Implementation of a momentum-based distance metric for motion graphs. Student: Alessandro Di Domenico (st.no ), Supervisor: Nicolas Pronost

Transcription

1 Implementation of a momentum-baed ditance metric for motion graph Student: Aleandro Di Domenico (t.no ), Supervior: Nicola Pronot April 3, 2014

2 Abtract Thi report preent the procedure and reult of the performed experiment relative to the contruction of a motion graph for character animation, uing a momentum-baed extenion of a kinematic-baed ditance metric. The graph i automatically generated from an input et of motion clip and ued to animate a human-like character in a virtual environment. The aim of the experiment i to generate motion graph with the new metric and evaluate their performance. The reult how that the new hybrid ditance metric offer improvement with repect to the kinematic-baed one, in term of average ditance between the poe in the tranition, while the kinematic-baed ditance metric perform better in term of graph denity.

3 Content 1 Introduction 2 2 Related work 3 3 Momentum-baed metric Method Experiment Point cloud ditance metric Momentum-baed ditance metric Hybrid ditance metric Concluion 17 1

4 1 Introduction Today technology allow the creation of extremely realitic and interactive virtual world. To preerve the illuion and enhance the ene of immerion in thee environment, the need come for the character that inhabit them to be animated in the mot authentic way poible. Animating human-like model repreent a huge challenge, becaue taking in conideration all the ubtletie of human movement, neceary to obtain atifying and convincing reult, i often hard and time-conuming. Many approache have been propoed to deal with thi problem; one of the mot widely ued animation technique i motion capture, which allow the recording of a peron performing pecific action. Motion capture i a reliable way to reproduce human movement but come with one big downide: editing the motion capture data, aide from mall change, ha proven to be very difficult. If the recorded data do not meet the quality requirement expected by the uer, the only olution i another motion capture eion; thi greatly increae cot and time conumption. Kovar et al. [2002] preented a method that allow the creation of new tream of motion baed on a et of pre-recorded movement while preerving the overall quality of the original data. The reult i a tructure called a motion graph, which allow the rearrangement of the original motion clip in different way, thu allowing the creation of new equence. In order to contruct thee new movement, the et of original data mut be examined for tranition point, i.e. point in the clip in which the character motion are ufficiently imilar between each other. One efficient way to determine uch imilarity i the comparion of the character poe: if the poe are cloe enough, blending technique can be ued to create a tranition between the data egment. In thi report we propoe a different method to find tranition point between clip: intead of jut uing a kinematic-baed metric to evaluate the poe imilaritie between the character, we implemented a phyic-baed extenion that will evaluate the angular and linear momentum of the character and etablih the preence of a imilarity according to the ditance between thee value. The purpoe i to compare the graph obtained uing the thee different metric and analyze the reult in term of connectivity and ditance between the poe in the tranition phae. The remainder of the report i organized a follow. In ection 2 we decribe related work. In ection 3 we decribe the method in detail and preent the experiment reult. In Section 4 we draw concluion and ugget area of improvement and future work. 2

5 2 Related work Motion graph contruction require the ue of a ditance metric to individuate good candidate for the tranition and many different metric have been propoed. In Kovar et al. [2002], detection of candidate tranition point i performed uing a point cloud ditance metric. In Gleicher et al. [2003], the author ue point cloud to compute ditance between frame of different clip. Point cloud are formed attaching marker to the keleton of the character in the two candidate frame and in a mall window of frame adjacent to them; ubequently, the weighted um of quared ditance between the point forming the point cloud i determined and tored in a databae. Tranition point are elected by earching for local minima in a function that aociate every frame with it ditance toward other frame of a motion clip. Thee minima are evaluated againt a uer-defined threhold and if they are below uch threhold, a tranition can be etablihed between the examined frame. In Arikan and Foryth [2002] a metric i defined by calculating the difference between joint poition and velocitie, a well a the difference between the toro velocitie and acceleration. Thee value are then ued to calculate a weight that i aigned to the edge of the motion graph; the weight repreent the degree to which two frame can be concatenated. A imilar metric i ued in Lee et al. [2002], where a ditance function i computed a the um of two term, decribing the weighted difference of joint angle and the weighted difference of joint velocitie. Subequently, the function i ued to calculate the expectation of tranition from one frame to another, by mean of a probability function which take the previouly calculated ditance function a input. Forbe and Fiume [2005] propoe a different way to calculate the ditance between frame. The animation dataet i tranformed with the Principal Component Analyi technique; the reulting principal component vector repreent the axe in pace. Point are repreented by a combination of thee principal component thu allowing the decription of different poe. The ditance i then computed a a caled L2 norm of thee PC coordinate. The ame concept i alo ued in Egge et al. [2004]. Li et al. [2008] take a different approach on the evaluation of tranition point in motion capture data. Uing the conjecture that in the comparion of two imilar trajectorie, the mot natural looking i the one that implie le effort, they develop a new metric to meaure aid effort. They are then able to compute an L-core for the different motion that allow them to elect the optimal tranition point. 3

6 The idea of uing linear and angular momentum for motion ynthei i not new; In Abe et al. [2004] the author introduce a ytem for the editing of highly dynamic character motion that allow animator to create mooth animation maintaining at the ame time a good amount of interactive control over the character. The preervation of the behaviour of the input motion i guaranteed by a technique that matche the linear and angular momentum pattern of the data; by olving a contrained optimization problem for every pline that i ued to define the momentum characteritic of a motion, a et of control point and knot are computed. The point and knot are ued to make ure that the edited motion follow the momentum path of the original data, thu reulting in phyically realitic movement. 4

7 3 Momentum-baed metric 3.1 Method The purpoe of thi experiment wa the evaluation of the performance of a motion graph contructed with a metric that wa not only baed on kinematic parameter and poe imilarity. We initially implemented one baed on the difference between a character linear and angular momentum in different poe and we tudied the quality of the reulting motion graph. The deciion to alo reearch on a metric that wa a combination of both the point cloud and the momentum-baed one came from the obervation that while two frame may have imilar value of momentum, it i not neceary true that they correpond to imilar movement of a character. In fact, angular and linear momentum are phyical propertie of moving bodie but, by themelve, they do not offer a viual decription of a performed movement; linear momentum i a vector quantity defined a the product of the body ma with it velocity. It quantifie the force that i neceary to top a moving body in the unit of time. It analogou quantity for patial rotation i the angular momentum; the angular momentum of a body i defined a the product of a body moment of inertia with it angular velocity; it meaure the tendency of a rotating body to perit in it rotary motion. Thu, it appear evident the need to incorporate ome ort of poe recognition phae while building the tranition between clip. The implementation wa followed by an experimentation phae in which we compared the outcome of the proce uing three different configuration: the firt ued the original point cloud ditance metric; the econd configuration wa purely baed on the momentum difference in the animation frame; finally, the third configuration wa the combination of the firt two, with the ue of different uer-defined threhold. The initial tage of the implementation proce wa focued on gathering and examining relevant tudie on animation technique and documentation about calculation of linear and angular momentum of human bodie. Thi wa followed by a tructural analyi of the given framework aimed at undertanding how to modify the already exiting ytem in order to ubtitute the former ditance metric with the new one. In fact, the framework wa already provided with a module for the creation of a motion graph generated from a et of motion clip, uing the point cloud ditance metric. Prior to the actual implementation of the momentum-baed metric, we pent ome time analyzing the point cloud ditance metric to better undertand it operating principle. Soon it appeared evident that the mot traightforward way to develop the new ditance metric would be to et up a tructure to attach to the character keleton in order to keep track of the phyical propertie that 5

8 were being conidered; thu a phyicskeleton tructure wa created. Thi tructure mainly acted a container of phyicbone; each bone wa approximated a a cylinder object with everal defining attribute attached to it. Namely, every bone had a name, a ma, a width and a length that were ued to calculate their moment of inertia and center of ma. Moment of inertia i a phyical property of a body that define it reitance to a variation of angular velocity about a certain rotation axi. The phyicskeleton contained a many bone a the number of body part of which the character keleton i compoed, a total of twenty-even; the inertia tenor of each bone i given by: 1 12 m(3r2 + h 2 ) + mrx I = m(3r2 + h 2 ) + mry mr2 + mrz 2 The ma and girth data for the limb were extrapolated by experimental data found in Plagenhoef et al. [1983]. The whole character had a ma of 75 kg, ditributed along the part that compoed it; every part had a weight aociated to it that wa ued to determine it ma. We ued the pre-exitent code that already took care of aligning the character poition in order to create a coherent animation, with a major ditinction. In the original code, the tranition were generated only if the computed point cloud ditance wa below a certain uer-defined threhold; thi mechanim acted a a filter and allowed the creation of a relatively mall databae. The new verion, intead, created every poible tranition between all the input frame and the reulting databae wa much bigger than before; thi erved for the contruction of a unique databae that could be ued with the three different metric, intead of building a databae for each of them. The filtering wa thu moved to a later phae. In detail, the whole experimental proce can be divided in the following four point: 1. The motion file were loaded into the framework, which wa alo ued to et the virtual environment in which the character moved and it correponding phyicskeleton object. 2. Subequently, the file were examined frame by frame: for each of them, linear and angular momentum of the character were computed. Each frame of the firt motion wa compared to each frame of the econd one and the meaured difference wa temporary tored. 3. After thi tage, the point to build the point cloud were gathered, for every frame of both the file, and the point cloud ditance wa tored. 6

9 4. Finally, the tranlation and rotation data for the model alignment were calculated and the point cloud ditance, the linear momentum ditance and the angular momentum ditance were tored in the databae. Formula ued for the calculation of the momentum are: and p = mv L = I ω The firt one refer to the linear momentum, while the econd one refer to the angular momentum. Since the linear and angular momentum are trictly dependent on the linear and angular velocity,repectively, they were calculated frame by frame a : and v(t 2 ) = x(t 2 ) x(t 1 ) t ω(t 2 ) = θ(t 2 ) θ(t 1 ) t The final databae contained, a precedently tated, all the poible tranition between the frame of the two motion file; once it wa completed, we built the motion graph from it. In order to do o, we ued different combination of three different threhold; thee threhold, applied on the databae, filtered out the tranition and the reulting ubet wa ued to build the graph. 7

10 3.2 Experiment The framework run on a pc that mount a 2.40 GHz dual core proceor, 8 GB of ram, in a Window environment with the Viual Studio IDE. The virtual environment ued for the imulation i compoed of a quare plane on which a tylized human model i able to walk. Figure 1 how u the environment and the model ued in the experiment. Figure 1: A cloe-up of the model ued in the experiment In order to obtain a motion graph of adequate dimenion, a total of 10 animation were ued; the animation how a character walking at different gait peed. The output databae i compoed by a total of 910 MB worth of data. It tructure i hown in Figure 2 and Figure 3. The former how the animation table with the max timekey column being the duration time of each video; the latter how the tranition table in which we can ee the column ditance, which i referring to the point cloud ditance, and the column that indicate to the linear momentum ditance and the angular momentum ditance. The column from timekey and to timekey point out the keyframe in the motion in which the tranition happen; from thi image it can be noticed that each tranition i repeated in two direction, becaue it can happen in both way. Thi deign choice wa already part of the framework and we decided not to modify it. A previouly pecified, the caue for the big ize of the databae i 8

11 that it contain all the poible tranition between the different frame compoing the motion clip. We choe the approach of filtering valid tranition in a econd tage for reaon of computing peed. In fact, the creation of different databae for the three different metric (point cloud, momentum-baed and hybrid) would have taken more time than the about 150 hour of calculation required for one big comprehenive databae. Figure 2: The tranition table of the output databae Figure 3: The animation table of the output databae The experiment wa conducted a follow: after the creation of the motion graph, the reulting animation ran while a background proce calculated the average ditance between the frame choen for the tranition. Thi wa neceary becaue the framework did not include any motion blending, making the evaluation of naturalne of the motion very difficult. The average ditance allowed u to judge the performance of the different metric with objective data. Secondly, the graph denity wa meaured and compared for the different 9

12 threhold and metric; a dene graph i a graph in which the number of edge i cloe to the maximal number of edge and graph denity i the property that define if a graph i a dene or pare one, with value in the interval [0,1]. The dener the graph i, the more connected are the different motion clip. Graph denity i calculated a: 2 E D = V ( V 1 ) where E i the number of edge and V i the number of vertice of the graph. The reult howed an advantage of the hybrid metric over the other two that were teted in term of average ditance. The time needed to build a motion graph varied greatly according to the threhold value elected by the uer and by the type of metric utilized, ranging between 80 and econd. The average ditance conidered for the reult repreent the ditance between the poe, in meter; the maller it i, the more imilar are the poe, which lead to better blending reult. 10

13 3.2.1 Point cloud ditance metric Firt of all, the experiment wa carried out on the point cloud ditance metric; thu the only parameter that wa modified wa the point cloud ditance threhold. The value of thi threhold ranged between 0.003m and m, with a pace of m, for a total of 10 different value. The average time to build the motion graph wa 2977 econd, with the value toward the end of the interval taking coniderably more time than thoe at the beginning; a bigger threhold mean more tranition and thu a bigger build time for the graph. The reult oberved uing thi metric howed that the average ditance between the tranition frame increaed with the growth of the threhold value, ranging from m to m. The graph denity i alo in a relationhip of direct proportionality with the threhold, with higher threhold having a higher value for the edge denity parameter, a a reult of the le retrictive condition to create a tranition. The reult are howed in the following chart Average Ditance AverageDitance [m] T hrehold[m] 10 2 Figure 4: Average ditance chart that how the growth of the average ditance with the threhold for the point cloud ditance metric. From the reult we can deduce that low value of the threhold will guarantee better reult in term of viual quality, but with more retriction on the number of animation that can be created. 11

14 Edge Denity 0.15 EdgeDenity T hrehold[m] 10 2 Figure 5: Edge denity chart that how the growth of the edge denity with the threhold for the point cloud ditance metric Momentum-baed ditance metric The point cloud ditance metric experiment wa followed by the experiment on the new momentum-baed metric. In thi cae, the variable were repreented by the linear momentum ditance and the angular momentum ditance threhold. The former varied between 10 Kgm2 and 100 Kgm2, with a pace of 10 Kgm2, while the latter went from 1.5 Kgm to 15 Kgm, with a pace of 1.5 Kgm. Every value of the linear momentum threhold wa teted with every value of the angular momentum threhold. The average time to contruct the graph wa about 6448 econd, more than twice the time required by the point ditance metric. Additionally, an unforeeen low memory error wa raied when the value of the linear momentum ditance threhold wa et between 80 Kgm2 and 100 Kgm2. For thi reaon, ome of the cae were conidered incomplete and dicarded in the final evaluation. The reult oberved with thi metric can be conidered quite unatifactory; the following chart how the reult when the angular momentum threhold i fixed at 1.5 Kgm. 12

15 Average Ditance Momentum-baed metric AverageDitance [m] LinearMomentumT hrehold[ Kgm2 ] Figure 6: Average ditance chart that how the reduction of the average ditance with the growth of threhold for the momentum-baed ditance metric. From the chart it can be noticed that, in thi cae, the average ditance between the poe decreae a the momentum threhold grow. In addition to that, if we look at the number, we can actually ee that even for the threhold that have the lowet average ditance, the actual value i much higher with repect to the point cloud ditance metric value. Obviouly thi i not the behaviour that we were looking for and it confirm the idea that a pure momentum-baed ditance metric i not adequate to build motion graph that reult in atifying animation. On the contrary, if we take a look at the data for the edge denity, we can ee that the beaviour doe not differ from the expected one, with bigger threhold having bigger denity value. In thi chart we how the reult when the angular momentum threhold i fixed at 1.5 Kgm Hybrid ditance metric Finally, we teted the lat ditance metric, a combination of the firt two. In practical term, thi mean that the tranition were created uing the point cloud ditance threhold and the momentum ditance threhold together. A in the previou experiment, the value of the threhold varied between 0.003m and m for the point cloud ditance threhold, between 10 Kgm2 100 Kgm2 15 Kgm and for the linear momentum ditance threhold and between 1.5 Kgm and for the angular momentum ditance threhold. Thi brought the final number of teted value to 100. An excerpt of the collected data can be een in Figure 8. 13

16 10 2 Edge Denity 2.1 EdgeDenity LinearMomentumT hrehold[ Kgm2 ] Figure 7: Edge denity chart that how the growth of the edge denity with the linear momentum threhold and a fixed threholf for the angular momentum. The average time for the graph contruction wa econd. Generally we noticed a reduction of the average ditance between tranition; comparing it with the point cloud ditance metric, we found that in the bet cae, in which the reult howed the lowet average ditance for both the metric, the percentage reduction wa up to 19%. Comparing the hybrid metric with the momentumbaed metric howed that, in the bet cae, the contraction wa up to 97%. The following chart how the reult when the linear momentum threhold i fixed at 10 Kgm2 and the angular momentum threhold i at 1.5 Kgm, and compare them with the point cloud ditance curve howed before: 14

17 Figure 8: Value for the threhold ued in the generation of the motion graph for the hybrid ditance metric Average Ditance Hybrid Metric Average Ditance Point Cloud Metric AverageDitance [m] P ointcloudt hrehold[m] 10 2 Figure 9: Average ditance chart that compare the point cloud ditance metric curve with the hybrid ditance metric curve. We can clearly ee the advantage of the hybrid ditance metric curve that alway tay under the point cloud ditance metric one. Unfortunately, we did not find the ame poitive reult in term of graph denity: if we look at the chart for the graph denity with the ame parameter, we can ee omething 15

18 unexpected: 0.15 Edge Denity Point Cloud Metric Edge Denity Hybrid Metric EdgeDenity T hrehold[m] 10 2 Figure 10: Edge denity chart that compare the edge denity curve for the point cloud ditance metric and the hybrid ditance metric. We can immediately notice that the denity curve for the hybrid ditance metric i decreaing, intead of increaing like it would be expected. In it maximum value, the hybrid curve i till well under the maximum denity value reached by the point cloud ditance, which put the hybrid ditance metric at diadvantage. Comparing the number, we actually have a cutback of about 18.85%. The ame can be oberved if we compare the hybrid ditance metric with the pure momentum-baed one: the contraction reache the value of 70%, in the bet cae, when the denity value are the highet for both of the metric. The reaon for thi behaviour are till not clear and are being invetigated. 16

19 4 Concluion In concluion, a ditance metric baed on phyical propertie of a moving character wa developed and teted to evaluate poible improvement againt one of the metric that are currently ued to generate motion graph. In particular, it wa compared to the point-cloud ditance metric, a technique alo dicued by Mémoli and Sapiro [2004]. The analyi howed that a pure momentum-baed metric by itelf did not produce intereting reult; the main problem wa that angular and linear momentum are propertie that, by themelve, do not decribe movement in a way that i uitable to the tak of building a motion graph. However, the implementation of a metric that take into account both the ditance between the poe and the imilaritie in momentum howed promiing reult that could help improving the final quality of the animation. Further reearch hould be aimed at improving the graph connectivity, which would guarantee more freedom in the creation of different animation. Moreover, further phyical propertie could be ued to increae the preciion of the metric, like torque and force; alo, it would be worth exploring different kind of technique for poe recognition and doing a comparative tudy on the quality of the reulting animation. 17

20 Reference Yeuhi Abe, C. Karen Liu, and Zoran Popović. Momentum-baed parameterization of dynamic character motion. In Proceeding of the 2004 ACM SIGGRAPH/Eurographic ympoium on Computer animation, SCA 04, page , Aire-la-Ville, Switzerland, Switzerland, Eurographic Aociation. ISBN doi: / URL Okan Arikan and D. A. Foryth. Interactive motion generation from example. ACM Tran. Graph., 21(3): , July ISSN doi: / URL Arjan Egge, Tom Molet, and Nadia Magnenat-Thalmann. Peronalied realtime idle motion ynthei. In Computer Graphic and Application, PG Proceeding. 12th Pacific Conference on, page IEEE, K. Forbe and E. Fiume. An efficient earch algorithm for motion data uing weighted pca. In Proceeding of the 2005 ACM SIGGRAPH/Eurographic Sympoium on Computer Animation, SCA 05, page 67 76, New York, NY, USA, ACM. ISBN doi: / URL Michael Gleicher, Hyun Joon Shin, Luca Kovar, and Andrew Jepen. Snaptogether motion: aembling run-time animation. SIGGRAPH 08, page 52:1 52:9, New York, NY, USA, ACM. doi: / URL Luca Kovar, Michael Gleicher, and Frédéric Pighin. Motion graph. SIGGRAPH 02, page 51:1 51:10, New York, NY, USA, ACM. doi: / URL Jehee Lee, Jinxiang Chai, Paul S. A. Reitma, Jeica K. Hodgin, and Nancy S. Pollard. Interactive control of avatar animated with human motion data. ACM Tran. Graph., 21(3): , July ISSN doi: / URL Lei Li, Jame McCann, Chrito Falouto, and Nancy Pollard. Lazine i a virtue: Motion titching uing effort minimization. Computer Science Department, Facundo Mémoli and Guillermo Sapiro. Comparing point cloud. In Proceeding of the 2004 Eurographic/ACM SIGGRAPH ympoium on Geometry proceing, SGP 04, page 32 40, New York, NY, USA, ACM. ISBN doi: / URL 18

21 S. Plagenhoef, F.G. Evan, and T. Abdelnour. Anatomical data for analyzing human motion,