# Hand gesture analysis

In this tutorial we will learn how to use Kendallβs shape space to analyze hand gesture data.

using Manifolds, CSV, DataFrames, Plots, MultivariateStats

Our first function loads dataset of hand gestures, described here.

function load_hands()
hands_url = "https://raw.githubusercontent.com/geomstats/geomstats/master/geomstats/datasets/data/hands/hands.txt"
hand_labels_url = "https://raw.githubusercontent.com/geomstats/geomstats/master/geomstats/datasets/data/hands/labels.txt"

hands = reshape(hands, size(hands, 1), 3, 22)
return hands, hand_labels
end
load_hands (generic function with 1 method)

The following code plots a sample gesture as a 3D scatter plot of points.

hands, hand_labels = load_hands()
scatter3d(hands[1, 1, :], hands[1, 2, :], hands[1, 3, :])

Each gesture is represented by 22 landmarks in $βΒ³$, so we use the appropriate Kendallβs shape space

Mshape = KendallsShapeSpace(3, 22)
KendallsShapeSpace(3, 22)

Hands read from the dataset are projected to the shape space to remove translation and scaling variability. Rotational variability is then handled using the quotient structure of KendallsShapeSpace

hands_projected = [project(Mshape, hands[i, :, :]) for i in axes(hands, 1)]

In the next part letβs do tangent space PCA. This starts with computing a mean point and computing logithmic maps at mean to each point in the dataset.

mean_hand = mean(Mshape, hands_projected)
hand_logs = [log(Mshape, mean_hand, p) for p in hands_projected]

For a tangent PCA, we need coordinates in a basis. Some libraries skip this step because the representation of tangent vectors forms a linear subspace of an Euclidean space so PCA automatically detects which directions have no variance but this is a more generic way to solve this issue.

B = get_basis(Mshape, mean_hand, ProjectedOrthonormalBasis(:svd))
hand_log_coordinates = [get_coordinates(Mshape, mean_hand, X, B) for X in hand_logs]

This code prepares data for MultivariateStats β mean=0 is set because weβve centered the data geometrically to mean_hand in the code above.

red_coords = reduce(hcat, hand_log_coordinates)
fp = fit(PCA, red_coords; mean=0)
PCA(indim = 59, outdim = 19, principalratio = 0.991345477862632)

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PC1           PC2           PC3           PC4           PC5           PC6           PC7           PC8           PC9          PC10          PC11          PC12          PC13          PC14          PC15          PC16          PC17          PC18          PC19
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1    0.0225725    -0.025823      0.00247119   -0.00194504    0.011187      0.00865122   -0.00432       0.00232697   -0.00133113   -0.00603743    0.00279173    0.00459607    0.0026104    -0.00194346   -0.0010809     0.00371279   -0.0026941    -0.00336716   -4.78115e-6
2   -0.0298018     0.012855     -0.0253788    -0.0107526    -0.000642301   0.00243343   -0.0129186     0.0117729     0.0159723     0.00107763   -0.00131555    0.00285026    0.00463989   -0.00216125    0.00161956    0.00105094    0.00029935   -0.000818863   0.000226847
3   -0.00496432   -0.0149723    -0.00377204    0.0160204    -0.0052388     0.00212934   -0.00568703   -0.00338036    0.000889778  -0.00214546   -0.000896225  -0.00137907   -0.00247179    0.00239153   -0.000377108  -0.00215105    0.0027909     0.000117751  -0.00127213
4    0.0056335    -0.0129395     0.00752466    0.00555847   -0.00280322    0.00782194    0.00111858    0.0085054    -0.00273334   -0.00581609    0.00450121   -0.00038663    0.00167242    0.00231527    0.000307966  -0.00131181    0.000720744   0.00111114    0.00361935
5   -0.0113838     0.0232298    -0.00547126   -0.0179074    -0.00225183   -0.0123001    -0.00379069    0.00375553   -0.0110292     0.00304722   -0.00459228   -0.00477702   -0.000442113   0.00166295    0.00238507    0.00489784   -0.00261756    0.00101093   -0.00178457
6   -0.0345859    -0.0233477     0.0296546     0.00689836    0.003326     -0.00135883   -0.0119614     0.00329535   -0.00698649   -0.00493814    0.00376636    0.000613432  -0.00149234   -0.000250127   0.00623361    0.00254364   -0.000999917   9.66511e-6   -0.000225613
7   -0.0374342    -0.0209963     0.00235653    0.000683237   0.00758501   -0.00999398    0.00657121    0.00782893   -0.00406903   -0.00581627   -0.00079722    0.0041202     0.00319899    0.00585783   -0.00167581    0.0030249     0.00322473   -0.00108314   -0.00116042
8    0.02496       0.00943876    0.011668     -0.00772333    0.00960598   -0.00142074    0.00211964   -0.000840499  -0.000234754  -0.000749657   0.000307986   0.0066857    -0.000133384   0.00300887    0.0022704     0.00120301    0.00203841    0.000966472   0.00120561
9    0.0296003     0.0472131     0.00210213   -0.00393545   -0.00142636   -0.00236192   -0.00533011   -0.00345383   -0.00439341   -0.00852948   -0.000396924  -0.000150678   0.00330386    0.00089967    0.00388721   -0.00199243   -0.00161544    0.00329972   -0.00160867
10   0.0187554     0.00337061    0.0254247     0.00182716   -0.00726025    0.00345359    0.0118753     0.00392614   -0.00737372   -0.000261283  -0.0044591     0.000623762  -0.00123177   -0.00187434    0.00347221    0.00122414    0.00162647    0.000179093  -0.000639519
11  -0.0276189    -0.000868697   0.00124921   -0.0271846    -0.0182645    -0.00654734   -0.00377689   -0.00309552    0.00263585    0.0083263     0.0117277    -0.000277824  -0.0013572    -0.00171603   -0.00248708    0.0002054    -0.00129968    0.000538015   0.000635364
12   0.0742252    -0.0111949    -0.00393113   -0.00375025   -0.00466852    0.00718482   -0.00421926   -0.000133097  -0.00285935    0.00220137   -0.0063239     0.00591546   -0.00391862   -0.00220676   -0.000179801   0.00446115   -0.000848733  -0.00206532    0.00226783
13   0.0426582     0.00910716    0.0199886     0.00477317    0.000139353   0.00719425   -0.0105414    -0.00511285   -0.00217195    0.0106759    -0.000717452   0.00304458   -0.00227829    0.00118983    0.00302885    0.00259039    0.000590182  -0.00113679   -0.0014884
14   0.0473028     0.00651493    0.00140138    0.00225381    0.00439764   -0.00750531    0.0193882    -0.00406276    0.0131133    -0.00935677    0.00405362   -0.00109819   -0.000160323   0.00193398    0.00109956    0.000814692   0.000914761   0.00198064    0.00125932
15   0.0332915     0.0221761     0.0200686    -0.000418236  -0.00238077    0.0136492    -0.00572773   -0.00525892    0.00565712   -0.00186374    0.00193716   -0.00251317    0.00147158   -0.00144082   -0.0036298     0.00233316   -0.000552749  -0.00117945   -0.00143965
16  -0.019766     -0.0126687    -0.0327939    -0.00786485    0.00374512    0.00537198   -0.0105409    -0.0120939    -4.87631e-5   -0.000200912   0.00649485    0.00341607    0.00856351   -0.000627434   0.00512775    0.00157364    0.0025359    -0.000340485   0.00302362
17  -0.0163038     0.00392205    0.00368172   -0.00108868   -0.00403097   -0.00250602   -0.00154047   -0.0091664     0.012208     -0.00212529   -0.00249079    0.00425261   -0.0009888     0.000699026   0.00159838    0.00182519   -0.00268461    0.0024567    -0.00287173
18   0.0315265     0.014339     -0.0281659     0.0188983     0.00116313    0.00517276   -0.012306      0.00964101    0.00378792    0.003776     -0.00471276   -0.00276278    0.00073812   -0.0018049    -0.00110501   -0.00142815   -0.00324852    0.00125496    0.00036833
19  -0.01342      -0.0266179     0.00692541   -0.00394567    0.00696648    0.00546101    0.0107503     0.00161672    0.00243154    0.0111542    -0.00859616   -0.0022637    -0.00125188   -0.00218111    0.00418941   -0.00245702    0.00109089    0.00262105    0.000894704
20  -0.000739408   0.00263129    0.00124919    0.000518372  -0.00405283   -0.00980204   -0.00494474   -0.00280603   -0.00543177   -0.00590718    0.00579551    0.00110744   -0.00812888    0.00441085   -0.00391783    0.00174295   -0.000252033  -0.00209523    0.00204457
21   0.0504932    -0.0110797    -0.0026189    -0.00962761   -0.0153567    -0.00273023    0.00361822    0.00201177   -0.00445139   -0.000379839   0.00688118   -0.000391294  -0.000934333  -0.00118075   -0.00270666   -0.000350291   0.000905076   0.00117337   -0.000755983
22  -0.046736      0.00232599   -0.0170992    -0.00282647   -0.014517      0.00487345    0.00150701   -0.00101275   -0.00475473    0.00292961    0.00419816    0.0025307     0.00131901    0.00431956   -0.00200533   -0.00153549   -0.000850689  -0.000199973  -0.000797035
23  -0.0173763    -0.0135675    -0.0235186    -0.00469928   -0.010083      0.00622843    0.00612574    0.000719204  -0.00473516   -0.00170497   -0.0088133    -0.00339875    0.0030805     0.00313125   -0.00334306    0.00617814    0.00103001    0.00544743    0.000548909
24  -0.00826661    0.0167308     0.00865901   -0.00214922    0.0150335     0.00823526   -0.000297993  -0.00237168   -0.000986393   0.00220198   -0.000681713  -0.000197612   0.000283165   0.00363122    0.000841859  -0.000161184  -0.00456144    0.000377615   0.000890576
25  -0.00644559    0.010877      0.00278186   -0.00911345   -0.0107489     0.0243946     0.00709473    0.0105892    -0.00627612   -0.00400938    0.00573931   -0.000879342  -0.00421579   -0.00280443    0.00183861   -0.00101372    0.00165889   -0.00303974    0.0016519
26   0.0441278     0.0158065    -0.00400205   -0.000325081   0.00203837    0.0080049     0.00759341    0.00350391   -0.00106124    0.000470616   0.000231166   0.000781562   0.000640405   0.00106251   -0.00165851    0.0036525     0.00222514   -0.00223304   -0.00322942
27  -0.0185767     0.0125668     0.0141537     0.0144925     0.00211121   -0.0084411    -0.00693538    0.00588689    0.00509395   -0.000498211  -8.28981e-5   -0.00196856    0.000165433   0.00150428    0.00217742    0.00406355    0.00266332    0.000538881   0.000787166
28  -0.0288949     0.0285503    -0.00938241    0.00314241    0.00451199   -0.000322886   0.0102419    -0.0057291     0.00452652   -0.00434567    0.00333208    0.000768434   0.000697567  -0.000947317   0.00197396    0.00231527   -0.00262045   -0.0052142    -0.0022789
29   0.054652      0.016176      0.0233463    -0.00263117    0.0140462    -0.00251733    0.00410277    0.00636276    0.00827887    0.00558304    0.00735049    0.00273341    0.00209918   -0.00136746   -0.00493028    0.0020373     0.00130772    0.00142239   -0.00115467
30  -0.0234619     0.0175726     0.0109159     0.0120296    -0.00457707   -0.0145693    -7.70199e-5   -0.00427574   -0.00308155    0.0048407    -0.000529341  -0.00208163   -0.00219104   -0.00363319   -0.00608362    0.00641312    7.09267e-5    0.00118223    0.00326152
31  -0.00743661   -0.0210373    -0.0126597    -0.00541652    0.00933005    0.00815691    0.00468841    0.00631952    0.0111953     0.00090758    0.000217408  -0.0062275    -0.00622832    0.000955856   0.00072591    0.00169066   -0.00115975   -0.00147563   -0.00250162
32   0.00942946    0.0348065    -0.00344763    0.00870651   -0.0074523    -0.00813785    0.00974278    0.00503949   -0.0115295     0.00778616    0.00187608   -6.06739e-5    0.0101328    -0.00379749    0.00144605   -0.000102658  -0.00132698   -0.00276075    0.00106114
33   0.0632112    -0.0391171    -0.00200189   -0.0220903    -0.0144734    -0.0113132     0.0077086    -0.000969037   0.00503636    0.00706564    0.00437077    0.00130097    0.00340908    0.0051326     0.00253346    0.000507903  -0.000707207  -0.00056295   -0.00194977
34   0.0207818    -0.0135932     0.000771381   0.0036835     0.0213596    -0.00486239    0.00157098   -0.00771602   -0.00359822   -0.00378203   -0.0016533    -0.0110685    -0.000753667  -0.00229644   -0.00150167   -0.000303523   0.00158846   -0.000607838   0.000615078
35  -0.000379201  -0.0120639     0.0297885    -0.00270901    0.00650958   -0.000380978  -0.00121355    0.0101664    -0.00517949   -0.000558258   0.00197674    0.000277632   0.00797185   -0.000760715  -0.00264005   -0.00348748   -0.00231271    0.00137201    0.0015573
36   0.00115252    0.000667601  -0.000336516   0.0137512    -0.00182112    0.000312716   0.0107091    -0.0131901    -0.000686853   0.00429926   -0.00405865   -0.00152666    0.00779814    0.00289293    0.000712759  -0.00290687    0.000187923  -0.00341138   -0.00143035
37  -0.0710484    -0.0120054    -0.00742916    0.0315282     0.000899437   0.00447448    0.00873192   -0.00168979    0.00408006    0.0081241     0.0111211    -0.00380385   -0.0027588     0.000382961   0.000901206   0.00209142   -0.00297548    0.000369737   0.000194095
38   0.0173624    -0.00401998    0.00476487   -0.000229019   0.00277068    0.0121718    -0.00471613   -0.00762391    0.00490114    6.67001e-5   -0.00184396    0.00294105    0.00142187    0.00170863    0.00048986   -0.000522409   0.00089876    0.00154977   -0.00243131
39  -0.0132061     0.0340923    -0.00256571   -0.00165163   -0.0056104    -0.00854407   -0.00256856    0.00640797    0.00421851    0.00576642   -0.00326446    0.00247372   -0.00086705   -0.003962      0.00141905   -0.000994299   0.00542489    0.000229981  -0.00130346
40   0.0201995     0.00167282   -0.015568      0.000124091   0.00826228    0.00495587    0.0025755     0.0135085     0.0068398    -0.00135169   -0.000719346  -0.000414134   0.00126898   -0.00109902   -0.00309125    0.00299976   -0.00179919    0.000209681   0.00138728
41   0.0311951    -0.00349583   -0.000650444   0.00225274   -0.0202231     0.00133542    0.000267878  -0.00500205    0.00441379    4.99341e-6   -0.00226762   -0.00423897   -0.00153824   -0.00378025   -0.00351691   -0.000162936  -0.0024964     0.000454088   0.000183778
42   0.00295307   -0.00780823    0.00723101    0.0102236    -0.00344839    0.0073579    -0.00253667   -0.00139851   -0.00143191    0.00254803    0.00531096    0.00456018   -0.00235568   -0.00376306   -0.00340022   -0.00172833    0.00121904    0.00142326   -0.000685875
43   0.0194641    -0.0113322    -0.00277425    0.00886678    0.010623     -0.0172909    -0.00366068    0.00715197    0.00562361    0.00432699    0.00736201    0.00317588   -0.00247544   -0.00147035    0.00388555   -0.00220526    0.00103274    0.00163166    0.000317339
44  -0.0404226    -0.0150374     0.0122555    -0.00944404    0.0105142     0.000430679  -0.00492254   -0.0094228     0.0056596     0.000305474  -0.00401212    0.00405241    0.00754805   -0.0016228    -0.00720144    0.000628635   0.00159903   -0.00156151    0.00249425
45   0.019093      0.0161417    -0.0106445     0.00289789   -0.00179217   -0.00983076    0.00218895    0.00481717    0.0085834    -0.00607766    0.000501381  -0.00255306   -0.000130565   0.00217908    0.00346562   -0.000368367   0.00076262   -0.000891547   0.00282931
46   0.0260759     0.00663921   -0.0162865    -0.000305536   0.011383     -0.00196883   -0.00363319   -0.0036993    -0.00455917   -0.00290846    0.00888095   -0.00789423    0.00304331   -0.00740744    0.00050636   -0.000302411   0.0030843     0.00049445   -0.00284163
47  -0.0221746    -0.0199404     0.000534764   0.00477627    0.0141621     0.000433482   0.000167613   0.00463961   -0.00399233    0.00393828   -0.000758662  -0.00181117    0.00373518   -0.00279076   -0.000500963   0.00307397    0.0025922    -0.00118273   -0.00313347
48  -0.0550543     0.0363964    -0.00742737    0.00509457    0.00478327    0.00596863    0.0123141     0.00319395    0.00032336    0.00227062    0.00182639    0.0089813    -0.00131728    0.00282792   -0.00285173    0.000575806   0.000630176   0.00142055   -0.000739086
49   0.00913939   -0.0112506     0.00275407   -0.000108444   0.0175965    -0.00720219    0.0028736    -0.00852751   -0.00367575    0.00224067    0.00306352    0.00371249   -0.0013583    -0.00438014    0.00156735    0.00259369   -0.00342623    0.00252017   -0.000817948
50  -0.00233578   -0.0202756     0.0265616     0.0127293    -0.0118266     0.00893602    0.00197975    0.00795907    0.00501867    0.00222461    0.00222965   -0.00193209    0.00671279    0.000512806  -0.000123589   0.00127378   -0.00139443    0.001997     -0.0018316
51  -0.0153559    -0.0100082     0.0139632     0.00493122   -0.0222626     0.000493596   0.00833022   -0.0038709     0.00743823   -0.00699599   -0.0016895     0.00127742    0.00430974   -0.00836289    0.00714681    0.00356698   -0.000744286   0.000825278   0.00269039
52   0.0263224    -0.017408     -0.0229181    -0.00281498    0.00866802   -0.0019128     0.00195427    0.00439744   -0.00584011    0.00819845    0.000279562   0.000429365  -0.000174549   0.00240462    0.00358596    0.00083813   -0.00110688   -0.000486164   0.00155618
53   0.0340365     0.0249426    -0.000528411   0.00975803    0.0137844     0.0129551     0.00420257   -0.00700206   -0.00177581    0.00589381    0.00212311    0.00317137   -0.00186415    0.00236349    0.00271734   -0.000199331   0.000225671   0.00280694    0.00416741
54  -0.0117798     0.00903529    0.0387701     0.00454727   -0.00907697   -0.00488634   -0.00773384   -6.19644e-6    0.0089247     0.00618783   -0.00347535   -0.00495008   -0.000644665   0.00498122    0.00158708   -0.000299986   0.00137401   -0.00415685    0.00151692
55   0.040961     -0.0047418    -0.00467653    0.0299573    -0.010076     -0.00532661   -0.00871529    0.0040513    -0.00358455   -0.00462993    0.00250156    0.00223835    0.0054579     0.00519887   -0.000305873   0.000448611  -0.00211423    0.00125517   -0.00247527
56   0.0102925     0.013984      0.000812639  -0.00373706   -0.00349584    0.0124504    -0.00441857   -0.0044716     0.00140511    0.00328495    0.00684282   -0.00884618    0.0042292     0.00577289   -0.00200728    0.00248224    0.00354552    0.00261187    0.00147238
57   0.0477407     1.14678e-5   -0.00662024    0.00219729    0.00430952   -0.00449388    0.00119093    0.000191644   0.00325373    0.00379379   -0.000464058  -0.00390622    0.00283012    0.00361447   -0.000321092   0.000398221  -0.00180963   -0.00322221    0.00339463
58   0.0399973    -0.0189741    -0.0247896     0.0297376    -0.013145     -0.00537322    0.000798352  -0.00376624   -0.00048942    0.0013669    -0.00261489    0.00622029   -0.00186308   -0.00227817   -0.00329716    6.72505e-6    0.00269157   -0.00133317    0.000372921
59  -0.00330724    0.000903053   0.00809613   -0.00496936    0.00629712   -0.00809498    0.00396068    0.0024341     0.00266899   -0.00182288   -0.00357433    0.00138473    0.00143361   -0.00171682   -0.00717317   -0.00423229   -0.00252162    0.00019209    0.000617265
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Importance of components:
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
PC1        PC2        PC3         PC4         PC5         PC6         PC7         PC8         PC9        PC10        PC11         PC12         PC13         PC14         PC15         PC16         PC17         PC18        PC19
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
SS Loadings (Eigenvalues)  0.0571665  0.0189413  0.0127985  0.00692343  0.00559569  0.00374975  0.00277789  0.00221675  0.00204783  0.00136863  0.00118197  0.000853869  0.000786129  0.000575358  0.000561562  0.000342726  0.000253164  0.000229252  0.00020354
Variance explained         0.477945   0.15836    0.107003   0.0578838   0.0467831   0.0313501   0.0232248   0.0185333   0.017121    0.0114425   0.00988197  0.00713883   0.00657249   0.00481032   0.00469498   0.00286538   0.00211659   0.00191668   0.00170171
Cumulative variance        0.477945   0.636305   0.743308   0.801192    0.847975    0.879325    0.90255     0.921083    0.938204    0.949647    0.959528    0.966667     0.97324      0.97805      0.982745     0.98561      0.987727     0.989644     0.991345
Proportion explained       0.482117   0.159743   0.107937   0.0583891   0.0471916   0.0316238   0.0234275   0.0186951   0.0172705   0.0115424   0.00996824  0.00720116   0.00662987   0.00485231   0.00473597   0.0028904    0.00213507   0.00193341   0.00171657
Cumulative proportion      0.482117   0.64186    0.749797   0.808186    0.855378    0.887002    0.910429    0.929124    0.946395    0.957937    0.967905    0.975106     0.981736     0.986589     0.991325     0.994215     0.99635      0.998283     1.0
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ

Now letβs show explained variance of each principal component.

plot(principalvars(fp), title="explained variance", label="Tangent PCA")

The next plot shows how projections on the first two principal components look like.

fig = plot(; title="coordinates per gesture of the first two principal components")
for label_num in [0, 1]
cur_t = MultivariateStats.transform(fp, cur_hand_logs)
scatter!(fig, cur_t[1, :], cur_t[2, :], label="gesture " * string(label_num))
end
xlabel!(fig, "principal component 1")
ylabel!(fig, "principal component 2")
fig

The following heatmap displays pairwise distances between gestures. We can use them for clustering, classification, etc.

hand_distances = [
distance(Mshape, hands_projected[i], hands_projected[j]) for
i in eachindex(hands_projected), j in eachindex(hands_projected)
]
heatmap(hand_distances, aspect_ratio=:equal)