Hierarchical Object Modeling

1
Hierarchical Object Modeling

• Construct complex models from a set of simple
  geometric objects
• Extend the use of transformations to include
  hierarchical relationships
• Useful for characterizing relationships among
  model parts in applications such as robotics and
  figure animations.

2
Symbols and Instances

• A non-hierarchical approach which models our
  world as a collection of symbols.
• Symbols can include geometric objects, fonts and
  other application-dependent graphical objects.
• Use the following instance transformation to
  place instances of each symbol in the model.
• The transformation specify the desired size,
  orientation and location of the symbol.

3
Example
4
Hierarchical Models

• Relationships among model parts can be
  represented by a graph.
• A graph consists of
• A set of nodes or vertices.
• A set of edges which connect pairs of nodes.
• A directed graph is a graph where each of its
  edges has a direction associated with it.

5
Trees and DAG Trees

• A tree is a directed graph without closed paths
  or loops.
• Each tree has a single root node with outgoing
  edges only.
• Each non-root node has a parent node from which
  an edge enters.
• Each node has one or more child nodes to which
  edges are connected.
• A node without children is called a terminal node
  or leaf.

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Trees and DAGS Trees
root node
edge
node
leaf
7
Example
8
Trees and DAG DAG

• Storing the same information in different nodes
  is inefficient.
• Use a single prototype for multiple instances of
  an object and replace the tree structure by the
  directed acyclic graph (DAG).
• Both trees and DAG are hierarchical methods of
  expressing relationships.

9
Example A Robot Arm

• The robot arm is modeled using three simple
  objects.
• The arm has 3 degrees of freedom represented by
  the 3 joint angles between components.
• Each joint angle determines how to position a
  component with respect to the others.

10
Robot Arm Base

• The base of the robot arm can rotate about the y
  axis by the angle ?.
• The motion of any point p on the base can be
  described by applying the rotation matrix

11
Robot Arm Lower Arm

• The lower arm is rotated about the z-axis by an
  angle ?, which is represented by
• It is then shifted to the top of the base by a
  distance , which is represented by
  .
• If the base has rotated, we must also rotate the
  lower arm using .
• The overall transformation is
  .

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Robot Arm Upper Arm

• The upper arm is rotated about the z-axis by the
  angle ?, represented by the matrix
  .
• It is then translated by a matrix
  relative to the lower arm.
• The previous transformation is then applied to
  position the upper arm relative to the world
  frame.
• The overall transformation is

13
OpenGL Display Program
display() glRotatef(theta, 0.0, 1.0,
0.0) base() glTranslatef(0.0, h1,
0.0) glRotatef(phi, 0.0, 0.0,
1.0) lower_arm() glTranslatef(0.0, h2,
0.0) glRotatef(psi, 0.0, 0.0,
1.0) upper_arm()
14
Tree Representation

• The robot arm can be described by a tree data
  structure.
• The nodes represent the individual parts of the
  arm.
• The edges represent the relationships between the
  individual parts.

15
Information Stored in Tree Node

• A pointer to a function that draws the object.
• A homogeneous coordinate matrix that positions,
  scales and orients this node relative to its
  parent.
• Pointers to children of the node.
• Attributes (color, fill pattern, material
  properties) that applies to the node.

16
Trees and Traversal

• Perform a tree traversal to draw the represented
  figure.
• Two possible approaches
• Depth-first
• Breadth-first
• Two ways to implement the traversal function
• Stack-based approach
• Recursive approach

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Example A Robot Figure

• The torso is represented by the root node.
• Each individual part, represented by a tree node,
  is positioned relative to the torso by
  appropriate transformation matrices.
• The tree edges are labeled using these matrices.

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Stack-Based Traversal

• The torso is drawn with the initial model-view
  matrix
• Trace the leftmost branch to the head node
• Update the model-view matrix to
• Draw the head
• Back up to the torso node.
• Trace the next branch of the tree.
• Draw the left-upper arm with matrix
• Draw the left-lower arm with matrix
• Draw the other parts in a similar way.

19
OpenGL Implementation Head
Figure() glPushMatrix() torso() glTranslate
glRotate3 head() glPopMatrix()
20
OpenGL Implementation Left Arm
glPushMatrix() glTranslate glRotate3 Left_upper_a
rm() glTranslate glRotate3 Left_lower_arm() glPo
pMatrix()
21
Push and Pop Operations

• glPushMatrix puts a copy of the current
  model-view matrix on the top of the
  model-view-matrix stack.
• glTranslate and glRotate together construct a
  transformation matrix to be concatenated with the
  initial model-view matrix.
• glPopMatrix recovers the original model-view
  matrix.
• glPushAttrib and glPopAttrib store and retrieve
  primitive attributes in a similar way as
  model-view matrices.

22
Recursive Approach

• Use a left-child right-sibling structure.
• All elements at the same level are linked left to
  right.
• The children are arranged as a second list from
  the leftmost child to the rightmost.

root
23
Example

24
FirstChild pointer arrow that points
downward NextSibling pointer arrow that goes
left to right

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Data Structure for Node
Typedef struct treenode GLFloat m16 void
(f)() struct treenode sibling struct
treenode child treenode
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Description of Data Structure Elements

• m is used to store a 4x4 homogeneous coordinate
  matrix by columns.
• f is a pointer to the drawing function.
• sibling is a pointer to the sibling node on the
  right.
• child is a pointer to the leftmost child.

27
OpenGL Code Torso Node Initialization
glLoadIdentity() glRotatef(theta0, 0.0, 1.0,
0.0) glGetFloatv(GL_MODELVIEW_MATRIX,torso_node.m
) Torso_node.ftorso Torso_node.siblingNULL To
rso_node.child head_node
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OpenGL Code Upper Left Arm Node Initialization
glLoadIdentity() glTranslatef(-(TORSO_RADIUSUPPE
R_ARM_RADIUS), 0.9TORSO_HEIGHT,
0.0) glRotatef(theta3, 1.0, 0.0,
0.0) glGetFloatv(GL_MODELVIEW_MATRIX,
lua_node.m) Lua_node.fleft_upper_arm Lua_node.s
ibling rua_node Lua_node.child lla_node
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OpenGL Code Tree Traversal
Void traverse (treenode root) if (rootNULL)
return glPushMatrix() glMultMatrixf(root-gtm)
root-gtf() if (root-gtchild!NULL)
traverse(root-gtchild) glPopMatrix() if
(root-gtsibling!NULL) traverse(root-gtsibling)
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Animation

• Objective to model a pair of walking legs.

31
Animation Walking Legs

• Rotation of the base will cause rotation of all
  the other parts.
• Rotation of other parts will only affect
  themselves and those lower parts attached to them.

32
Definition of Body Metrics

• The relative body metrics are defined in the file
  model.h.
• The main parts include the base and the two legs.

define TORSO_HEIGHT 0.8 define TORSO_WIDTH
TORSO_HEIGHT0.75 define TORSO_DEPTH
TORSO_WIDTH/3.0 define BASE_HEIGHT
TORSO_HEIGHT/4.0 define BASE_WIDTH
TORSO_WIDTH define BASE_DEPTH TORSO_DEPTH
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Drawing the Base
void Draw_Base(void) glPushMatrix()
glScalef(BASE_WIDTH,BASE_HEIGHT, BASE_DEPTH)
glColor3f(0.0,1.0,1.0) glutWireCube(1.0)
glPopMatrix()
34
Drawing the Upper Leg
void Draw_Upper_Leg(void) glPushMatrix()
glScalef(UP_LEG_JOINT_SIZE,UP_LEG_JOINT_SIZE,UP_L
EG_JOINT_SIZE) glColor3f(0.0,1.0,0.0)
glutWireSphere(1.0,8,8) glPopMatrix()
glColor3f(0.0,0.0,1.0) glTranslatef(0.0,-
UP_LEG_HEIGHT 0.75, 0.0) glPushMatrix()
glScalef(UP_LEG_WIDTH,UP_LEG_HEIGHT,UP_LEG_WIDTH
) glutWireCube(1.0) glPopMatrix()
35
Drawing the Whole Leg
void Draw_Leg(int side) glPushMatrix()
glRotatef(walking_anglesside3,1.0,0.0,0.0)
Draw_Upper_Leg() glTranslatef(0.0,-
UP_LEG_HEIGHT 0.75,0.0) glRotatef(walking_a
nglesside4,1.0,0.0,0.0) Draw_Lower_Leg()
glTranslatef(0.0,- LO_LEG_HEIGHT 0.625,
0.0) glRotatef(walking_anglesside5,1.0,0.
0,0.0) Draw_Foot() glPopMatrix()
36
Drawing the Complete Model
void Draw_Base_Legs(void) glPushMatrix()
glTranslatef(0.0,base_move,0.0) Draw_Base()
glTranslatef(0.0,-(BASE_HEIGHT),0.0)
glPushMatrix() glTranslatef(TORSO_WIDTH
0.33,0.0,0.0) Draw_Leg(LEFT)
glPopMatrix() glTranslatef(-TORSO_WIDTH
0.33,0.0,0.0) Draw_Leg(RIGHT)
glPopMatrix()
37
Modeling the Walking Motion
38
Codes for Calculating Vertical Displacement
double find_base_move(double langle_up, double
langle_lo, double rangle_up, double rangle_lo)
double result1, result2, first_result,
second_result, radians_up, radians_lo
radians_up (PIlangle_up)/180.0 radians_lo
PI(langle_lo-langle_up)/180.0 result1
(UP_LEG_HEIGHT 2UP_LEG_JOINT_SIZE)
cos(radians_up) result2 (LO_LEG_HEIGHT2(L
O_LEG_JOINT_SIZEFOOT_JOINT_SIZE) FOOT_HEIGHT)
cos(radians_lo) first_result LEG_HEIGHT -
(result1 result2)
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Codes for Calculating Vertical Displacement
radians_up (PIrangle_up)/180.0
radians_lo PI(rangle_lo-rangle_up)/180.0
result1 (UP_LEG_HEIGHT 2UP_LEG_JOINT_SIZE)
cos(radians_up) result2 (LO_LEG_HEIGHT
2(LO_LEG_JOINT_SIZEFOOT_JOINT_SIZE) FOOT_HEI
GHT) cos(radians_lo) second_result
LEG_HEIGHT - (result1 result2) if
(first_result lt second_result) return (-
first_result) else return (-
second_result)
40
Key-Frame Animation

• The more critical motions of the object is shown
  in a number of key frames.
• The remaining frames are filled in by
  interpolation (the inbetweening process)
• In the current example, inbetweening is automated
  by interpolating the joint angles between key
  frames.

41
Interpolation Between Key Frames

• Let the set of joint angles in key frame A and
  key frame B be
• and
  respectively.
• If there are M frames between key frame A and B,
  let
• The set of joint angles in the m-th in-between
  frame is assigned as

42
Implementation of Key-Frame Animation
switch (flag) case 1
l_upleg_dif 15 r_upleg_dif 5
l_loleg_dif 15 r_loleg_dif 5
l_upleg_add l_upleg_dif / FRAMES
r_upleg_add r_upleg_dif / FRAMES
l_loleg_add l_loleg_dif / FRAMES
r_loleg_add r_loleg_dif / FRAMES
walking_angles03 l_upleg_add
walking_angles13 r_upleg_add
walking_angles04 l_loleg_add
walking_angles14 r_loleg_add
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Implementation of Key-Frame Animation
base_move find_base_move(walking_angles0
3, walking_angles04, walking_angles13
, walking_angles14 )
frames-- if (frames 0)
flag 2 frames FRAMES break
case 2
44
Snapshots of Walking Sequence
45
Solid Modeling

• How would you interpret the above object?

46
Solids

• Definition
• A model which has a well-defined inside and
  outside
• For each point, we can determine whether the
  point is inside, outside, or on the solid
• Purpose
• Model more complicated shapes
• Mass properties calculations
• Volume and Moment of Insertia
• CAD/CAM manufacturing

47
Representations

• Constructive Solid Geometry (CSG)
• Spatial Partitioning representation
• Cell Decomposition
• Spatial-occupancy Enumeration
• Quadtrees and Octrees
• Binary Space Partitioning (BSP)

48
Constructive Solid Geometry (CSG ) Trees (1)

• Data Structure Binary tree
• Nodes Boolean Operations
• Union, Intersection, Difference
• Nodes Transformation
• To position and scale objects
• Leaves Primitives
• Blocks and wedges
• Quadrics spheres, cylinders, cones, paraboloids
• Deformed solids

49
CSG Trees (2)

• Constructive solid geometry (CSG) operates on a
  set of solid geometric entities instead of
  surface primitives.
• Uses three operations union, intersection, and
  set difference
• A?B consists of all points in either A or B.
• A?B consists of all points in both A and B.
• A-B consists of all points in A which are not in
  B.

50
CSG Trees (3)
51
CSG Trees (3)

• The algebraic expressions are stored and parsed
  using expression trees.
• Internal nodes store operations.
• Terminal nodes store operands.
• The CSG tree is evaluated by a post-order
  traversal.

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CSG Trees (4)

• Example

Extracted from Foley et al.s book
53
CSG Trees (5)

• Example

CSG does not provide unique representation
Extracted from Foley et al.s book
54
Spatial Partitioning Representation (SPR) Cell
Decomposition

• Each cell-decomposition system defines a set of
  primitives cells that are typically parameterized
• Gluing them together

Extracted from Foley et al.s book
55
SPR Spatial-Occupancy Enumeration

• Special case of cell decomposition
• Decomposed into identical cells arranged in a
  fixed, regular grid

Extracted from Foley et al.s book
56
SPR Quadtrees and Octrees (2)

• A hierarchical variant of spatial-occupancy
  enumeration
• Designed to address approachs demanding storage
  requirements
• Use separating planes and lines parallel to the
  coordinate axes.
• For a quadtree, each parent node has four child
  nodes.
• For an octree (derived from quadtrees), each
  parent node has eight child nodes.

Extracted from Foley et al.s book
57
SPR Quadtrees and Octrees (3)

• Successive subdivision can be represented as a
  tree with partially full quadrants (P), full (F)
  and empty (E)
• No standard for assigning quadrant numbers (0-3)

Extracted from Foley et al.s book
58
SPR Quadtrees and Octrees (4)

• Octree is similar to the quandtree, except that
  its three dimensionals are recursively subdivided
  into octant.

Extracted from Foley et al.s book
59
SPR Quadtrees and Octrees (5)

• Boolean set operations

S
T
Extracted from Foley et al.s book
60
SPR Binary Space-Partitioning (BSP) Trees (1)

• Recursively divide space into a pairs of
  subspaces, each separated by a plane of arbitrary
  orientation and position.
• Originally, used in determining visible surface
  in graphics

View point
61
SPR Binary Space-Partitioning (BSP) Trees (2)

• Rendering of a set of polygons using binary
  spatial-partition tree (BSP tree)
• Plane A separates polygons into two groups
• B,C in front of A
• D,E and F behind A.
• In the BSP tree
• A is at the root.
• B and C are in the left subtree.
• D, E and F are in the right subtree.

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SPR Binary Space-Partitioning (BSP) Trees (3)

• Proceeding recursively
• B is in front of C
• D separates E and F
• In the 2 BSP sub-trees
• B is the left child of C
• E and F are the left and right of D respectively

63
SPR Binary Space-Partitioning (BSP) Trees (4)

• Later to represent arbitrary polyhedra.
• Each internal node is associated with a plane and
  has two child pointers one for each side
• If the half-space on a side of the plane is
  subdivided further, its child is the root of a
  subtree
• If the half-space is homogenous, its child is a
  leaf, representing a region either entirely
  inside or entirely outside, labeled as in or
  out

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Comparison of Representations

• Accuracy
• Spatial-partitioning only an approximation
• CSG high
• Uniqueness
• Octree and spatial-occupancy-enumeration unique
• CSG not