CS 450: COMPUTER GRAPHICS

BASIC ILLUMINATIONSPRING 2015

DR. MICHAEL J. REALE

INTRODUCTION

• Let’s say we have a triangle on the screen

• We’ve already projected it and let’s assume it’s entirely within the clipping window

• The rasterizer needs to fill in each of the fragments/pixels inside of this triangle

• If we need to fill in one color straightforward, but not realistic-looking

• How do we make surfaces look realistic?

VERTEX ATTRIBUTES PIXEL/FRAGMENT VALUES

• Most of the time, a model is broken up into a triangle mesh

• For a given triangle, each vertex can be given different values (attributes):

• Color

• Texture coordinates

• Normal vector

• Etc.

• These values are interpolated across the triangle to get the individual pixel/fragment values

BARYCENTRIC COORDINATES

BARYCENTRIC COORDINATES

• Barycentric coordinates – parameterizes the space that can be formed as a weighted combination of a set of reference points

• In other words any location is a weighted average of the reference points

• Weights all sum to 1

• For 2 reference points:

• (u,v) = barycentric coordinates of P with respect to A and B

• P is on segment AB IFF 0 <= u <= 1 AND 0 <= v <= 1

• Barycentric coordinates of A = (1,0)

• Barycentric coordinates of B = (0,1)

1

vu

vBuAP

BARYCENTRIC COORDINATES:TRIANGLES• For 3 reference points (triangles):

• (u,v,w) = barycentric coordinates of P with respect to A, B, and C

• A (1,0,0)

• B (0,1,0)

• C (0,0,1)

• Inside or on triangle IFF 0 <= u,v,w <= 1

• Centroid or barycenter = (1/3, 1/3, 1/3)

http://en.wikipedia.org/wiki/Barycentric_coordinate_system#/media/File:TriangleBarycentricCoordinates.svg

1

wvu

wCvBuAP

BARYCENTRIC COORDINATES: WHY DO WE CARE?

• Given a point P in a triangle, the barycentric coordinates give us a weight for each vertex in the triangle can use to interpolate vertex attribute values!

• Example: if each vertex has a different color (C0 , C1 , C2 , respectively), and the barycentric coordinates for P are (u,v,w), the interpolated color for P is given by:

210 wCvCuCC p

BARYCENTRIC COORDINATES AS A PLANE

• One can rearrange the previous barycentric equations to form a point and two axes (effectively defining a plane):

• Note: We don’t really need u

1

wvu

wCvBuAP

)()(

)1(

11

ACwABvAP

wCvBwAvAAP

wCvBAwvP

wCvBuAP

wvuwvu

SOLVING FOR BARYCENTRIC COORDINATES• Let’s say we have a point P and we want to find the barycentric coordinates for it

• Let the following be true:

• Then we can rewrite our previous equation:

APVACVABV 210

210

10

10

)()(

)()(

)()(

)()(

VVwVv

APVwVv

VwVvAP

ACwABvAP

SOLVING FOR BARYCENTRIC COORDINATES

• We can turn this into a 2 x 2 system of linear equations by taking the dot product of both sides first with V0 and then with V1:

210 )()( VVwVv

121110121110

020100020100

)()()()(

)()()()(

VVVVwVVvVVVVwVVv

VVVVwVVvVVVVwVVv

SOLVING FOR BARYCENTRIC COORDINATES

• Fortunately, this can be solved using Cramer’s Rule!

wvu

DDw

DDv

dd

dd

VVVV

VVVVD

dd

dd

VVVV

VVVVD

d

d

VV

VVE

dd

dd

VVVV

VVVVD

w

v

wv

1

/

/

)()(

)()(

)()(

)()()(

)(

)()(

)()(

2101

2000

1210

0200

1121

0120

1112

0102

21

20

12

02

1101

0100

1110

0100

121110

020100

)()(

)()(

VVVVwVVv

VVVVwVVv

SOLVING FOR BARYCENTRIC COORDINATES

• The following code is taken from “Real-Time Collision Detection” by Christer Ericson (pg. 47-48)

void Barycentric(Point a, Point b, Point c, Point p, float &u, float &v, float &w) {Vector v0 = b – a, v1 = c – a, v2 = p – a;float d00 = Dot(v0,v0);float d01 = Dot(v0,v1);float d11 = Dot(v1,v1);float d20 = Dot(v2,v0);float d21 = Dot(v2,v1);float denom = d00 * d11 – d01 * d01;v = (d11 * d20 – d01 * d21) / denom;w = (d00 * d21 – d01 * d20) / denom;u = 1.0f – v – w;

}

LIGHT SOURCES

LIGHTING MODEL AND LIGHTS

• Lighting model (or shading model) = used to calculate the color of an illuminated position on the surface of an object

• Light source (or “light”) = emits light (radiant energy)

TYPES OF LIGHT SOURCES

• There are three basic types of light sources:

• Point lights

• Directional lights

• “Spotlights”

POINT LIGHTS

• Point light

• Located at a single point in space

• Emit light in all directions

DIRECTIONAL LIGHT (“SUN”)

• Directional Light

• Emits light in one direction

• Position of light doesn’t matter only the direction matters

• Simulates a light “infinitely” far away (e.g., the sun)

“SPOTLIGHTS”

• Spotlight

• Has a position, direction, and angle θ defining how wide the cone of light is

• If Vlight = direction of the light and Vobj = direction to object, then:

cos lightobj VV

conelight inside Otherwise,

conelight outsidecos lightobj VV

RADIAL INTENSITY ATTENUATION

• Let’s say we have a point light source

• The farther an object is from the light the less light reaches the object (less intense)

• Officially: attenuation factor = 1/d2

• In practice, however, this doesn’t look realistic:

• Objects close to light too much intensity variation

• Objects far from light very little variation

• Reason why it doesn’t look right: lights aren’t REALLY infinitesimal points only an approximation

• Alternative: use inverse quadratic function with tunable parameters:

2210

1)(

dadaadAradial

ATTENUATION AND DIRECTIONAL LIGHTS

• Attenuation doesn’t make sense with directional lights since light is “infinitely” far away, all points in scene equidistant from light

ANGULAR INTENSITY ATTENUATION

• For spotlights, as angle between object and light direction gets larger cosine gets smaller

• Can use as part of attenuation factor

otherwiseVV

VVA a

lightobj

lightobjangular )(

coscos if0

BASIC LIGHTING MODEL

INTRODUCTION

• When light hits an opaque surface, part of it reflected and part of it absorbed

• For transparent surfaces, light is also transmitted through the surface

• We’ll start by assuming a monochrome world (i.e., only black-and-white-and-shades-of-gray)

• So, we’ll refer to the intensity of the light as a single value (we’ll deal with RGB color later)

• For simplicity, we often determine the shading of an object based on three factors:

• Diffuse reflection

• Ambient light

• Specular reflection

DIFFUSE REFLECTION

• Diffuse reflection = when white light hits an object, what we see as the “color” of an object

• Example: apple absorbs all frequencies except red has red diffuse color

• Underlying physics: surfaces with microfacets (bumpy, grainy, matte) reflects light in lots of different directions

• Ideal diffuse reflectors or Lambertian reflectors

• Incident light scattered with equal intensity in ALL directions, INDEPENDENT of viewing angle

• Depends on angle between NORMAL and DIRECTION-to-LIGHT angle of incidence

DIFFUSE REFLECTION: SIMPLE VERSION

• kd = diffuse intensity (“color”) of surface

• I = light intensity

• N = normal

• L = vector to light from point

• The larger the angle between N and L the less light hits the surface (and gets reflected off)

• Use dot product of N and L (as long as it’s greater than 0)

0)(if 0

0)( if )(cos

LN

LNLNIkIkDiffuse d

d

DIFFUSE REFLECTION: DIFFERENT LIGHT SOURCES

• Point lights L = vector from current position to light position

• Directional light L = direction vector for light

AMBIENT LIGHT

• Ambient light = sets the overall brightness of the scene

• Applied to ALL objects equally

• Approximating global illumination (in real world, light bouncing around different objects)

• To compute ambient light contribution to point on surface:

• Sometimes graphics packages will use a separate ambient coefficient for the surface:

intensitylight ambient aI

ad IkAmbient

aa IkAmbient

DIFFUSE + AMBIENT LIGHT

• Note: all these factors vary from 0 to 1

0)( if

0)( if )(

LNIk

LNLNIkIkAmbientDiffuse

aa

daa

SPECULAR REFLECTION

• Specular reflection = when all (or almost all) of the incident light is reflected back

• A shiny or reflective spot

• Depends on:

• Normal N

• Light vector L

• View vector V = vector from point to camera

• R = ideal reflection angle

• When R is in line with V most reflective

SPECULAR REFLECTION: DULL AND SHINY

• Different surfaces reflect light over finite range of viewing positions

• Shiny surfaces narrow range

• Duller surfaces wider range

SPECULAR RELFECTION:PHONG SPECULAR-REFLECTION MODEL

• Phong specular-reflection model

• Angle φ = angle between R and V

• ns = specular-reflection exponent

• Sometimes called “shininess”

• If we use a constant coefficent for specular reflection, then:

sns IkSpecular cos

SPECULAR REFLECTION: GETTING R

• L projected onto N is the same as R projected onto N; therefore:

• Our specular component can then be computed by:

LNLNR

NLNLR

)2(

)2(

0)(or 0)( if 0

0)( and 0)( if )(

LNRV

LNRVRVIkSpecular

sns

SPECULAR REFLECTION: A SIMPLER WAY• A simplified way to calculate specular reflection is to look at the

halfway vector H between L and V:

• Now, we just look at the angle between N and H (or rather the dot product between them):

• Problem: if V, L, and N are not coplanar slightly off

• Another problem: need to check if V and L are on same side of N (L · V > L · N) if so, don’t use specular effect

VL

VLH

sns HNIkSpecular )(

GETTING THE VIEW VECTOR

• To get the vector from the current point to the viewer, you need:

• 3D WORLD position P of the point on the polygon

• PROBLEM: In vertex/pixel shader, have NORMALIZED DEVICE COORDINATES

• SOLUTION: Also pass in INVERSE Model-View-Projection transform to UNDO transformation

• P = invMVP*position

• 3D position of camera E

• To get the vector to the camera:

PE

PEV

)(

DIFFUSE + AMBIENT + SPECULAR

• So, for a single point light source, our combined diffuse, ambient, and specular reflections from a position on an illuminated surface are given by:

• NOTE: If light is behind surface, only use ambient light

• ALSO NOTE: If V and L are on the same side of N, no specular effects

snsdaa HNIkLNIkIk

SpecularDiffuseAmbient

)()(

MULTIPLE LIGHT SOURCES

• To deal with multiple light sources, just sum all values up for diffuse and specular components (ambient light is only added once, however):

n

i

nsdiaa

sHNkLNkIIkFinal1

)()(

SURFACE LIGHT EMISSIONS

• In addition to lights, one can also have surfaces that emit light

• Examples: fluorescent light bulbs, neon signs, anything from Tron,…

• Have addition term ke that is handled like an ambient term (except that it is specific to the surface)

• TECHNICALLY, light coming from the surface should affect other objects in the world:

• Use directional light with cone/box around it to determine area of effect

• Use lots of point lights

• Ignore it to save time (more common)

• Radiosity model (covered later)

http://thedisneyblog.com/wp-content/uploads/2012/06/tron-uprising-art.jpg

DIFFUSE + AMBIENT + SPECULAR + SURFACE EMISSION + ATTENUATION

• The general, monochromatic illumination model for surface reflection that includes:

• Multiple light sources

• Attenuation factors

• Point lights, directional lights, spotlights, and surface emissions

• …is given by:

n

ispeculardiffuseangularradialaae IIAAIkkFinal

1

)(

0)(if 0

0)( if )(

LN

LNLNIkI ddiffuse

otherwise 0

0.0)( if }))(,0(max{ LNHNIkI

sns

specular

2210

1

dadaaAradial

otherwiseVV

VVA a

lightobj

lightobjangular )(

coscos if0

THE ILLUMINATION FROM THIS PIXEL IS TOO HIGH!

• The final output intensity has to be between 0 and 1 how to do we keep it in that range when we have lots of lights, etc.?

• Set a max value for each term in equation

• If exceed max value, set to max

• Normalize individual terms by dividing each by magnitude of largest term

• More complicated: get ALL pixel intensities for scene, and then scale set of intensities to range [0,1]

n

ispeculardiffuseangularradialaae IIAAIkkFinal

1

)(

RGB COLOR

• For color lights and surfaces, our intensity values now become vectors:

• Examples:

• Our equations now become vector equations

aBaGaRa

dBdGdRd

IIII

kkkk

,,

,,

COLOR REFLECTIONS?

• In the original specular-reflection model, ks was set to a constant value INDEPENDENT of the surface color usually same color as light

• Gives surface a “plastic” appearance

• For nonplastic surfaces specular color may be different from both light color and diffuse color of object!

POLYGON SURFACE RENDERING

INTRODUCTION

• Given a surface’s color information (diffuse, specular, etc.) and the light color information, how do we actually render the polygon?

• There are three major approaches for this, and the difference between them is at what level we compute the lighting equation:

• Flat shading per primitive (triangle)

• Gouraud shading per vertex

• Phong shading per pixel/fragment

FLAT SHADING

• Flat shading = compute shading per primitive (triangle)

• All pixels/fragments have the same color

• Basically use the same normal for the whole triangle

• Advantages:

• Fastest

• Accurate if surface that the polygon is part of is NOT a curved surface

• Disadvantages:

• Curved surfaces look terrible transitions between polygons is sudden

•

GOURAUD SHADING

• Gouraud shading = compute shading per-vertex

• Interpolate COLOR values to fill in pixels

• Use per-vertex normal

• Pronounced “guh-row”

GOURAUD SHADING: PER-VERTEX NORMAL?• Usually, we think of the POLYGON as having a normal

• For a CURVED surface:

• To get the per-vertex normal, we get the average of all normals from the polygons the vertex is a part of:

• For a NON-CURVED surface:

• Sometimes we actually don’t want the surface to smoothly transition from one polygon to another

• To address this, duplicate the vertex so that you get two different average normals:

n

kk

n

kk

v

N

NN

1

1

Curved surface Duplicate vertex to create crease

GOURAUD SHADING: PROS AND CONS

• Advantages:

• Looks better than flat shading for curved surfaces

• Fairly efficient

• Disadvantages:

• Depending on vertex tessellation (i.e., how many vertices/polygons), specular highlights don’t look right

• Kind of smearing the highlight across the polygon

Fewer polygons More polygons

PHONG SHADING

• Phong shading = compute shading per-pixel/fragment

• Interpolate NORMALS across surface (from vertex normals)

• Advantages:

• Highest quality rendering

• Disadvantages:

• More computationally intensive (have to compute shading formula at each pixel)

• Overkill for non-curved surfaces (and we will have to duplicate vertices to get the creases we want)

FLAT VS. GOURAUD VS. PHONG

Flat shading- Fastest- Lowest quality- Bad for curved surfaces

Gouraud shading- Not as fast- Better quality- Specular highlights may not look right

Phong shading- Slowest - Highest quality

OPENGL?

• In legacy, handling lights, what kind of shading you were using, etc. was a LOT more complicated

• In modern OpenGL, you pretty much handle this yourself with the vertex/pixel shaders:

• Gouraud shading:

• Compute per-vertex color

• Pass to fragment shader to be interpolated

• Use interpolated color directly for fragment

• Phong shading:

• Pass per-vertex NORMAL to fragment shader to be interpolated

• Compute fragment color in fragment shader

• For flat shading, pass in same normal for all vertices that are part of the same triangle