CS 376 Introduction to Computer Graphics 03 / 26 / 2007 Instructor: Michael Eckmann.

CS 376Introduction to Computer Graphics

03 / 26 / 2007

Instructor: Michael Eckmann

Michael Eckmann - Skidmore College - CS 376 - Spring 2007

Today’s Topics• Questions?

• Visible Surface determination– Scanline algorithm– Raycasting

• Illumination modeling


Scanline Algorithm• The Scanline algorithm for visible surface determination is an image

precision algorithm. It is an extension to the scan conversion algorithm

we discussed for filling a polygon. But now, we need to deal with a set

of polygons, not just one.

• The Scanline algorithm processes one scanline of the frame buffer at a

time.

• One assumption we are making (which can be overcome with

modifications to the algorithm) is that no polygons penetrate each other.


Scanline Algorithm• 1) Create an edge table for all edges that are not horizontal (ignore those) that

are projected onto the view plane.

• 2) Given the x,y,z coordinates of the 2 endpoints of the edge,– Sort the edges in the edge table according to the smaller y (the y of the

lower endpoint) of its two endpoints and then within those with the same smaller y, sort by increasing x of the lower endpoint.

• 3) Each entry in the edge table contains the following information about each

edge– a) x coordinate of the endpoint with smaller y– b) y coordinate of the other endpoint– c) x increment – the value to increment x when moving from one scanline to

the next– d) polygon ID

• 4) A polygon table, keyed on polygon ID contains coefficients of the plane that

polygon is on, the color info for it, and a boolean in-out flag used during the

algorithm.


Scanline Algorithm• 5) An Active Edge Table is also stored during the running of the algorithm. For

the current scanline, the active edge table contains the edges, in order by

increasing x, whose projections onto the view plane intersect with the current

scanline.

• Pseudocode for this algorithm:– Initialize frame buffer to background color– Add edges to edge table– Add polygons to polygon table– For (each scanline)

• Update active edge table • Starting at the edge with the lowest x• For (each pixel in scanline)

– Determine if went in or out of any polygons (set appropriate in-out flags)

– Test in-out flags, if “in” only one polygon, draw that color– If not “in” any polygons do nothing– If “in” multiple polygons then determine which polygon is closest

(make depth calculations based on plane coefficients) and draw that color


Scanline Algorithm• Modifications can be made for efficiency by exploiting different coherence

properties.

• To reduce depth computations -– Within a scanline, the same polygon influences the color of the pixels

between two edges, regardless of how many polygons we are “in”, so depth computations are only necessary when entering a polygon or leaving a polygon.

– Assuming we have two or more overlapping (in z) polygons and as we scan across the scan line we leave (go past right edge of) one of the polygons. If the polygon we just left was just obscured then we don't need to compute the depth among the other polygons, we can just draw the color of the polygon that was closest (because it must still be closest). Only when leaving a closest polygon are depth calculations necessary.

• Example on the board.– Also if the same edges are active and in the same order from one scanline to

the next, no depth computations are necessary because no depth relationship changes have occurred. Just draw the same colors in the same order (but possibly a different number of pixels each.)

• Example on the board.


Scanline Algorithm• This algorithm, although specified as working only for polygons can be

extended to work for arbitrary (curved) surfaces. This is accomplished

(partially) by replacing the edge table and active edge table with a surface table

and an active surface table.


Raycasting method• Raycasting is an image precision visible surface detection

method.

• Consider line of sight from a pixel position on a viewplane

through a scene. We are tracing the paths of light rays.

• Light rays emanate from objects into our eyes. To determine

which object influences the color at a pixel, we trace the light

ray path backwards from the pixel out into the world and

determine which object it passes through first.

• Works well for scenes with curved surfaces, like spheres, etc.

• Ray-tracing is an extension of ray casting that we will discuss

in detail in the weeks to come.


Looking ahead• The topics that I plan to cover next are mostly in chapter 10:

– Illumination models and Shading models– Texture mapping, bump mapping – to add details– Ray tracing and radiosity


Illumination• Until now we have been dealing with solid colored polygons.

Solid colors make objects look flat --- do not give a 3d effect.

• There are many schemes to shade and to simulate light and

shadows in a scene.

• Depth cueing (discussed at the end of chapter 9) is one way to

achieve some sense of depth / clarity in a scene.– Imagine a polygon that is not perpendicular to the x-y

plane (that is, it spans a range of depths (z) in the scene).– The nearer points of the polygon can be made more

intense than the further points of the polygon. This can be done gradually to get the effect of fading into the distance.

– The intensity can be multiplied by a factor. The factor is a function of the depth of the point.

• Depth cueing is a light source independent model of

illumination.


Illumination• Illumination models are used to calculate the color of a point on

the surface of an object.

• A surface rendering method uses the calculations from the

illumination model to determine the pixel colors of all the

projected points in a scene.– Can be done by doing illumination model calculations at each

pixel position (like you'll see is done in ray tracing).OR– Can be done by doing a small number of illumination

calculations and interpolating between pixels (like is often done in scan line algorithms.)

• Two main areas that influence the realism in computer graphics are– Accurate representations of surface properties (e.g. How shiny

or dull, or textured, etc.)– Good descriptions of lighting (e.g. Color of the light, position,

etc.)


Illumination• Besides the lighting model and the surface properties of an

object, the position of the viewer has an effect on what an

object looks like.

• Light source dependent models of illumination– The color of a point on an object is influenced by light

sources• Some of the properties of light sources include

– Color– Intensity– Position– Direction

– It is also influenced by surface properties of the object like– Color– How shiny or dull it is (how much light is reflected

from it.)– It is also influenced by where the viewer is viewing the

scene.


Illumination• When light hits an object, it is either

– Reflected (light “bounces” off the object, this is what we see)

• Diffuse• Specular

– Absorbed (what light is not seen after hitting the object)– Transmitted (refracted) passes through if the object is

translucent or transparent– Or some combination of these

• Surface properties and lighting models are complex entities.


Illumination• The simplest illumination model (the one we've been using

during the course so far) is one where no lights are modelled,

objects are therefore nonreflective and are self-luminous (they

emit their own colors without light hitting them.) Each

object's surface has a constant single color.

• Illumination models can be expressed with illumination

equations. This particular simple model is expressed with

I = ki

where I is the resulting intensity andk

i is the object's constant intensity

– I in this case does not depend on the position of the point of which we are determining intensity

Illumination• To take illumination one step further from that simple

method, assume – Instead of self luminous objects we have objects that

reflect light and– We have a nondirectional, diffuse light source

(ambient/background light) that equally effects all surfaces in all directions

• It represents the amount of light equally reflected from all surfaces ---- Think daylight/sunlight

• This illumination model can be expressed with illumination

equation:I = I

ak

a

where I is the resulting intensityk

a is the ambient reflection coefficient of the object (a material property of the surface that doesn't correspond directly to a physical property of materials (it's sort of a catch-all))

Ia is the intensity of the ambient light


Illumination• The illumination model with just the ambient light results in

about the same kind of look to the scene as the self luminous

objects --- uniform intensity across surfaces.

• To go further in our illumination model discussion, let's

consider point light sources. A point light source is a light

source whose light rays emanate in all directions from a point,

uniformly.

• The point light source has a position in 3d space. Therefore, a

surface's intensity will vary with the distance from the

position of the point light source.


Illumination• Diffuse reflection reflects light of equal intensity in all

directions. Contrast this with specular reflection which

reflects light in unequal intensities at different directions.

• Shiny surfaces result in specular reflection and dull, matte

surfaces result in diffuse reflection.

• A Lambertian surface is one that reflects light, equally

intense in all directions, independent of the viewing position.

A Lambertian surface reflects light diffusely.

• Examples of Lambertian surfaces are matte photographic

prints (as opposed to glossy prints), chalk, etc.


Illumination• Lambert's cosine law is:

– The intensity of the light reflected from a surface is proportional to the cosine of the angle between the surface normal and the direction to the light source.

• From the diagram on the last slide– If L is a vector from the surface towards the light source

and N is a normal vector to the surface and A is the angle between them, then

cos A = L ● N / |L||N|

– If L and N are of unit length then cos A = L ● N

• Ip is the intensity of a point white light source, k

d is the diffuse

reflection coefficient of the surface

• Our Illumination equation for a lambertian surface with a point light source is: I = I

pk

d L ● N


Illumination• To simulate both ambient light and point light sources on

diffuse reflecting surfaces, we can simply add in the ambient

light term to our previous illumination equation and end up

with:

• I = Iak

a + I

pk

d L ● N

• This creates a more realistic look. Without the ambient light

term objects will look harshly lit.


Illumination• Light is attenuated as it moves through space. The

attenuation factor is 1 / dL

2

– where dL

2 is the square of the distance from the light source

• Light hitting surfaces closer to the light source receive higher

energy than further surfaces.

• Without taking this into account, two equal surfaces in the

world at different distances (that overlap eachother in the

viewplane), will appear to be one surface in the view plane.

Example on the board.


Illumination• But, because actual lights are not point light sources, and

ambient light isn't exactly accurately modelling light in the world we do not use this attenuation factor exactly as 1 / d

L2

• Instead, to get a more realistic look (than if we had used the

inverse square law) we use an inverse quadratic function

• fradatten

(dL) = 1/ (a

0 + a

1d

L + a

2d

L2 ) where the a coefficients can

be adjusted for realism.

• We use this factor as a multiplier of the point light source

term:

• I = Iak

a + f

radatten(d

L) I

pk

d L ● N


Illumination• Specular reflection as stated before is a property of a surface

where light is reflected in unequal intensities at different

directions.

• A perfect mirror reflects light in only one direction. Less

perfectly shiny surfaces reflect light more in one particular

direction than other directions.

• Either way, consider the angle the light and the surface

normal makes. The direction that the most intense light is

reflected is at an angle which is equal to that angle but on the

other side of the surface normal.

• The next slide has a diagram for specular reflection.

Illumination• L is a vector in the direction of the light source. N is the surface normal.

R is the vector which is in the direction of the maximum specular

reflection. V is the vector in the direction toward the viewer.

• The angle between L and N is the same as the angle between N and R.

• The angle phi, between R and V affects how much light is reflected in

V's direction.

• If the surface was a perfect mirror, then light is only reflected along R. If

the surface is shiny, but not perfect, light is reflected mostly in R's

direction but decreases reflection as we go in directions at angles away

from R.

• An illumination model that takes into account non perfect specular

reflections is the Phong model.

Illumination• The Phong model sets the intensity of the specular reflection proportional

to cos ns (phi), where phi is the angle between V and R.

• ns is the specular reflection exponent and is specified based on the type

of surface you are modelling. Very shiny surfaces have ns values of 100

or more and less shiny surfaces can have ns values closer to 1.

• The Phong equation is:

fradatten

(dL) I

pW(theta)cos ns (phi)

where W(theta) is the fraction of specularly reflected light, which is typically set to some constant, otherwise it is a function on theta which is the angle between L and N.

Illumination

Illumination• Our illumination model with the Phong term added in now becomes

I = Iak

a + f

radatten(d

L) I

pk

d L ● N +

fradatten

(dL) I

pW(theta)cos ns (phi)

which can be simplified to:

I = Iak

a + f

radatten(d

L) I

p (k

d L ● N + W(theta)cos ns (phi))

Illumination• I = I

ak

a + f

radatten(d

L) I

p (k

d L ● N + W(theta)cos ns (phi))

Notice the following:– the Phong term is dependent on the point light source and the attenuation factor– our whole illumination model at this point is influenced by ambient light and a point light source. Further, our object has diffuse reflection properties and specular reflection properties. And the phi angle is dependent on where the viewer is.

Illumination• If we have more than one point light source (which is white

light), then we sum up over all the light sources i:

I = Iak

a + Sum

all i [f

radatten(d

L) I

pi (k

d L ● N + W(theta)cos ns (phi))]

Illumination• If the light sources are not white light sources, then we can

seperate out the red, green and blue components. Also, if

objects are to be colored, then they reflect different

wavelengths and absorb others.

• Specify ambient light now as 3 components: IaR

IaG

IaB

• Specify point light now as 3 components: IpR

IpG

IpB

• Specify objects with color: OR

OG

OB

I R = I

aRk

aO

R + Sum

all i [f

radatten(d

L) I

pRi (k

d L ● N + W(theta)cos ns (phi))]

I G and I

B can be expressed similarly.

Illumination• We typically want our Intensities to be in the range 0 to 1. With many

light sources that are specified as too intense, computed intensities of

objects can easily be too large (> 1).

• We can normalize the range of calculated intensities into the range 0 to 1.

• Or we can specify our light source intensities better and if any

calculations go over 1, then clip them to intensity 1.

Directional light source• The light sources that we have discussed so far were point light sources

which radiated light in all directions. Real lights in the world do not

have this property. Instead, real lights typically are directional.

• A directional light source can be described as having – a position– a directional vector – an angular limit from that vector– a color

• The vector will be the axis in a cone shaped volume. This volume is the

only part of space that will be illuminated by this light. Anything outside

the cone will not be illuminated by this light.

• See diagram next slide.

Directional light source• To determine how intense the light is from a directional light source on

an object we determine the angle between

– the vector Vo from the position of the light to the point on the object

we're illuminating and

– the cone axis vector Vl– So, we take the dot product: V

o • V

l = cos alpha, alpha is the angle

between them.

– If Vo is in the same direction as V

l, then the light should be the most

intense (alpha = 0). The light should get less intense as we go in

larger angles away from Vl.

– We can attenuate the light as function of alpha like so: • cosal alpha• where al is the attenuation exponent (a property of the light

source) --- if al is 0, it's a point light source, the smaller al is,the larger the attenuation is (because the cos is less than 1.)

CS 376 Introduction to Computer Graphics 03 / 26 / 2007 Instructor: Michael Eckmann.

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Transcript of CS 376 Introduction to Computer Graphics 03 / 26 / 2007 Instructor: Michael Eckmann.