ny nz

x1, y1, z1, …. , ,X2, y2, z2 …….,

/* where (fr, fg, fb) contains front face colors, (br, bg, bb) are background colors (x,y,z): 3D vertex position

(nx,ny,nz) : vertex normal

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Hw#1 requirements

• Deadline: to be announced

• Output: shaded objects with at least two lights

• Rotation, Scaling, Translation, Shear

• Clipping (front and back, left and right, top and bottom)

• Camera: two different views

– Object view and camera view

• C, C++, Java, etc.

– Limited WebGL library calls

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Polygon file format used

• e.q.

Triangle fr fg fb br bg bb x1 y1 z1 nx1 ny1 nz1

x2 y2 z2 nx2 ny2 nz2 x3 y3 z3 nx3 ny3 nz3 Triangle

– Where

fr, fg, fb are foreground colors (Red Green Blue) nx, ny, nz are vertex normal

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Other formats (more efficient): winged-edge representation

• Vertices 1, (x, y, z)

2, (x1, y1, z1) 3, …

23, … 890, ….

1010

• Triangle 1010, 23, 890

• Triangle 1, 2, 800

Winged –edge: in which each edge points to two vertices, two faces, and the four (clockwise and counterclockwise) edges that touch them. Winged-edge meshes allow constant time traversal of the surface, but with higher storage requirements.

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HW#1: expected results

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Visible-Surface Determination

• The painter's algorithm

• The Z-buffer algorithm (chap 5.8, textbook)

– The point nearest to the eye is visible,...

– Very easy both for software and hardware.

– Hardware Implementation: Parallel ---> fast display

• Scan-line algorithms

– One scan line at a time

• Area-subdivision algorithm

– Divide and conquer strategy

• Visible-surface ray tracing

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List-priority algorithms

• Depth-sort algorithm

sort by Z coord.(distance to the eye),

resolve conflicts(splitting polygons), scan convert·

---v.s.---painter's algorithm Binary Space PartitionTrees(BSP tree) (Chap 9.10, p505,

textbook)

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Determine the depth order!

1. Mountain, 2. bridge, 3. people 4. Hat

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Depth Sort: Easy Cases

• A lies behind all other polygons

– Can render

• Polygons overlap in z but not in either x or y

– Can render independently

E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-

Wesley 2012

31

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Hard Cases

Overlap in all directions but can one is fully on one side of the other

cyclic overlap

penetration

Wesley 2012

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BSP: Binary Space PartitionTrees(BSP tree)

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The Display Order of Binary Space Partition Trees(BSP tree)

if Viewer is in front of root, then

• Begin {display back child, root, and front child}

• BSP_displayTree(tree->backchild)

• displayPolygon(Tree->root)

• BSP_displayTree(tree->frontchild)

• end else

• Begin

• BSP_displayTree(tree->frontchild)

• displayPolygon(Tree->root)

• BSP_displayTree(tree->backchild)

• end

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Test: please give the BSP binary tree, and display order of this diagram. (Choose

smaller number as the new root)

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Visibility determination(2): Z-buffer algorithm

Initialize a Z-buffer to infinity (depth_very_far) {

Get a Triangle, calculate one point's depth from three vertices by linear interpolation

If the one point's depth depth_P(x,y) is smaller than Z- Buffer(x,y)

Z-Buffer (x,y) = depth_P(x,y), Color_at(x,y) = Color_of_P(x,y) else

DO_NOTHING }

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Complexity of shading

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Homework #2 (could be) Unity 3D games

• Modify an exisiting game

• Change the way a canon ball is fired (weapon system), tank motion control etc.

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HW#1: expected results

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HW#1: formula

•

– Ia is object color

– Ip is the color of light, and can have multiple lights

• Note

– color overflow problem (integer color up to 255) – MAX = max(R, G, B) = 255 etc.

• Output format

– RGBx RGBx.... 256*256 pixels

– better results: 32<=R,G,B<=230, each 1 byte binary data

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Visible line determination

1. Assume that visible surface determination can be done fast (by hardware Z-buffer or software BSP tree)

– This method is used in most high performance systems now!

(hollow triangle, mesh)

2. Depth cueing is more effective in showing 3D (in vector graphics machine, e.g. PS300). see sec.

14.3.4

– depth cueing: intensity interpolation

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Visible - line determination:

Appel's algorithm

• quantitative invisibility of a point=0 --> visible

• quantitative invisibility changes when it passes

"contour line".

• contour line:(define)

• vertex traversing

– EF: contour line

– AB: whether this line

segment is partially visible?

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Standard Graphics Pipeline

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How to transform a plane? a surface normal?

plane equation Ax + By + Cz + D = 0 N^T*P=0 since p is transformed by M, How should we transform N?

find Q, such that (Q*N)^T*M*P=0 (N')^T.(P')=0 i.e. N^T*Q^T*M*P=0, i.e. Q^T*M=I

∴Q^T=M^-1 Q=(M^-1)^T

similar, the surface normal is transformed by Q, not M!!

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Aliasing, anti-aliasing

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E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Aliasing

• Ideal rasterized line should be 1 pixel wide

• Choosing best y for each x (or visa versa) produces aliased raster lines

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Antialiasing by Area Averaging

• Color multiple pixels for each x depending on coverage by ideal line

original antialiased

magnified

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Polygon Aliasing

• Aliasing problems can be serious for polygons

- Jaggedness of edges

- Small polygons neglected - Need compositing so color of one polygon does not

totally determine color of pixel

All three polygons should contribute to color

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Anti-aliasing results:

sharp lines and triangles

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What is Volume Rendering?

• The term volume rendering is used to describe techniques which allow the visualization of

three-dimensional data. Volume rendering is a technique for visualizing sampled functions of three spatial dimensions by computing 2-D

projections of a colored semitransparent volume.

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• There are many example images to be found which illustrate the capabilities of ray casting.

These images were produced using IBM's Data Explorer: (left) Liver, (right) Vessels

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Volume Rendering: result images

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Ming’s brain vessels, MRI

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How to calculate surface normal for scalar field?

• gradient vector

• D(i, j, k) is the density at voxel(i, j, k) in slice k

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Marching cubes (squares)

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Surface normal calculation for cube corners

• Gx (i, j, k) = ( D(i+1, j, k) – D(i-1, j, k))/2

• Gy (I, j, k) = (D (I, j+1, k) –D(I, j-1,k))/2

• Gz(I, j, k) = ((D(I, j, k+1) –D (I, j, k-1))/2

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Marching Cubes Algorithm

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Ray casting for volume rendering

• Theory

– Currently, most volume rendering that uses ray casting is based on the Blinn/Kajiya model. In this model we have a volume which has a density

D(x,y,z), penetrated by a ray R.

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• Rays are cast from the eye to the voxel, and

the values of C(X) and (X) are "combined" into single values to provide a final pixel intensity.

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Transparency formula

• For a single voxel along a ray, the standard transparency formula is:

where:

– C_out is the outgoing intensity/color for voxel X along the ray – C_inis the incoming intensity for the voxel

• Splatting for transparent objects: back to front rendering

– Eye  Destination Voxel  Source Voxel – C_d’= (1-α_s) C_d +α_s C_s

α_d’= (1-α_s) α_d + α_s

C_s : Color of source (background object color)

α_s:Opaque index (opaque = 1.0, transparency = 0.0)

when background α_s= 1.0, destination α_d= 0.0, C_{d’ =}C_s,α_d’ =α_s,

similarly, when foreground (destination) is NOT transparent, α_d= 1.0, C_{d’ =}C_d (color of itself)

 

 _i     _i _i

in

out C x c x x

C  1  

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More details on transparency

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Texture mapping

1. What is texture?

2. How to map a texture to an object surface?

<- : direction of mapping

pixel value = sum of weighted texels within the four corners mapped from a pixel

3. See pictures

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3D Modeling Methods

• Creation of 3D objects

– Revolving – 3D polygon

– 3D mesh, 3D curves

– Extrusion from 2D primitives (set elevation in Z-axis) An example (new CS building construction)

(step bye step demo) of AutoCAD – Features that are useful

• VPOINT, LIMITS, LINE, BREAK, Elevation, SNAP, GRID, etc

– 3D digitizers

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Driverless Car (LiDar etc.)

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Curves and surfaces

• Used in airplanes, cars, boats

• Patch (補片)

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How to model a teapot?

• How to get all the triangles for a teapot?

• What kind of curved surfaces?

• How to display (scan convert) these surfaces?

• What's the surface normal?

• Discuss ways to "define" a curved surfaces.

• Can we show an implicit surface equation easily?

e.g. f(x,y,z) = ax² + by² + cz² + 2dxy + 2eyz + 2fxz + 2gx + 2hy + 2jz + k = 0

• Given (x,y), find z value

• Double roots, no real roots?

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Curves and Surfaces

(Textbook Chap 11)

• Topics

– Polygon meshes

– Parametric cubic curves

– Parametric bicubic surfaces – Quadric surfaces

Parametric cubic curves x(t) = a_xt³ + b_xt² + c_xt + d_x y(t) = a_yt³ + b_yt² + c_yt + d_y z(t) = a_zt³ + b_zt² + c_zt + d_z

• Continuity conditions

– Geometric continuity (G₀): join together – Parametric continuity (C¹) (see below) – Cⁿcontinuity: dⁿ / dtⁿ[Q(t)]continuous

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(Curves) Geometry and drag equation

• In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is:

• This equation is attributed to Lord Rayleigh, who originally used L² in place of A (with L being some linear dimension).^[2]

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Form Drag

• Form drag or pressure drag arises because of

the shape of the object. The general size and shape of the body are the most important factors in form drag;

bodies with a larger presented cross-section will have a higher drag than thinner bodies; sleek ("streamlined") objects have lower form drag. Form drag follows

the drag equation, meaning that it increases with velocity, and thus becomes more important for high- speed aircraft.

• Form drag depends on the longitudinal section of the body. A prudent choice of body profile is essential for a low drag coefficient. Streamlines should be continuous, and separation of the boundary layer with its

attendant vortices should be avoided.

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B'ezier Curve

Q'(0) = 3 (p1-p0) Q'(1) = 3 (p3-p2)

Why choosing "3" ?

Q(t) = (1-t)³p0 + 3t(1-t)²p1 + 3t²(1- t)p2 + t³p3 ...e.q.11.29

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Bezier curve(2)

in matrix form T*M_B*G_B

Note: Q'(0) = -3(1-t)²p0 + 3(1-t)²p1|_t=0 =3(p1-p0) if p1 - p4 is equally spaced, the curve Q(t) has

constant velocity! (that's why to choose 3)

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Subdividing B'ezier curves

Advantage of B'ezier curves

1. explicit control of tangent vectors -->interactive design

2. easy subdivision

-->decompose into flat (line) segments

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Subdividing B'ezier curves(2)

• new control points: pa, pb, pc, pe, pf,

• in addition to p1, p2, p3, p4

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Splitting a Cubic Bezier

p₀, p₁, p₂, p₃determine a cubic Bezier polynomial and its convex hull

Consider left half l(u) and right half r(u)

Angel and Shreiner: Interactive Computer Graphics 7E © Addison-

Wesley 2015

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l(u) and r(u)

Since l(u) and r(u) are Bezier curves, we should be able to find two sets of control points {l₀, l₁, l₂, l₃} and {r₀, r₁, r₂, r₃} that determine them

Wesley 2015

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Efficient Form

l₀= p₀ r₃= p₃

l₁= ½ (p₀+ p₁) r₁= ½ (p₂+ p₃)

l₂= ½ (l₁+ ½ ( p₁+ p₂)) r₁= ½ (r₂+ ½ ( p₁+ p₂)) l₃= r₀= ½ (l₂+ r₁)

Requires only shifts and adds!

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Convex Hulls

{l₀, l₁, l₂, l₃} and {r₀, r₁, r₂, r₃}each have a convex hull that that is closer to p(u) than the convex hull of {p₀, p₁, p₂, p₃} This is known as the variation diminishing property.

The polyline from l₀to l₃ (= r₀) to r₃is an approximation

to p(u). Repeating recursively we get better approximations.

Wesley 2015

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Every Curve is a Bezier Curve

• We can render a given polynomial using the recursive method if we find control points for its representation as a Bezier curve

• Suppose that p(u) is given as an interpolating curve with control points q

• There exist Bezier control points p such that

• Equating and solving, we find p=M_B^-1M_A

p(u)=u^TM_Aq

p(u)=u^TM_Bp

Wesley 2015

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Bezier

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Pierre Etienne B’ezier Introduction

• Pierre Etienne Bezier was born on September 1, 1910 in Paris. Son and grandson of engineers, he chose this profession too and enrolled to study mechanical engineering at the Ecole des Arts et Metiers and received his degree in 1930. In the same year he entered the Ecole Superieure

d'Electricite and earnt a second degree in

electrical engineering in 1931. In 1977, 46 years later, he received his DSc degree in mathematics from the University of Paris.

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In 1933, aged 23, Bezier entered Renault and worked for this company for 42 years

• Bezier's academic career began in 1968 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers. He held this position until 1979. He wrote four books,

numerous papers and received several distinctions including the "Steven Anson Coons" of the

Association for Computing Machinery and the

"Doctor Honoris Causa" of the Technical University Berlin. He is an honorary member of the American Society of Mechanical Engineers and of the Societe Belge des Mecaniciens, ex-president of the Societe

des Ingenieurs et Scientifiques de France, Societe des Ingenieurs Arts et Metiers, and he was one of the first Advisory Editors of "Computer-Aided Design".

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Parametric bicubic surfaces

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Parametric bicubic surfaces

• First consider parametric cubic curve Q(t) = T*M*G

∴Q(s) = S*M*G

• To add the second dimension, G becomes G(t) G_i(t) = T*M*G_i, where G_i = [g_i1, g_i2, g_i3, g_i4]^T

∴Parametric bicubic surfaces => S*M*G*M^T*T^T where S = [1, s, s², s³]

T = [1, t, t², t³]

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Parametric bicubic surfaces (cont.)

• Therefore

– X(s, t) = S*M*G_x*M^T*T^T – Y(s, t) = S*M*G_y*M^T*T^T – Z(s, t) = S*M*G_z*M^T*T^T

• Normals to surfaces

How to calculate?

B'ezier surfaces

– X(s, t) = S*M_B*G_Bx*M_B^T*T^T – Y(s, t) = S*M_B*G_By*M_B^T*T^T – Z(s, t) = S*M_B*G_Bz*M_B^T*T^T

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A typical Bezier Patch: with control points

ctrl_pts = [

[[0, 0, 20], [60, 0, -35], [90, 0, 60], [200, 0, 5]], [[0, 50, 30], [100, 60, -25], [120, 50, 120], [200, 50, 5]], [[0, 100, 0], [60, 120, 35], [90, 100, 60], [200, 100, 45]], [[0, 150, 0], [60, 150, -35], [90, 180, 60], [200, 150, 45]] ];

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B'ezier patches display

• How to display B'ezier patches efficiently?

– Brute force iterative evaluation is very expensive Why? elaborate

– Subdivide into smaller polygons

need flatness test to stop subdivision – Adaptive subdivision is more practical

• How to avoid it?

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Splines (

^{平滑曲線,樣條}

)

• A B-spline is a

generalization of the Bézier curve. Let a vector known as the knot vector be defined

where T is a nondecreasing sequence with ti  [0, 1]

and define control points P₀, ..., P_n . Define the degree as

The "knots" t_p+1, ..., t_m-p-1 are called internal knots.

^t ^t ^t_m

T  ₀, ₁,...,

.

1



 m n p

(1)

(2)

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Splines

• Define the basis functions as

• Then the curve defined by is a B-spline.

 

    ^N  ^t

t t

t t t

t N t

t t t

N

otherwise

t t and t

t t t if

N

p i i

p i

p i p

i i p i

i p

i

i i i

i i

1 , 1 1

1 1 1

, ,

1 1

0

, 0

1











 



 



   

 　

    



 ⁿ

i

p i

iN t

P t

C

0

,

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Every Curve is a Bezier Curve

• We can render a given polynomial using the recursive method if we find control points for its representation as a Bezier curve

• Suppose that p(u) is given as an interpolating curve with control points q

• There exist Bezier control points p such that

• Equating and solving, we find p=M_B^-1M_I

q

p(u)=u^TM_Iq

p(u)=u^TM_Bp

Wesley 2015

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Every segment of a curve (B-spline, etc.) can be defined as a Bezier curve: II

P(u) = u^T M_s p (p is the control point vector of B-Spline)

P(u) = u^T M_b q (q is the control point vector of Bezier curve)

Therefore q = M_b ^-1 M_s p (so the conversion is done)

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Comparison of Bezier and B-spline curves

Bezier B-spline

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About the difference between Bezier and cubic B-spline curve

1. Bezier curve will pass through the two end control points, while B spline does not.

2. Each B spline curve has a small segment defined, almost between the second and the third control points, and of course, will usually NOT pass through them.

3. Because of the limitation of 1 above, in order to have the specific starting point and end points, we let the first 4 control points the same.

Similarly, the last 4 control points are the same.

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About the difference between Bezier and cubic B-spline curve II

4. The Bezier curve is C¹ continuous, since the limitation of 1 above.

5. B-spline is designed to be C² continuous, since it does not need to pass any control points.

6. However, one small segment of B-spline can be “simulated” by a Bezier curve, given new 4 control points, as proved below.

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B-spline ( basis spline) :

for 4 control points, given B0, B1, B2, and B3 as the weighting function

0 <= U <= 1.0

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B-spline matrix: Bs,

with 4 control points Pi

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Cubic B-Spline Curve

• Cubic B-Spline Curve, C² continuous

• P(u) = u^TM p, where P is control points [p^i-2, p^i-1, pⁱ, pⁱ⁺¹]^T

• At first define it to be C²continuous, set up boundary conditions, and we can get

if u is defined as [1,u, u², u³] in this order, P(u) = u^TM_S p

basis function = Ms^Tu = (1/6) [ (1-u)³, 4-6u²+3u³, 1+3u+3u²-3u³, u³]^T















 



 





1 3 3

1

0 3 6 3

0 3 0

3

0 1 4

1

6 Ms 1

(97)

Curve DEMO

• DEMO 1: Bezier curve

http://math.hws.edu/eck/cs424/notes2013/canvas/bezier.ht ml (connected curve)

http://blogs.sitepointstatic.com/examples/tech/canvas- curves/bezier-curve.html (basic curve)

DEMO 2: B-spline curve demo

https://www.cs.utexas.edu/~teammco/research/bsplines/

DEMO 3: Bezier , Bsline, NURBS DEMO

(note: manually add more control points)

(98)

Bezier Patch demo

• DEMO 3:

Youtube video:

https://www.youtube.com/watch?v=2tLC0oIbKS 0

DEMO 4: Bezier Surfaces (patch) http://www.ibiblio.org/e-

notes/Splines/bezier3d.html Fighter plane demo:

http://www.ibiblio.org/e-

notes/webgl/deflate/yf23.html

(99)

WebGL Interactive models

• http://www.ibiblio.org/e-

notes/webgl/models.htm#spline

• (including animated horse, fox, eagle, shark,

human elbow bones, and more static 3D models, such as bike, etc.)

DEMO 5: more control points for B-spline curve http://nurbscalculator.in/ (need to add more control points)(Q1: make the first three control

points to be at the same position, what happens?)

(100)

Ray tracing: Turner Whitted

• Key to success, from light to eye or from eye to screen?

(101)

Concept: how to win? (shooting ball game below)

(102)

The Rendering Equation: Jim Kajiya, 1986

(渲染方程式)

I(x, x’): light (intensity) from patch x’ to patch x g(x,x’): visibility geometry, o: invisible, 1: visible

e(x, x’): the rate at which light is emitted from patch x’ to x, when x’ is an emitter.

ρ (x, x’, x’’): patch’s reflectivity,

(103)

Definitions

• where

I(x, x’): light (intensity) from patch x’ to patch x g(x,x’): visibility geometry, o: invisible, 1: visible e(x, x’): the rate at which light is emitted from patch x’ to x, when x’ is an emitter.

ρ (x, x’, x’’): patch’s reflectivity, light from x’’ to x’

and the ratio reflected to x

(104)

Another detailed definition of the Rendering Equation

The rendering equation describes the total amount of light emitted from a point x along a particular viewing direction, given a function for incoming light and a BRDF.

(105)

Explanation

1. Conservation of energy.

2. Assuming that L denotes radiance, we have that at each particular position and direction, the outgoing light (Lo) is the sum of the emitted light (Le) and the reflected light.

3. The reflected light itself is the sum from all

directions of the incoming light (Li) multiplied by the surface reflection and cosine of the incident angle.

(106)

The Rendering Equation

(107)

Terminology

Radiance: 輻射, 光芒, light or heat that comes from something.

Illumination, ^{照明, light}

Illuminance: 照度: 單位面積上的光通量,

單位: LUX, 1 LUX = 1 流明/平方米, Lumen (lm) :流明, luminous flux, 1 lm = 1 cd.sr,

1 cd: Candela, 燭光 (一瓦的白燈光, 約等於一燭光),

1 sr: steradian, 球面度, 立體角 (半徑 , 若張開的面積為 r², 則角度為 1立體角, 1 sr)

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(109)

(110)

Ray tracing(1)

(111)

Simple recursive ray tracing

L_i: shadow ray R_i: reflected ray N_i: normal

T_i: transmitted ray whether

1. L₁=R₁+T₁? or

2. f¹(L₁)=f(R₁)+f(T₁)? or 3. Color=f(L₁,R₁,T₁ )

(112)

Shadow in ray tracing

(113)

Faster: ray tracing

• halfway vector

(114)

From known data to unknown

(115)

Ray Tracing Algorithm

Trace(ray)

For each object in scene Intersect(ray, object) If no intersections

return BackgroundColor For each light

For the object in scene

Intersect(ShadowRay, object) Accumulate local illumination Trace(ReflectionRay)

Trace(TransmissionRay)

Accumulate global illumination

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Ray Tracing Algorithm

(117)

Code example: A simple ray tracer

• Author: Turner Whitted

– famous for his implementation of recursive ray tracer.

• Simplified version:

– input: quadric surfaces only

i.e. f(x,y,z)=ax² + by² + cz² + 2dxy + 2eyz + 2fxz + 2gx + 2hy + 2jz + k = 0

– Shading calculation: as simple as possible

• Surface normal

[df/dx, df/dy, df/dz]

= [ 2ax+2dy+2fz+2g, 2by+2dx+2ez+2h, 2cz+2ey+2fx+2j ]

(118)

Sample program

Color trace_ray( Ray original_ray ) {

Color point_color, reflect_color, refract_color Object obj

obj = get_first_intersection( original_ray ) point_color = get_point_color( obj )

if ( object is reflective )

reflect_color = trace_ray( get_reflected_ray( original_ray, obj ) ) if ( object is refractive )

refract_color = trace_ray( get_refracted_ray( original_ray, obj ) ) return ( combine_colors( point_color, reflect_color, refract_color )) }

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Code example: A simple ray tracer

• The simple ray tracer is complete and free to copy [need modification to be term project]

• Input surface properties

– r, g, b, relative_index_of_refraction, reflection_coef, transmission_coef, object_type

– number_of_objects, number_of_surfaces, number_of_properties

• How to calculate the intersection of a ray and a quadric surface?

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Ray to quadric surface intersection

• intersection calculation:

– let direction=(D_x, D_y, D_z), origin=(O_x, O_y, O_z) line ==> (x,y,z)=(O_x, O_y, O_z) + t*(D_x, D_y, D_z) – quadric surface

f(x,y,z) = ax² + by² + cz² + 2dxy + 2eyz + 2fxz + 2gx + 2hy + 2jz + k = 0

– replace (x,y,z) in (2) by (1),

acoef*t² + bcoef*t + ccoef = 0, solve for t

• for example:

acoef = a*D_x² + b*D_x*D_y + c*D_x*D_z + e*D_y² + f*D_y*D_z + h*D_z²

(1)

(2)

(121)

Special notice:

1. Avoid to intersect a surface twice within a tiny triange

– e.g. t1=100, t2=100.001

– This may happen because of numeric percision

2. If a ray doesn't hit anything, give it a non-offensive background color, (20,92,192).

– This is the sky color(assume it is day time, of course).

– Otherwise, choose twilight or dark sky color.

3. How to modify this program to accept triangles?

Grid methods?

– Each grid center contains a pointer to the list of triangles which are(partly) contained in the grid.

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Special notice:

4. Shading model

5. ^I ^I



^N ^L



^KS ^S ^KT ^T n

j

a      





1

(123)

• Ultimately, this yields the following pseudo code:

• For more info, please see my document Ray_Tracing.bw

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Ray-object intersection acceleration

(125)

More about Ray-Tracing: Photon Mapping

• Photon mapping is a two-pass global

illumination rendering algorithm developed by Henrik Wann Jensen between 1995 and 2001^[1] that approximately solves the rendering equation for integrating light radiance at a given point in space. Rays from the light source (like photons) and rays from the camera are traced independently until some termination criterion is met, then they are connected in a second step to

produce a radiance value.

• Progressive photon mapping (PPM) starts with ray tracing and then adds more and more photon mapping passes to provide a progressively more accurate render.

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Photon Mapping: result

Global illumination using photon maps, EuroRendering 1996, by Henrik Wann Jensen

•

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More about Ray-Tracing: Photon Mapping

Photon: A Modular, Research-Oriented Rendering System, Siggraph 2019 poster

Figure 1: Several visualization techniques implemented using the building blocks provided by our system. In the center, the buddha features Belcour’s BSDF model [Belcour 2018], while the left one, the glass brick induces caustics that can be rendered efficiently using particle tracing methods.

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Radiosity (熱輻射法)

Donald Greenberg and Tomoyuki Nishita (1985)

What is Radiosity in Computer Graphics?

(DEMO)

(129)

Turner Whitted (

upper middle

, ray tracing), Donald Greenberg(

lower right,

radiosity)

129

(130)

Tomoyuki Nishita,

Rendering, 2011/11

Steven Coons Award 2005, major Asian CG researcher

130

(131)

What is still missing in ray-traced images?

• Diffuse to diffuse reflection?

(132)

(133)

The Rendering Equation: Jim Kajiya, 1986

(渲染方程式)

I(x, x’): light (intensity) from patch x’ to patch x g(x,x’): visibility geometry, o: invisible, 1: visible

e(x, x’): the rate at which light is emitted from patch x’ to x, when x’ is an emitter.

ρ (x, x’, x’’): patch’s reflectivity,

(134)

134

Wesley 2012

Rendering Equation: Another version

Consider light at a point p arriving from p’

i(p, p’) = υ(p, p’)(ε(p,p’)+ ∫ ρ(p, p’, p’’)i(p’, p’’)dp’’

occlusion = 0 or 1/d² emission from p’ to p

light reflected at p’ from all points p’’ towards p

(135)

135

Wesley 2012

Radiosity

• Consider objects to be broken up into flat patches (which may correspond to the

polygons in the model)

• Assume that patches are perfectly diffuse reflectors

• Radiosity = flux = energy/unit area/ unit time leaving patch

(136)

136

Wesley 2012

Patches

(137)

Radiosity

A

^k

(138)

Definitions

• where

Bⁱ: B are the radiosity of patches i and j Eⁱ: the rate at which light is emitted from patch i

Pⁱ: patch i's reflectivity

F^ji:formfactor (configuration factor ) , which specifies the fraction of energy leaving the patch j that arrives at patch i.

Aⁱ, A^j areas of patch i and j.

(139)

Reciprocity in radiosity

^{(互換, 互惠)}

Total energy

passing through Ai and Aj should be the same!

(2)

(3)Simplified Eq from (1)

互惠互惠

A

ⁱ

^A

^j

(140)

140

Wesley 2012

Radiosity Equation

energy balance

b

_i

a

_i

= e

_i

a

_i

+ ρ

_i

∑ f

_ji

b

_j

a

_j

reciprocity

f

_ij

a

_i

= f

_ji

a

_j

radiosity equation

b

_i

= e

_i

+ ρ

_i

∑ f

_ij

b

_j