Hierarchical Modeling and Scene Graphs

Hierarchical Modeling

• In computer graphics, the main goal of hierarchical modeling is organizing and efficiently manipulating objects using a structured hierarchy.
• Local coordinate systems and parent-child relationships are used for transformations like translation, rotation, and scaling during hierarchical modeling.
• A scene graph is a data structure often used to represent hierarchical models in computer graphics
• Transformations applied to a parent node are inherited by its child nodes in a hierarchical model
• Modeling a robotic arm, where each joint's movement relies on the previous joint, exemplifies hierarchical modeling
• Hierarchical modeling in animation is advantageous because it allows complex objects to be animated more efficiently by manipulating sub-parts
• The key benefit of hierarchical modeling is organizing objects into a structured hierarchy, which allows for better transformations

Data Structure

• Scene Graph is commonly used to represent hierarchical models
• Scene Graph describes data structure commonly used to represent hierarchical models.

Object Manipulation

• It enables nested objects to inherit transformations from parent objects.
• A matrix stack in computer graphics is intended to store transformation matrices in a hierarchical manner
• The matrix stack facilitates the pushing and popping of transformations for managing nested structures within hierarchical transformations
• A copy of current transformation matrix is stored for later use when transformation matrix is pushed onto the stack
• The "matrix stem" is the root transformation matrix in a hierarchy.
• The matrix stem serves as the origin of transformations propagating through a hierarchy

Transformation Propagation

• Transformations to parent objects propagate to child objects in hierarchical models
• The order of transformations during propagation depends on the scene graph
• Transformation propagation example: character's arm moves with the body in an animation
• Transformation propagation ensures consistency in movement and transformations across connected objects

Rendering in 3D Computer Graphics

• The primary goal of rendering in 3D computer graphics is converting 3D models into 2D images
• Lighting, shading, and textures are used in 3D computer graphics to convert 3D models into 2D images
• Ray tracing simulates real-world lighting, reflections, and shadows in 3D rendering with greater accuracy
• Phong lighting model is a lighting model that considers ambient, diffuse, and specular components
• Shadow mapping simulates light interaction and shadow casting in 3D rendering

Projection in 3D Graphics

• The core purpose of projection in 3D graphics is to transform a 3D scene into a 2D image for display
• Objects appear smaller as they move farther from the camera when using perspective projection
• Matrix transformations is a mathematical concept that is part of perspective projection
• Perspective convergence is a phenomenon in 3D graphics where parallel lines seem to converge at a distance
• The vanishing point is the point where parallel lines appear to converge in perspective projection.
• Orthographic projection preserves object proportions without perspective distortion
• Oblique projection differs from orthographic projection because it distorts depth
• The view volume in 3D rendering is the region of space visible after projection.
• Frustum and orthographic are common view volumes types

Rendering Techniques

• Converting 3D scenes into 2D images defines primary goal of rendering
• Ray tracing simulates light interaction with objects by tracing rays from the camera
• Shadow rays in ray tracing tests if a point is in shadow
• Point light emits light source that emits light uniformly
• Penumbra shadow is created using area lights producing softer, more realistic shadows
• Lambertian reflectance approximates light reflection off diffuse surfaces
• Perspective projection gives a realistic view where distant object appear smaller
• The point where parallel lines converge refers to vanishing point in perspective projection

Matrices and Coordinates

• Projection matrix transforms 3D world coordinates to 2D screen coordinates.
• The perspective divide in homogeneous coordinates converts from clip space to Normalized Device Coordinates (NDC).
• Perspective projection's z coordinate becomes a depth value in Normalized Device Coordinates (NDC) post-perspective divide
• The w component to divide x, y, and z after applying perspective matrix
• Increasing the field of view (FOV) makes more of the scene visible, while objects appear smaller
• A truncated pyramid is the geometric shape that defines perspective view frustum

View Frustum

• top, bottom, left, right, near, and far defines boundaries of the view frustum
• The near clipping plane in a view frustum defines the minimum render distance
• Objects outside the view frustum are clipped, or not rendered
• The perspective projection matrix converts 3D eye-space coordinates into clip-space
• Camera position isn't a parameter that influences perspective projection matrix
• Aspect ratio scales the x-coordinate to prevent distortion.
• Setting the far clipping plane too far causes z-fighting (depth precision issues).
• gluPerspective() / glm::perspective() are functions in OpenGL/GLM that creates a perspective projection matrix

Transformations

• The correct order of transformations in the graphics pipeline is: Model → View → Projection
• Translation matrix is used to move an object in 3D space.
• Transforming scale, rotate, translate in order of Scale → Rotate → Translate
• Scaling changes the size of an object
• A 4x4 transformation matrix allows combined rotations, translations, and scaling
• Color grading is not a basic 3D transformation
• A 3D rotation has 3 degrees of freedom: roll, pitch, and yaw
• Y-axis is typically used for "yaw" rotation.
• Quaternion rotation allows for faster interpolation than Euler angles

Euler Angles and Gimbal Lock

• Quaternions avoids gimbal lock
• Euler angles defined as, set of three angles (roll, pitch, yaw)
• A disadvantage of Euler angles: suffer from gimbal lock
• Gimbal lock occurs when two axes align, which results to losing degree of freedom.
• Commonly used for aerospace by NASA : ZYX
• Quaternions or axis-angle representations helps in Gimbal lock mitigation
• Gimbal lock is defines the loss of rotational degree of freedom due to aligned axes
• XYZ Euler angle sequence is most prone to gimbal lock
• Spacecraft attitude control relies heavily on the importance of gimbal lock
• Sudden jumps in rotation is a visual artifact is cause by gimbal lock

Optimizations & Algorithms

• Discarding objects outside the camera’s visible volume defines primary purpose of view frustum culling
• Checks if an object’s bounding volume intersects view frustum
• Objects failing the view frustum test, are excluded from rendering
• Painter’s Algorithm determines visibility by sorting objects from back to front and drawing them in order
• Overlapping or cyclic occlusions represents a major drawback of the Painter’s Algorithm
• Z-buffering is more efficient and accurate compared to the Painter's Algorithm
• Z-buffer stores Depth (distance from the camera) for each pixel
• Z-buffer resolves visibility between overlapping objects by keeping the object with smallest Z-value, closest to the camera
• Z-buffering key advantage over Painters algorithm, consists of working with transparency and cyclic overlaps correctly
• Z-fighting is caused by insufficient Z-buffer precision for overlapping surfaces
• Using a higher precision depth buffer (e.g., 24-bit instead of 16-bit) that reduces Z-fighting
• Z-fighting creates Flickering or shimmering surfaces
• Backface culling removes polygons facing away from the camera
• Dot product of view vector and face normal determines backface culling

Clipping and Bounding volumes

• Eliminating objects outside camera's field of view defines view volume testing in 3D rendering
• View volume takes form of a frustum when using perspective projection
• Objects culled to improve performance is result of view volume testing
• Bounding volume refers to simplified geometric shapes, approximating object's spatial extent
• Clipping ensure only visible portions of objects are processed for rendering
• Discarding geometry outside the view frustum is the primary purpose of clipping during rasterization
• Triangle that intersects it's near clipping plane is split into smaller triangles
• Clipping is performed in clip space
• Sutherland-Hodgman algorithm clips polygons against a plane

Scene graphs, trees and rendering

• Hierarchically organize and transform objects refers to primary purpose of a scene graph
• Binary Space Partitioning (BSP) tree is optimized for static scenes with axis-aligned geometry.
• Expensive recomputation when objects move is a disadvantage of BSP trees for dynamic scenes
• Octree partitions space into nested cubes for efficient spatial queries
• Quad-tree have lower memory usage in 2D applications compared to octree
• BSP tree constructs by recursively splitting space with hyperplanes
• In a BSP tree, the polygon that best divides the remaining geometry evenly defines the splitting plane at each node
• Key Limitation of BSP trees: known for Poor handling of dynamic objects
• Healthcare : Industry uses 3D rendering for pre-visualizing surgical procedures
• Creating photorealistic building walkthroughs is a common use of 3D rendering in architecture
• Virtual Reality (VR) : Technology relies heavily on 3D rendering for immersive experiences
• Scanline rendering: Algorithm commonly used for real-time rasterization
• Light reflections and refractions : Ray tracing simulates more accurately than rasterization
• NVIDIA’s RTX (DLSS + ray tracing) is a hybrid rendering technique combines rasterization and ray tracing
• To accelerate collision detection and visibility tests is primary use of an octree in 3D rendering
• Dynamic tree (e.g., loose octree) improves upon static trees, allows runtime updates to moving objects
• Bounding Volume Hierarchy (BVH): Optimization used for dynamic scenes in modern engines
• Ignoring the Z-axis : Drawback of quadtrees for 3D scenes