Motion Graphs By Lucas Kovar, Michael Gleicher, and Frederic Pighin Presented by Phil Harton

Overview  What  Why  Creating  Using

What are motion graphs?  Directed graph representing a roadmap of motion data for a character Edges are clips of motion, some from the original motion capture, some generated as transitions A vertex defines 2 sets of motion clips where motions from one set can flow seamlessly into motions from the other

A simple motion graph Walk2 Walk1 Run2/Jog Jog/Walk1 Run1 Run2 Walk1/Walk1 Run2/Run2 Jog Walk1Walk2 Jog Run1 Run2 Original motion data Walk2/Walk1

Why do we want them?  To better utilize motion capture data because it’s: Difficult to modify Time consuming and expensive Hard to use in animations without having captured the exact motion desired

Building motion graphs  Identify transition candidates  Select transition points  Eliminate problematic edges

Identify transition candidates  For each frame A, calculate its distance to each other frame B by basically measuring volume displacement  Use a weighted point cloud formed over a window of k frames ahead of A and behind B, ideally from the character mesh  Calculate the minimal weighted sum of squared distances between corresponding points, given that a rigid 2D transformation may be applied to the second point cloud

Identify transition candidates

Select transition points  The previous step gave us all the local minima of the distance function for each pair of points  Now we simply define a threshold and cut transition candidates with errors above it  May be done with or without intervention  Threshold level depends on type of motion – eg. walking vs. ballet

Create transitions  For each pair of frames A i and B j which fell under the distance error threshold, blend A i through A i+k-1 with B j through B j-k+1 Align frames with appropriate rigid 2D transformation Use linear interpolation to blend root positions Use spherical linear interpolation to blend joint rotations  Treat constraints as binary flags – frames in first half of transition use A’s constraints, second half use B’s  Transition tagged with union of A and B’s labels

Create transitions  Blend weight function  Root position interpolation  Joint rotation interpolation

Eliminate problematic edges  We want to get rid of: Dead ends – not part of a cycle Sinks – part of one or more cycles but only able to reach a small fraction of the nodes Logical discontinuities – eg. boxing motion forced to transition into ballet motion  Goal is to be able to generate arbitrarily long streams of motion of the same type

Eliminate problematic edges  Each frame is associated with a set of 0 or more labels  For each unique set of labels, form a subgraph of edges whose frames have exactly this set  Find the strongly connected components (SCCs) – maximal set of nodes where there is a connecting graph walk between each ordered pair  Discard all edges that aren’t in the largest SCC  Give warning if: A set of labels has below a certain threshold of frames For any ordered pair of SCCs there is no way to get from the first to the second

Using motion graphs  We have a database of motion segments and mappings between them, now we want to find motion streams that conform to user specifications  This is approached as a search problem, where the user specifies a non-negative scalar error function g(w,e) as well as a halting condition  Total error of a path w defined as:

Searching  Goal is to find a complete graph walk that minimizes f(w)  Use branch and bound – keep track of the best complete graph walk w opt and cut current branch when the error exceeds f(w opt )  Works best when a tight lower bound is found early  Use a greedy ordering heuristic – for a set of unexplored child nodes, select the one that minimizes g(w,c)

Searching  Even with branch & bound and ordering heuristic, search is still exponential  Trade some optimality for speed by searching incrementally  Find optimal graph walk of n frames, retain first m nodes and search again from the last retained node  Their implementation used for n (2.67 to 4 sec), for m (~1 sec)

Defining optimization criteria  So how do you define g(w,e) to find your desired motion??  First, two guidelines: g should give some sort of guidance throughout the motion, not just evaluate the end result g should be no more restrictive than necessary – balance guiding search to a particular result with allowing it to consider many options

Path synthesis  The example application they present is to make a character travel along a specified path on the ground  Basic strategy for g is to measure difference between actual path traveled and desired path  P(s) is the point on P whose arc-length distance from the start is s  s(e i ) is the arc-length from frame 0 to frame i in edge e

Path synthesis  Potential problem if character stands still with no incentive to move, thereby accruing zero error Fix this by replacing s(e i ) with t(e i ), which forces a small amount of progress with each frame  If we wish to require different types of motion, break the path into segments by type of motion desired during each part  If character is within threshold distance from end of current path, allow search to consider motion edges of both types, otherwise just the type of the current path  Allow only one type switch per path

Path synthesis applications  Interactive control User controls a character, motion clips are selected in real time  High-level keyframing Animator decides what the character should do and where, rather than tedious keyframing  Motion dumping Same as the first, but with AI characters  Crowds Paths are generated for multiple characters to avoid collisions

Questions?