Procedural Modeling Slides for DDM by Marc van Kreveld 1.

transcript

Procedural Modeling

Slides for DDMby Marc van Kreveld

Procedural modeling

• Creating 3D models from a set of rules– L-systems– Fractals– Generative modeling

• Procedural modeling gives procedural content• Procedural modeling uses– many parameters that can be set– often randomness

Procedural modeling

• Cheaper and faster than manual construction, especially when many similar models are needed

• Possible to generate– the geometry– the texture– the placement of the model in a scene– whole scenes

(e.g. urban: street pattern, buildings, trees, …)

Procedural modeling

Procedural terrain modeling

• Oldest, best known, for height maps: midpoint displacement algorithm

• New vertices get an elevation that is the interpolation of its neighbors, +/- some random term

• The random term decreases in every iteration, other-wise the terrain gets more and more steep/rugged

• Ruggedness can beglobally controlled by the standard deviation of the random term

• Looks good, but is not natural:– symmetric with respect to high and low, but real terrains

are not– real terrains are formed by erosion from wind, rain, water

flow, …– ridges and valleys are not realistic

• Possible solutions:– apply wind and rain erosion models to the generated

terrain– compute where rivulets/rivers would form based on local

gradient and apply river erosion models

• Overhangs and cliffs are not supported• No global user control

• A similar approach can be used for coastlines, but there are different coastline types

• Constraints are needed to ensure that the coastline does not self-intersect

• For rivers, one must be aware that rivers and terrain height cannot be generated independently– generate a height map and determine where the rivers are– generate a river network and compute a consistent height

map– with little height influence, meandering must be modeled

• River deltas, waterfalls, cascades, …

• For urban environments:– start with generating a dense road network– partition the resulting blocks into lots– (a) use the lots are building footprints and raise them

to random heights (skyscrapers, office buildings), or(b) place a building footprint of a house on the lot and generate the house by some scheme• extrusion plus roof generation• L-system or other grammar

L-systems

• Named after Aristid Lindenmayer• Parallel grammar: every non-terminal symbol is

rewritten simultaneously to get the next object• Models the growth of plants• Typically self-similar at different scales (like fractals)

L-systems, definition

• Grammar G = (V, , P) where– V is a set of symbols (terminals or non-terminals)– is a symbol or sequence of symbols that is the start– P is a set of production rules (rewriting rules)

Symbols that do not occur on the left-hand side of a production rule are terminals (constants)

L-systems, example 1

• Algae:– Variables: A B– Start: A– Rules: A AB , B A

• Start: A• 1st generation: AB• 2nd generation: ABA• 3rd generation: ABAAB• 4th Generation: ABAABABA• 5th Generation: ABAABABAABAAB

Intuition: - A is mature and spawns a young (B), but also stays- It takes young B one generation to mature (become A)

tree L-system

• Non-deterministic version of algae:– Variables: A B C– Start: A– Rules: A AB , B A , B C , C A

• Start: A• 1st generation: AB• 2nd generation: ABA or ABC• 3rd generation: ABAAB or ABCAB or ABAA or

ABCA• 4th Generation: …

Intuition: - It takes B one or two generations to mature

L-systems, bracket notation

• A tree can be written using brackets [ and ] that enclose branches

Gbranch

branchsubbranch

AB[CD[F]E]G[I]H

• Simple tree:– Variable: S– Start: S– Rule: S S[S]S[S]S

• Start: S• 1st generation: S[S]S[S]S• 2nd generation:

S[S]S[S]S[S[S]S[S]S]S[S]S[S]S[S[S]S[S]S]S[S]S[S]S

• We can interpret the generated strings as drawing instructions like in Turtle Graphics:– S : draw line segment– [ : rotate by 45 degrees– ] : rotate by -45 degrees and use the

state from before the matching [

• S[S]S[S]S

• S[S]S[S]S[S[S]S[S]S]S[S]S[S]S[S[S]S[S]S]S[S]S[S]S

• Another tree:– Variables: A, S– Start: A– Rules: A S[A]S[A]A , S SS

• Start: A• 1st generation: S[A]S[A]A• 2nd generation:

SS[S[A]S[A]A]SS[S[A]S[A]A]S[A]S[A]A

• Example of subapical growth mechanism:– new branches are created at apices (A) only– stems (S) only become longer

L-systems, control mechanisms

• Lineage: transfer of genetic information from an ancestor cell to its descendant cells

• Interactive: information (or nutrients) is exchanged between neighboring cells

• Lineage corresponds to context-free L-systems• Interaction corresponds to context-sensitive

L-systems

L-systems, context-sensitivity

• Context-sensitive L-systems have production rules with not just a single variable on the left side

• A<B>CD … is a production rule to rewrite B if it is preceded by A and succeeded by CD

• If a context-sensitive production rule can be applied and also a context-free rule, the context-sensitive one takes precedence

L-systems, context-sensitivity

• With brackets it gets complicated:a production rule BC<S>G[H]M … can be applied to the string ABC[DE][SG[HI[JK]L]MNO] because bracketed parts to the right may be skipped/ignored

ABC[DE][SG[HI[JK]L]MNO]

B C D EI

LM N O 41

• Context-sensitive L-system:– Start: J [ I ] I [ I ] I [ I ] I– Rule: J < I J

• Models a signal from the root upwards to the apices

• Context-sensitive L-system:– Start: I [ I ] I [ I ] I [ I ] J– Rule: I > J J

• Models a signal from an apex downwards to the root

L-systems, signals

• Example where the root structure and branch structure develop simultaneously (top)

• Same L-system but starting with a severed branch

L-systems, geometry

• Branching as given by production rules is topological; no coordinates are involved yet

• We add symbols in the production rules that steer the process, they are ignored when applying rules

• The symbol “[” pushes a situation onto a stack, the symbol “]” recovers it (meaning: the branch has been drawn; continue with the stem)

L-systems, examples 3a and 3b

• Example: + rotates by +45 degrees, - by -45 degrees, variable S moves forward a fixed distance and draws

• Recall the rule S S[S]S[S]S

S S[+S]S[+S]S S S[-S]S[+S]S

• Grassy plant:– Variables: X F– Start: X – Rules:

X F-[[X]+X]+F[+FX]-X , F FF

• + uses an angle of 25°; - uses an angle of -25°;F draws; X does not

6th generation

L-systems, 3D models

• The drawing state in 3D requires a position and a yaw, pitch and roll

• In 2D we had 2 rotation symbols, in 3D we need 6

• The symbol “[” pushes a state with position, yaw, pitch and roll onto a stack, the symbol “]” recovers it

L-systems, appearance

• More symbols can be used to control color, and increase/decrease edge length and edge thickness

length

thickness

Extensions to L-systems

• Stochastic L-systems: – A rule is applied with a certain probability in every

generation– When multiple rules can be used for the same symbol, we

can assign relative probabilities describing which one to apply (cp. non-deterministic L-systems)

• Example:– Variable: A– Start: A– Rules: A(0.6) A[+A][-A]A , A(0.4) AA

50full sparse

Extensions to L-systems

• Differential L-systems (Prusinkiewicz, Hammel, Mjolsness 1993): – Rules as in L-systems combined with differential equations– Separates discrete changes (new branch) from continuous

changes (lengthening, thickening, bending)– Allows continuous growth, instead of discrete– Allows easier animations

Model synthesis

• Procedural modeling algorithm by Merrell and Manocha (2011)

• Useful for man-made structures• User-assisted generation• Starts with a simple model, user-defined input• Generates similar models with larger complexity• Uses neighborhoods and constraints

Model synthesis: approach

• A model is a set of closed polyhedra• The input model is denoted by E, the generated

model by M• Vertices in E have a neighborhood; all vertices in M

should have a neighborhood that is the same as some vertex in E

Neighborhood of a vertex: adjacency of edges and facets around the vertex, directions of those edges and facets, and side of the polyhedron w.r.t. the vertex

• Also points on edges and facets have neighborhoods54

input model E generated model M

• The neighborhood of a vertex defines its state• Given an example model E:– Determine all different occurring states in E– Generate sets of parallel planes (planes supporting facets

from E), these form a 3D grid G– Loop: pick a vertex of G and assign it a possible state

(using states that occur in E and the previously assigned states of adjacent vertices in G)

– Until all vertices with an assigned state form a proper model in G

parallel lines (planes)

At a yellow vertex, we can have 16 different states in principle,2 of which do not give a vertex (all 4 incident cells inside/outside)

4 states occur in model E (like a’, b’, c’, ..) as vertices, 2 of them as edges (like e’, ..)

The remaining 8 states are not allowed at vertices in M

• States described by half-planes: h1 = below green facet; h2 = behind blue facet; h3 = right of red facet

next vertex along this edge must have state that includes h1 h2

• The algorithm again:– Determine all different occurring states in E– Generate sets of parallel planes (planes supporting facets

from E), these form a 3D grid G– Loop: pick a vertex of G and assign it a possible state

(using states that occur in E and the previously assigned states of adjacent vertices in G)

– Until all vertices with an assigned state form a proper model in G

a model growing approach with backtracking when the algorithm finds a vertex where no state is possible

• Constraints are needed to capture the user’s intent• Constraints limit what the algorithm can generate

• Dimensional constraints: limits on the lengths of features (width of roads, height of chairs, …)– all lengths will be multiples of the chosen plane spacing

• Algebraic constraints: limits on ratios of lengths

• Connectivity constraints: roads must all connect into a network; rooms in a house must all be connected– number of seeds influences number of components

• Large-scale constraints: more global constraints like the arrangement of buildings in a city

• Large-scale constraints– make locations in the space to be occupied– the state of a location is the object that is there– states are assigned with equal probabilities, but this can

also be influenced by the user

Model synthesis: output

Model synthesis: efficiency

• Number of planes depends on number of plane normals in input and on plane spacing vs. extent

• Backtracking is limited, and local in practice• m normals and n planes per normal gives

proportional to m3 n3 vertices in the grid• Models shown have 60–400 polygons user-made

(in E) and 1000–8000 polygons generated (in M)• Computation time a few seconds to a few minutes

Model synthesis: limitations

• Needs a grid structure for the model• Not good for organic models (plants, trees)• Less good for curved models, but possible• Less good for a combination of small (detailed) and

large objects• Semi-automatic

Summary

• Procedural generation allows models to be generated rather than built manually by artists or reconstructed from scans

• There are considerable time and cost savings• L-systems work well for organic shapes, but they also

work for buildings, road networks, …• Model synthesis is a model growing technique that

can be used for semi-automatic generation of man-made structures

Summary

• There is more to be said on placement of a model• Clouds, fire, movement like leaves blowing in the

wind, etc., has also been studied• Procedural generation of textures exists• There are various books on the topic• There are several software packages for procedural

generation• Convenient user control remains an issue; there is a

lot of trial-and-error in the generation

Procedural Modeling Slides for DDM by Marc van Kreveld 1.

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