B. SC CSIT Computer Graphics Unit 4 By Tekendra Nath Yogi

Unit 4:Visible surface detection methods,
Illumination models and surface rendering methods
Presented By : Tekendra Nath Yogi
Tekendranath@gmail.com
College Of Applied Business And Technology

Outline
• Unit 4. 12 Hrs.
– Visible Surface Determination:
• Various Techniques
• Algorithms for Visible Surface Detection, (Z- Buffer, List priority, Scan Line
Algorithms)
– Illumination models.
• Ambient light
• Diffuse reflection
• Specular reflection
• Light intensity attenuation
• Color consideration
• Transparency
• shadows
– Surface Shading Methods/surface Rendering Methods
• Constant intensity shading
• Gouraud
• Phong Shading
2By: Tekendra Nath Yogi2/9/2019

Visible surface determination/ Hidden surface Elimination methods
• For the generation of realistic graphics display. We have to identify those
parts of a scene that are visible from a chosen viewing position.
• To solve this problem different algorithms are used called the visible
surface detection or hidden surface elimination algorithms.
• These algorithms determines the lines, edges, or surfaces that are visible or
invisible to an observer located at a specific point in space.
• So, that we can display only the visible lines or surfaces to produce
realistic display.

Contd….
• Visible surface detection algorithms are broadly classified according to
whether they deal with object definitions directly or with their projected
images. These two approaches are called:
– Object space methods and
– Image space methods.

Contd….
Pseudo code for Image space Method
for(each pixel in the image)
determine the object closest to the viewer that is penetrated by the projector through
the pixel;
draw the pixel in the appropriate color;
Ex: Depth buffer method
Pseudo code for Object space Method
for(each object in the world)
determine those parts of the object whose view is not blocked by other parts of it or
any other object;
draw those parts in the appropriate color;
Ex: Back-face detection method
Image space methods are by far the more common!

Contd….
• Algorithms for Visible surface detection:
– Back face detection (plane equation method)
– z-buffer Algorithm(Depth buffer method)
– List Priority algorithm
– Scan Line Algorithm

Back face detection (plane equation method)
• It is fast and simple object-space method for identifying the back-faces of
polyhedron which is based on the inside –outside test.
• A point (x, y, z) is inside a polygon surface if
Ax+By+Cz+ D <0
• When an inside point is along the line of sight to the surface , the polygon must be a
back face
• We simplify this test by considering the normal vector N to a polygon surface
which has Cartesian component (A,B,C).
• If V is vector in viewing direction from the eye position then this polygon is back
face if
2/9/2019 7
faceback:0NV
VN
By: Tekendra Nath Yogi

2/9/2019 8
• If the object descriptions have been converted to a projection
coordinates and viewing direction is parallel to the viewing Zv axis
then V= (0,0,Vz) and
V.N= Vz . C
• If viewing direction along negative Zv axis the polygon is back
face if C<0
• If viewing direction along positive Zv axis , then polygon is back
surface if C>0
Limitation:
• It cannot be applicable for all case like overlapping case such as two
faces of single object is overlapping
Contd….

Depth Buffer (Z- buffer)Method
• A commonly used image-space approach to detecting visible surfaces . It is
developed by CutMull in 1974.
• This method compares surface depths at each pixel position on the
projection plane
• Each surface of a scene is processed separately, one point at a time across
the surface.
• object depth is measured from the view plane along the z axis of a viewing
system.
• The Z buffer algorithm is usually implemented in the normalized
coordinates , so that Z values range form 0 at the back clipping plane to 1
at the front clipping plane.

Contd….
• Two buffers required
– Depth buffer:
• to store depth value for each (x, y) position as surfaces are
processed
• Initially set to 0, representing the z- value at the back clipping plane
.
– Refresh buffer:
• to store intensity values of each (x, y) position as surfaces
processed.
• Initially refresh buffer is initialized to background intensity

Contd…
• At viewing plane position (x, y), surface S1 has the smallest depth( means have
highest Z value in normalized coordinate.) from the view plane and so is visible
at top position

Contd….
• Algorithm:
1. Initialize the depth buffer and refresh buffer so that for all buffer positions (x,y)
depth(x, y) = 0,
refresh(x, y) = Ibackground
2. For each position on each polygon surface, compare depth values to previously
stored values in depth buffer to determine visibility.
● Calculate the depth z for each (x,y) position on the polygon.
● If z >depth(x,y), then
depth(x, y) = z,
refresh(x, y) = Isurf(x,y).
where, Ibackground is the value for the bacground intensity and Isurf(x,y), is the projected
intensity value for the surface at (x,y).
 After all surfaces have been processed, the depth buffer contains depth values for the visible
surfaces and the refresh buffer contains the corresponding intensity values for those
surfaces.

Contd….

Contd….
• Example: Let us consider two surfaces present in the scene as
shown in fig below

Contd….
• Step 1: Initialize the depth buffer

Contd….
• Step 2: Draw the polygon with depth 0.3

Contd….
• Step 3: Draw the polygon with depth 0.5

Contd….
• Advantages:
– For polygon surfaces, the depth buffer method is very easy
to implement, and it requires no sorting of the surfaces in a
scene.
– It can be implemented in hardware to overcome the speed
problem.
• Disadvantages:
– It requires an additional buffer and hence the large memory.
– It is a time consuming process as it requires comparison of
each pixel instead of for the entire polygon.

List Priority Algorithms
• List-Priority-Algorithms determine a visibility ordering for
objects ensuring that a correct picture results if the objects are
rendered in that order.
• E.g. if no object overlap another in z, then we need to sort the
objects by increasing z and render them in that order.
• The types of LPA algorithms
– Depth Sorting Algorithms
– Binary Space Partitioning Trees

Contd….
• Depth Sorting Algorithms (painter’s Algorithm): Each layer
of paint covers up the previous layer.
– i.e., first paints the background colors
– Next, the most distant objects are added, then the nearer objects, and so forth
– At the final step, the foreground objects are painted on the canvas

Contd….
• Depth sorting algorithm uses the similar technique: paint the
surface intensity into the frame buffer in order of decreasing distance from
the viewpoint.
• Three steps:
1. Sort all surfaces according to the smallest (farthest from view plane) z
coordinate value on each surface.
2. Resolve if any ambiguities( e.g., polygon’s z extents overlap->splitting
polygons if necessary).
3. Scan convert each polygon in ascending order of smallest z
coordinate.(i.e. back to front).

Contd….
• For scan conversion make the following tests for each surface that
overlaps with S.
• If any one of these tests is true, no reordering is necessary for that
surface.
– The bounding rectangles in the xy plane for the two surfaces do not
overlap
– Surface S is completely behind the overlapping surface relative to the
viewing position.
– The overlapping surface is completely in front of S relative to the
viewing position.
– The projections of the two surfaces onto the view plane do not overlap.

Test #1
• The boundary rectangles (minimum closed rectangles for the
surfaces) in the xy-plane for the two surfaces do not overlap.
• Test # 1 is passed, scan convert S2 and then S1. Because, S2
has the highest depth.

Test #2
• Surface S1 is completely behind the overlapping surface S2
relative to the viewing position.
• S1 is scan converted first, then S2 is scan converted.

Test #3
• The overlapping surface S2 is completely in front of S1 relative to the
viewing position.
• S1 is not completely behind S2. So, Test #2 fails.
• S1 is scan converted first.
25
In figure, S2 is in front of S1, but S1 is
not completely inside/behind S2 Test #2
is not true or fails, although
Test #3 is true.
By: Tekendra Nath Yogi2/9/2019

Test #4
• Bounding rectangles overlap but the projections of the two surfaces onto the
view plane do not overlap.
• Use plane equations and test for intersection if no intersection then scan convert
s1 first and s2 next.
• Note: all the tests are failed in second figure.
26
Test #4 passed for the first figure
but it fails for the second figure.

Continue Test #4..
• Case Study-1
Non of the tests
are true, so we have to
reorder the sequence
of the surfaces.
27
Initial Order:
S1->S2
Change the order:
S2->S1

Continue..
• Case Study 2:
Initial order:
S1->S2->S3
Because S1 has the Maximum
depth, then S2 has the maximum
depth and lastly S3 but painters
logic fails.
S1->S2:
Test #1 passed.
Check for S1 and S3 : All Tests fails.
Now, interchanging S3->S2->S1.
Check S2 and S3, all tests fail again.
Correct order is by swapping S2 and S3 position:
S2->S3->S1

Continue..
• Process of checking the order results in an infinite loop.
• So avoiding
infinite loop
any of the
polygon is
clipped or
split.

Contd….
• BSP-Tree Method:
– Binary space partitioning method is one of the efficient method for
determining object visibility
– Paint the surfaces onto the frame buffer from back to front.
– Particularly useful when the view reference point changes, but the
objects in a scene are at fixed positions.

Contd….
• To Build a BSP Tree :
1. Choose a polygon plane Pi, and compute the equation of the plane it
defines.
2. Test all the vertices of all the other polygons to determine if they are in
front of, behind, or in the same plane as Pi. If the plane intersects a
polygon, divide the polygon at the plane.
3. Polygons are placed into a binary search tree with Pi as the root.
4. Call the procedure recursively on the left and right sub-tree.
When the BSP Tree is complete, we process the tree by selecting the surfaces
for display in the order back to front.

Contd….

Contd….
• Scan Line Algorithm:
• Extension of the scan-line algorithm for filling polygon interiors.
• In this method, as each scan line is processed, all polygon surfaces
intersecting that lines are examined to determine which are visible.
• Across each scan line, depth calculations are made for each overlapping
surface to determine which is nearest to the view plane.
• When the visible surface has been determined, the intensity value for
that position is entered into the refresh buffer.

Contd…
Algorithm:
1. Initialize the necessary data structure
a. Edge table containing end point coordinates, slope and polygon pointer.
b. Surface table containing plane coefficients and surface intensity
c. Active Edge List, keep a trace of which edges are intersected by the given scan line
d. Flag for each surface (on entry of scan set on, on exit set off)
2. For each scan line repeat
a. update active edge list
b. determine point of intersection and set surface on or off.
c. If flag is on, store its Intensity value in the refresh buffer
d. If more than one surface is on, do depth sorting and store the intensity of surface
nearest to view plane in the refresh buffer

Contd…
• Illustration:
– For scan line 1
• The active edge list contains edges:
– AB,BC,EH, FG
• Between edges AB and BC, only flags for s1 = on and between edges
EH and FG, only flags for s2=on
• So, no depth calculation needed and corresponding surface intensities
are entered in refresh buffer

Contd..
• For scan line 2
– The active edge list contains edges:
• AD,EH,BC and FG
– Between edges AD and EH, only the flag for surface s1= on
– Between edges EH and BC flags for both surfaces = on
• Depth calculation (using plane coefficients) is needed.
• In this example ,say s2 is nearer to the view plane than s1, so intensities for surface s2 are
loaded into the refresh buffer until boundary BC is encountered
– Between edges BC and FG flag for s1=off and flag for s2 = on
• Intensities for s2 are loaded on refresh buffer

Contd..
• For scan line 3
– Same coherent property as scan line 2 as noticed from active list, so no
depth calculation needed between edges BC and EH

Contd…
• Problem: arise when surfaces cut through and cyclic overlap
• Solution: divide the surfaces (e.g. dashed lines) to eliminate overlap and cut
through
Fig: Intersecting and cyclically overlapping surfaces that alternately obscure one
another.

Contd….
• Area-subdivision Method:
– This visible surface detection algorithm is applied by
successively dividing the total view plane area into smaller and
smaller rectangles until each rectangle area contains either
• The projection of part of a single visible surface
• No surface projections
• Or the area has been reduced to the size of a pixel

Contd….
• Algorithm:
– divide image into 4 equal areas
– for each area (see tests below):
1. are all polygons disjoint from area? (test d)
• yes --> display background color
2. only one intersecting or contained polygon (tests b, c)
• yes --> fill with background color, and then draw contained polygon or intersecting portion
3. one single surrounding polygon, no intersecting or contained polygons (test a)
• yes --> draw area with that polygon's color
4. more than one polygon is intersecting, contained in, or surrounding, but only one
polygon is surrounding the area and is in front of others (test a)
• yes --> draw area with that front polygon's color
– ELSE --> subdivide the area into 4 equal areas and recurse

Contd….
E.g.,

Homework
• What are the object space and image space method of hidden surface removal?
Explain back surface detection method in detail with an example.
• Justify that hidden surface removal is required in computer graphics. Explain
in detail about depth buffer method.
• Explain in detail about depth buffer method. Justify that is better than plane
equation method.
• Explain in detail about scan line method for removing hidden surfaces. Just
that it is better than depth buffer method.
• Explain how BSP Tree method is used for hidden surface elimination
• Explain how area sub-division method is used for hidden surface elimination.
2/9/2019 42By: Tekendra Nath Yogi

Illumination models
• Introduction:
– A projection describes only where an object has to be drawn on the projection
plane.
– The determination of visible surfaces also focused only on the question which
objects should be drawn and which ones are hidden from view by others.
– The information where an object should be drawn on the projection plane is not
at all sufficient for a realistic representation of a 3D scene.
– Realistic displays of a scene are obtained by generating perspective projections
of objects and by applying natural lighting effects to the visible surfaces.

Contd….
• Illumination Model:
– Also called a lighting model or shading model
– Model for calculating light intensity at a single surface point.
– used to calculate the intensity of light that we should see at a given point on
the surface of an object.
• Surface Rendering methods:
– Also called surface-shading methods.
– defined as a procedure for applying a lighting model to obtain pixel
intensities for all the projected surface positions in a scene.

Contd….
• Light Sources:
• Light source is defined as an object that is emitting radiant
energy, such as a Light bulb or the sun.
– Two types of light sources:
• Point light source and
• Distributed light source

Contd..
• Point light source:
– The simplest model for a light emitter is a point source.
– Rays from the source follow radially diverging paths from the source position, as
shown in Figure below:
– This light-source model is a reasonable approximation for sources whose
dimensions are small compared to the size of objects in the scene.
– Sources, such as the sun, that are sufficiently far from the scene can be accurately
modeled as point sources.
Fig: Diverging ray paths from the point source

Contd..
• Distributed light source:
– A nearby source, such as the long fluorescent light as shown in figure below is more
accurately modeled as a distributed light source.
– In this case, the illumination effects cannot be approximated realistically with a
point source, because the area of the source is not small compared to the surfaces in
the scene.
– An accurate model for the distributed source is one that considers the accumulated
illumination effects of the points over the surface of the source.
Fig: An object illuminated with a distributed light source

Contd…
• Basic Illumination Models:
– Illumination models are used to calculate light intensities that we
should see at a given point on the surface of an object.
– Lighting calculations are based on:
• Optical properties of surfaces, such as glossy, matte, opaque, and
transparent. This controls the amount of reflection and absorption
of incident light.
• The background lighting conditions.
• The light-source specifications. All light sources are considered to
be point sources, specified with a coordinate position and intensity
value (color).

Contd..
• Ambient light :
– A surface that is not exposed directly to a light source still will be visible if
nearby objects are illuminated.
– The combination of light reflections from various surfaces produce a
uniform illumination called the ambient light, or background light.
– For example: Light in the Room.
– Ambient light has no spatial or directional characteristics.
– The amount of ambient light incident on each object is a constant for all surfaces and
over all directions.
– The amount of ambient light that is reflected by an object is independent of the
objects position or orientation and depends only on the optical properties of the
surface.

Contd..
• The level of ambient light in a scene is a parameter Ia , and each surface
illuminated with this constant value.
• Illumination equation for ambient light is:
I = kaIa
where
I is the resulting intensity
Ia is the incident ambient light intensity
ka ambient- reflection coefficient. Ka ranges from 0 to 1.

Contd…
• Ambient Light - Example
Fig. : An ambient illumination only.

Contd..
• Diffuse Reflection:
– When light is incident on an opaque surface part of it is reflected and
part of it is absorbed.
– Surface that are rough or grainy, tend to scatter the reflected light in all
direction which is called diffuse reflection.

Contd…
• The amount of Diffuse reflections(the brightness of the surface ) depends on:
– The amount of the incident light that is diffusely reflected (Il )
– diffuse-reflection coefficient, or diffuse reflectivity of the surface (kd).
• 0  kd  1;
• If kd is near 1 – then the surface highly reflective;
• If kd is near 0 – then the surface absorbs most of the incident light;
– orientation of the surface relative to the light source:
• As the angle between the surface normal and the incoming light direction()
increases, less of the incident light falls on the surface.
• If the incoming light from the source is perpendicular to the surface at a
particular point, that point is fully illuminated as shown in figure below.
• Thus, the amount of illumination depends on cos.

Contd…
(a) (b)
Fig.
A surface perpendicular to the direction of the incident light (a) is more
illuminated than an equal-sized surface at an oblique angle (b) to the incoming
light direction.

Contd…
• If Il is the intensity of the point Light source, then the diffuse reflection
equation for a point on the surface can be written As:
• Il,diff = kdIlcos or
• Il,diff = kdIl(N.L)
• where
• N is the unit normal vector to a surface and
• L is the unit direction vector to the point light source from a position on the surface.
Fig.
Angle of incidence between the unit light-source
direction vector L and the unit surface normal N.
N
L
To Light Source


Contd…
Figure below illustrates the illumination with diffuse reflection, using various
values of parameter kd between 0 and1.
Fig.: Series of pictures of sphere illuminated by diffuse reflection model only
using different kd values (0.4, 0.55, 0.7, 0.85,1.0).

Contd….
• Total Diffuse Reflection:
– We can combine the ambient and point-source intensity calculations to
obtain an expression for the total diffuse reflection.
– Idiff = kaIa+kdIl(N.L)
– where both ka and kd depend on surface material properties and are assigned
values in the range from 0 to 1.
Fig. Series of pictures of sphere illuminated by ambient and diffuse reflection model.
Ia = Il = 1.0, kd = 0.4 and ka values (0.0, 0.15, 0.30, 0.45, 0.60).

Contd…
• Specular Reflection:
– When we look at an illuminated shiny surface, such as polished metal,
we see a highlight, or bright spot, at certain viewing directions. This
phenomenon, called specular reflection.
– It is the result of total, or near total, reflection of the incident light in a
concentrated region around the specular reflection angle.
2/9/2019 58
Figure: Specular Reflection

Contd…
• Figure below shows the specular reflection direction at a point on the illuminated
surface.
In this figure,
– R represents the unit vector in the direction of specular reflection;
– L – unit vector directed toward the point light source;
– V – unit vector pointing to the viewer from the surface position;
– Angle  is the viewing angle relative to the specular-reflection direction R.
• shiny surfaces have a narrow specular- reflection range, and dull surfaces have a wider
reflection range. (i.e., for shiny surface = 0 and for rough surface  > 0 )
Fig. : Modeling specular reflection.
N
L
To Light Source
 
R
V

Contd…
• Phong Specular Reflection Model (phong model):
– Phong model is a model for calculating the specular-reflection range:
• Sets the intensity of specular reflection proportional to cosns;
• Angle  can be assigned values in the range 0o to 90o, so that cos varies from 0 to
1;
• Specular-reflection parameter ns is determined by the type of surface that we want to
display.
– Very shiny surface is modeled with a large value for ns (say, 100 or more);
– Small values (down to 1) are used for duller surfaces.
– For perfect reflector (perfect mirror), ns is infinite;
• Specular-reflection coefficient ks equal to some value in the range 0 to 1 for each
surface.

Contd…
N
L
R
Shiny Surface (Large ns)
N
L
R
Dull Surface (Small ns)
Fig. : Modeling specular reflection with parameter ns.

Contd…
• The intensity of Phong specular-reflection model depends on the intensity of
the incident light, material properties of the surface and the angle of
incidence, so that intensity of the reflected light can be expressed as:
Ispec = ksIl cosns
• Since V and R are unit vectors in the viewing and specular-reflection
directions, we can calculate the value of cosns with the dot product V.R.
Ispec = ksIl (V.R)ns
Fig. : Modeling specular reflection.
N
L
To Light Source
 
R
V

Contd…
• Phong Model Using Halfway vector:
– Simplified way of formulation of phong illumination model is the use of
halfway vector H.
– The halfway vector H = (L + V)/|(L + V)|
– The intensity for the specular reflection is:
Ispec = ksIl (N.H)ns

Contd…
• Combine Diffuse & Specular Reflections:
– For a single point light source, we can model the combined diffuse and
specular reflections from a point on an illuminated surface as:
I = Idiff + Ispec
I = kaIa + kdIl(N.L) + ksIl(N.H)ns

Contd….
• Combine Diffuse & Secular Reflections with Multiple Light
Sources :
– If we place more than one point source in a scene, we obtain the light
reflection at any surface point by summering the contributions from the
individual sources:
I = kaIa + n
i=1 Ili [kd (N.Li) + ks(N.Hi)ns]

Light Intensity Attenuation
• As the light emitted from a light source travels through space, its intensity
decreases. This is called attenuation.
– Example: a traffic light will not be able to illuminate an object 2km away.
• The intensity is attenuated by the factor: l/d2 where, d is the distance that the
light has traveled.
• This means that a surface close to the light source (small d) receives a higher
incident intensity from the source than a distant surface (large d).
• Therefore, to produce realistic lighting effects, take this intensity attenuation
into account.

Contd…
• Problem : Too much intensity variation for objects that are close to the light
source and too little intensity variation for object that are far away
– Solution: Use the following function instead
– The constants can be adjusted to produce better attenuation effects.
• Using this function, we can then write our basic illumination model as:

Color considerations
• Most graphics displays of realistic scenes are in color. But the illumination
model we have described so far considers only monochromatic lighting
effects.
• To incorporate color, we need to write the intensity equation as a function
of the color properties of the light sources and object surfaces.
• For an RGB description, each color in a scene is expressed in terms of red,
green, and blue components.
• We then specify the RGB components of light source intensities and
surface colors, and the illumination model calculates the RGB components
of the reflected light.

Contd..
• One way to set surface colors is by specifying the reflectivity
coefficients as three-element vectors.
• For Example:
– The diffuse reflection coefficient vector, have RGB components (kdR, k dC,
kdB).
– If we want an object to have a blue surface, we select a nonzero value in the
range from 0 to 1 for the blue reflectivity component, kdB while the red and
green reflectivity components are set to zero (kdR = kdG = 0).
– Any nonzero red or green components in the incident light are absorbed,
and only the blue component is reflected. The intensity calculation for this
example reduces to the single Expression:

Transparency
• Transparent surface produces both reflected and transmitted light as shown in
figure below:
• The relative contribution of the transmitted light depends on the degree of
transparency of the surface(kt) and whether any light sources or illuminated
surfaces are behind the transparent surface.

Contd…
• When a transparent surface is to be modeled, the intensity equations must be
include contributions from light passing through the surface.
• So, intensity equation becomes:
• Where, Itrans -> Transmitted intensity through a transparent surface.
• Irefl -> Reflected intensity from the transparent surface.
• Kt -> transparency coefficient , kt =1 for highly transparent objects and is 0
for highly opaque objects.
2/9/2019 By: Tekendra Nath Yogi 71

Contd..
• Transmitted intensity Itrans through a transparent surface from a background object
and Reflected intensity Irefl from the transparent surface with transparency
coefficient kt is given by:
• term (1 - kt) is the opacity factor. For highly transparent objects, we assign kt a
value near 1.
• Nearly opaque object transmits very little light from background object , and we
can set kt to a value near 0 for these materials (opacity near 1).

shadows
• A shadow is a dark area where light from a light source is blocked by
an opaque object.
• Why shadows?
– Make image more realistic
– Visual clues as how objects are positioned with respect to each other.
• How shadows are generated?
– Hidden surface method with light source at the view position can be used
to produce shadows
Fig: Objects modeled with shadow regions.
2/9/2019 73

Polygon (Surface) Rendering Method
• Rendering means giving proper intensities at each point in a
graphical object to make it look like a real world object.
• Application of an illumination model.
• Different Rendering methods are:
– Constant intensity shading
– Gouraud shading
– Phong shading

Constant-Intensity Shading
• A fast and simple method for rendering an object with polygon
surfaces is constant-intensity shading, also called flat shading.
• Procedure
– take a point on the object surface and calculate the intensity at this point by
using a illumination model.
– All points over the surface of the object are then displayed with the same
intensity value.
– Repeat above procedure for each object surface.

Contd..
• In general, flat shading of polygon facets provides an accurate rendering for an
object if all of the following assumptions are valid:
– The object is a polyhedron and is not an approximation of an object with a curved
surface.
– All light sources illuminating the objects are sufficiently far from the surface so that
N . L and the attenuation function are constant over the surface.
– The viewing position is sufficiently far from the surface so that V. R is constant over
the surface.
• Even if all of these conditions are not true, we can still reasonably approximate
surface-lighting effects using small polygon facets with flat shading and
calculate the intensity for each facet, say, at the center of the polygon.
• DRAWBACK: intensity discontinuity at the edges of polygons

Gouraud Shading(Intensity Interpolation shading)
• This intensity-interpolation scheme, developed by Gouraud.
– Renders a polygon surface by linearly interpolating intensity values across the
surface.
– Intensity values for each polygon are matched with the values of adjacent polygons
along the common edges, thus eliminating the intensity discontinuities that can
occur in flat shading.
• Each polygon surface is rendered with Gouraud shading by performing the
following calculations:
– Determine the average unit normal vector at each polygon vertex.
– Apply an illumination model to each vertex to calculate the vertex intensity.
– Linearly interpolate the vertex intensities over the surface of the polygon.

Contd..
• Step1: Unit Normal Vector calculation:
– The normal vector at vertex V is calculated as the average of the surface
normals for each polygon sharing that vertex.
– E.g.,

Contd..
• Step2: Vertex intensity calculation:
– Once we have the vertex normals, we can determine the intensity at the
vertices from a lighting model.

Contd..
• Step3: Linearly interpolate the vertex intensities over the surface of the
polygon.
• Two step Process:
• Bounding Intensity calculation: Intensity calculation at the intersection
of the scan line with a polygon edge
• Interior intensity calculation

Contd..
• Bounding Intensity calculation:
– For each scan line, the intensity at the intersection of the scan line with a
polygon edge is linearly interpolated from the intensities at the edge
endpoints.

Contd..
– For the example in Figure below, the polygon edge with endpoint
vertices at positions 1 and 2 is intersected by the scan line at point 4.
– A fast method for obtaining the intensity at point 4 is to interpolate
between intensities I1 and I2 using:
– Similarly, intensity at the right intersection of this scan line (point 5)
is interpolated from intensity values at vertices 2 and 3.

Contd..
• Interior intensity calculation: Once these bounding intensities are
established for a scan line, an interior point (such as point p in Fig.
above) is interpolated from the bounding intensities at points 4 and
5 as:

Contd..
• Incremental calculations are used to obtain successive edge intensity values
between scan lines and to obtain successive intensities along a scan line. As
shown in Figure below:
• if the intensity at edge position (x, y) is interpolated as:
• then we can obtain the intensity along this edge for the next scan line, y - 1, as:

Contd …
• Advantages:
– Gouraud shading removes the intensity discontinuities associated with the constant-shading
model.
• But it has some other deficiencies.
– Highlights on the surface are sometimes displayed with anomalous shapes,
and the linear intensity interpolation can cause bright or dark intensity
streaks, called Mach bands, to appear on the surface.
• Solutions: These effects can be reduced by using Phong shading, that require
more calculations.

Phong Shading(Normal Vector Interpolation shading)
• A more accurate method for rendering a polygon surface is to interpolate normal vectors,
and then apply the illumination model to each surface point.
• This method, developed by Phong Bui Tuong, is called Phong shading, or normalvector
interpolation shading.
• It displays more realistic highlights on a surface and greatly reduces the Mach-band
effect.
• A polygon surface is rendered using Phong shading by carrying out the following steps:
– Determine the average unit normal vector at each polygon vertex.
– Linearly interpolate the vertex normals over the surface of the polygon.
– Apply an illumination model along each scan line to calculate projected pixel intensities for the
surface points.

Contd..
• Interpolation of surface normals along a polygon edge between two vertices is
illustrated in Figure below:
Fig: Interpolation of surface normal along a polygon edge
• The normal vector N for the scan-line intersection point along the edge between
vertices 1 and 2 can be obtained by vertically interpolating between edge
endpoint normals:

Gouraud versus Phong Shading
• Gouraud shading is faster than Phong shading
• Phong shading is more accurate

Homework
• Explain in detail about diffuse reflection model.
• Difference between diffuse reflection and specular reflection. Why do we require shading
model? Explain it.
• Explain the visible effect that occurs when during animation of a Gouraud shading
polyhedron, the center of a highlight moves from one vertex to another along an edge.
• Why shading is required in the computer graphics? Explain in detail about constant intensity
shading.
• Explain in detail about Phong shading. How can you modify Phong shading model?
• Explain in detail about Gourand shading model. Compare it with Phong shading model.
• Explain in detail about Phong shading model. Compare it with Gouraud Shading model.

Thank You !

B. SC CSIT Computer Graphics Unit 4 By Tekendra Nath Yogi

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B. SC CSIT Computer Graphics Unit 4 By Tekendra Nath Yogi