Fitting 3D Data with a Cylinder

Transcription

1 Fittig 3D Data with a Cylider David Eberly, Geometric Tools, Redmod WA This work is licesed uder the Creative Commos Attributio 4.0 Iteratioal Licese. To view a copy of this licese, visit or sed a letter to Creative Commos, PO Box 1866, Moutai View, CA 94042, USA. Created: February 25, 2003 Last Modified: August 1, 2015 Cotets 1 Fittig by Miimizig a Error Fuctio The Miimax Approach Least-Squares Approach Represetatio of a Cylider 3 3 Fittig with a Cylider A Equatio for the Radius A Equatio for the Ceter A Equatio for the Directio Fittig for a Specified Directio 8 5 Pseudocode ad Experimets 8 1

2 This documet describes a algorithm for fittig a set of 3D poits with a cylider. The assumptio is that the uderlyig data is modeled by a cylider ad that errors have caused the poits ot to be exactly o the cylider. You could very well try to fit a radom set of poits, but the algorithm is ot guarateed to produce a meaigful solutio. 1 Fittig by Miimizig a Error Fuctio I geeral terms, a 3D poit set X i } is assumed to be o a surface whose structure is depedet o a set of m parameters. The surface is represeted implicitly by F (X; q = 0 for some fuctio F where q = (q 1,..., q m. Let Q deote the set of all relevat q. Although we expect X i to be exactly o the surface (F (X i ; q = 0, the measuremets that geerate X i have errors, ε i (q = F (X i ; q (1 We wish to select those parameters so that the error terms are as small as possible. Several approaches may be used to miimize the errors joitly. 1.1 The Miimax Approach Oe approach to miimizig the errors joitly is to fid ˆq for which the maximum absolute error is miimized, referred to as a miimax algorithm. For a selected q, the maximum error term is ad we seek a parameter vector ˆq for which E(q = max 1 i ε i(q (2 E(ˆq = mi q Q max 1 i F (X i; q (3 If the poits actually all lie o the surface with o experimetal errors, the error terms are all zero ad E(ˆq = 0. I practice, however, the errors are iheret i the measuremets, so we ca hope oly to make E(ˆq as small a positive umber as possible. The miimizatio of the maximum error does ot led itself to a relatively simple algorithm to compute ˆq. A alterative that does is a least-squares approach, described ext. 1.2 Least-Squares Approach A least-squares algorithm is desiged to miimize the total squared error, E(q = We seek a parameter vector ˆq for which [F (X i ; q] 2 (4 E(ˆq = mi q Q [F (X i ; q] 2 (5 2

3 Assumig F is differetiable with respect to the parameter compoets, the methods of calculus may assist us i locatig the global miimum of E(q. The global miimum occurs either at a poit where all the partial derivatives are zero, E/ q j = 0 for 1 j m, or at a boudary poit of the set of all parameters. The derivatives are E(q q j = 2 F (X i ; q F (X i; q, 1 j m q j Settig the partial derivatives to zero, we obtai a system of m oliear equatios i m ukows q j. I some cases the solutio may be obtaied i closed form by algebraic methods, but geerally root-fidig methods i multiple dimesios are ecessary. A differet attack o the problem is to miimize E(q usig a iterative scheme that searches the multidimesioal space of parameters. A iitial parameter tuple is chose. I may cases the iitial tuple is based o iformatio you kow about your specific applicatio. The costructio of the iitial tuple itself might be a complex process based o a least-squares algorithm. A lie is chose that cotais the iitial tuple, ad a search is performed for the miimum of E alog the lie. The parameter tuple at which the miimum occurs is used as the startig poit for aother liear search usig a lie of whose directio is differet from the previous directiot. Powell s Directio Set Method is oe such algorithm that searches alog a set of m lies whose directios are iitially the coordiate axis directios. After m lies have bee searched, the ext lie is costructed that cotais the iitial tuple ad the tuple obtaied from the m searches. This is desiged to give the best chaces of quickly gettig close to a miimum poit. The method does ot require derivative evaluatio, makig it a attractive algorithm whe the fuctio to be miimized is ot i a form that is ameable to differetiatio. The Cojugate Gradiet Method is a sophisticated method that also attempts to be smart about choice of lie directio. This method requires derivatives ad is usually the miimizer of choice i most applicatios. The Gradiet Descet Method choose the search lie i the directio associated with the largest decrease i the fuctio value; the directio is the egative of the gradiet of the fuctio. This method is regarded to be iferior i may situatios, though, because a lot of time ca be spet zigzaggig about the parameter space ad ot approachig a miimum i a reasoable amout of time. 2 Represetatio of a Cylider A ifiite cylider is specified by a axis cotaiig a poit C ad havig uit-legth directio W. The radius of the cylider is r > 0. Two more uit-legth vectors U ad V may be defied so that U, V, W} is a right-haded orthoormal set; that is, all vectors are uit-legth, mutually perpedicular, ad with U V = W, V W = U, ad W U = V. Ay poit X may be writte uiquely as X = C + y 0 U + y 1 V + y 2 W = C + RY (6 where R is a rotatio matrix whose colums are U, V, ad W ad where Y is a colum vector whose rows are y 0, y 1, ad y 2. To be o the cylider, we eed r 2 = y0 2 + y1 2 = (U (X C 2 + (V (X C 2 = (X C T (UU T + VV T (X C = (X C T (I WW T (X C (7 3

4 where I is the idetity matrix. Because the uit-legth vectors form a orthoormal set, it is ecessary that I = UU T + VV T + WW T. A fiite cylider is obtaied by boudig the poits i the axis directio, where h > 0 is the height of the cylider. y 2 = W (X C h/2 (8 3 Fittig with a Cylider Let X i } be the iput poit set. A error fuctio for a cylider fit based o Equatio (7 is E(r 2 [, C, W = F (Xi ; r 2, C, W ] 2 = [(X ( i C T I WW T (X i C r 2] 2 (9 where the cylider axis is a lie cotaiig poit C ad havig uit-legth directio W ad the cylider radius is r. Thus, the error fuctio ivolves 6 parameters: 1 for the squared radius r 2, 3 for the poit C, ad 2 for the uit-legth directio W. These parameters form the 6-tuple q i the geeric discussio preseted previously. For umerical robustess, it is advisable to subtract the sample mea A = ( X i/ from the samples, X i X i A. This precoditioig is assumed i the mathematical derivatios to follow, i which case X i = 0. I the followig discussio, defie P = I WW T, r 2 i = (C X i T P (C X i (10 The matrix P represets a projectio oto a plae with ormal W, so P 2 = P ad depeds oly o the directio W. The term ri 2 depeds o the ceter C ad the directio W. The error fuctio is writte cocisely as E = (r2 i r A Equatio for the Radius The partial derivative of the error fuctio with respect to the squared radius is E/ r 2 = 2 (r2 i r2. Settig this to zero, we have the costrait which leads to 0 = (ri 2 r 2 (11 r 2 = 1 ri 2 (12 Thus, the squared radius is the average of the squared distaces of the projectios of X i C oto a plae cotaiig C ad havig ormal W. The right-had side depeds o the parameters C ad W. 4

5 Observe that r 2 i r2 = r 2 i 1 j=1 r2 j = (C X i T P (C X i 1 j=1 (C X j T P (C X j = C T P C 2X T i P C + X T i P X i 1 j=1 (C T P C 2X T j P C + X T j P X j ( = C T P C 2X T i P C + X T i P X i C T P C + 2 j=1 XT j P C 1 j=1 XT j P X j ( ( 1 = j=1 XT j X T i 2P C + X T i P X i 1 j=1 XT j P X j ( = X T i 2P C + X T i P X i 1 j=1 XT j P X j (13 The last equality is based o the precoditio j=1 X j = A Equatio for the Ceter The partial derivative with respect to the ceter is E/ C = 4 (r2 i r2 P (X i C. Settig this to zero, we have the costrait 0 = (r2 i r2 P (X i C = (r2 i r2 P X i [ (r2 i r2 ] P C = (r2 i r2 P X i (14 where the last equality is a cosequece of Equatio (11. Multiply equatio (13 by P X i, sum over i, ad use equatio (14 to obtai ( 0 = 2P X ix T i P C + ( (X T 1 i P X i P X i j=1 XT j P X j P X i ( = 2P X ix T i P C + (15 (X T i P X i P X i where the last equality is based o the precoditio X i = 0. We wish to solve this equatio for C, but observe that C + tw are valid ceters for all t. It is sufficiet to compute a ceter that has o compoet i the W-directio; that is, we may costruct a poit for which C = P C. It suffices to solve Equatio (15 for P C writte as the liear system A(P C = B/2 where Y i = P X i ( A = P X ix T i P = Y iy T i (16 B = (X T i P X i P X i = (Y T i Y i Y i The projectio matrix is symmetric, P = P T, a coditio that leads to the right-had side of the equatio defiig A. We have used P = P 2 to itroduce a additioal P factor, X T i P X i = X T i P 2 X i = X T i P T P X i, which leads to the right-had side of the equatio defiig B. The covariace matrix of the samples is M = X ix T i. If the samples are approximately distributed o a cylider, we expect M to be osigular. However, the matrix A = P MP is sigular because the projectio matrix P is sigular. We caot directly ivert A to solve the equatio. The liear system ivolves terms 5

6 that live oly i the plae perpedicular to W, so i fact the liear system reduces to two equatios i two ukows i the projectio space ad is solvable as log as the coefficiet matrix is ivertible. Choose U ad V so that U, V, W} is a right-haded orthoormal set; the P X i = µ i U + ν i V ad P C = k 0 U + k 1 V, where µ i = U X i, ν i = V X i, k 0 = U P C, ad k 1 = V P C. The matrix A becomes ( ( ( ( A = UU T + µ i ν i UV T + VU T + VV T (17 ad the vector B becomes µ 2 i B = 2 (µ 2 i + νi 2 (µ i U + ν i V (18 The vector A(P C becomes ( ( A(P C = k 0 µ 2 i + k 1 µ i ν i U + k 0 µ i ν i + k 1 ν 2 i ν 2 i V (19 Equatig this to B ad groupig the coefficiets for U ad V leads to the liear system µ2 i µ iν i k 0 = 1 2 (µ2 i + ν2 i µ i µ iν i k 1 2 (20 2 (µ2 i + ν2 i ν i ν2 i The coefficiet matrix is the covariace matrix of the projectio of the samples oto the plae perpedicular to W. Ituitively, this matrix is ivertible as log as the projectios do ot lie o a lie. If the matrix is sigular (or early sigular umerically, the origial samples lie o a plae (or early lie o a plae umerically. They are ot fitted well by a cylider or, if you prefer, they are fitted by a cylider with ifiite radius. The matrix system of Equatio (20 has solutio k0 k 1 1 = ( 2 µ2 i ν2 i ( 2 µiνi ν2 i µiνi µiνi µ2 i (µ2 i + ν2 i µi (µ2 i + ν2 i νi (21 which produces the cylider ceter P C = k 0 U + k 1 V; use this istead of C i Equatio (7. Although the solutio appears to deped o the choice of U ad V, it does ot. Let W = (w 0, w 1, w 2 ad defie the skew symmetric matrix 0 w 2 w 1 S = w 2 0 w 0 (22 w 1 w 0 0 By defiitio of skew symmetry, S T = S. This matrix represets the cross product operatio: Sξ = W ξ for ay vector ξ. Because we have a right-haded orthoormal set, it follows that SU = V ad SV = U. It may be also show that S = VU T UV T. Defie matrix  by ( ( ( (  = UU T µ i ν i UV T + VU T + VV T = SAS T (23 ν 2 i 6 µ 2 i

7 Effectively, this geerates a 2 2 matrix that is the adjugate of the 2 2 matrix represetig A. It has the property ÂA = δp where δ = µ2 i ν2 i ( µ iν i 2. The trace of a matrix is the sum of the diagoal etries. Observe that Trace(P = 2. Takig the trace of ÂA = δp, we obtai 2δ = Trace(ÂA. The cylider ceter is  P C = Trace(ÂA (X T i P X i P X i (24 where we have used ÂP =  because ST P = S T. This equatio is idepedet of U ad V but depedet o W. It may be show that ( δ = 1 2 Trace(ÂA = WT Adj X i X T i W (25 where Adj(M is the traspose of the cofactor matrix of M. Also observe that SP = S ad P S T = S T, so (  = SAS T = S X i X T i S T (26 These imply that P C is a ratioal fuctio of the compoets of W. The umerator has degree 4 ad the deomiator has degree A Equatio for the Directio Let the directio be parameterized as W(s = (w 0 (s, w 1 (s, w 2 (s, where s is a 2-dimesioal parameter. For example, spherical coordiates is such a parameterizatio: W = (cos s 0 si s 1, si s 0 si s 1, cos s 1 for s 0 [0, 2π ad s 1 [0, π/2], where w 2 (s 0, s 1 0. The partial derivatives of E are E s k = 2 (ri 2 r 2 (C X i T P (C X i (27 s k Solvig E/ s k = 0 i closed form is ot tractable. It is possible to geerate a system of polyomial equatios i the compoets of W, use elimiatio theory to obtai a polyomial i oe variable, ad the fid its roots. This approach is geerally tedious ad ot robust umerically. Istead we may substitute ito Equatio (27 the expressio of Equatio (13 ad the cylider ceter of Equatio (24 to obtai two equatios ivolvig the three compoets of W. The coditio W = 1 provides a third equatio for the compoets. A multidimesioal root fider may be used to locate cadidate directios. Alteratively, we may skip root fidig for E/ s k = 0 ad istead substitute Equatios (13 ad (24 directly ito the error fuctio E = (r2 i r2 2 to obtai a oegative fuctio, G(W = Y T i Y i 1 ( Y T j Y j 2Y T  i Trace(ÂA j=1 (Y 2 T i Y i Y i where Y i = P X i, A = Y iy T i, ad  = SAST. A umerical algorithm for locatig the miimum of G may be used. Or, as is show i the sample code, the domai for (s 0, s 1 may be partitioed ito samples (28 7

8 at which G is evaluated. The sample producig the miimum G-value determies a reasoable directio W. The ceter C ad squared radius r 2 are iheret i the evaluatio of G, so i the ed we have a fittig cylider. The evaluatios of G are expesive for a large umber of samples, but it is possible to implemet the evaluatios o a GPU for speed. 4 Fittig for a Specified Directio Although oe may apply root-fidig or miimizatio techiques to estimate the global miimum of E, as show previously, i practice it is possible first to obtai a good estimate for the directio W. Usig this directio, we may solve for C = P C i Equatio (24 ad the r 2 i Equatio (12. For example, suppose that the X i are distributed approximately o a sectio of a cylider so that the leastsquares lie that fits the data provides a good estimate for the directio W. This vector is a uit-legth eigevector associated with the largest eigevalue of the covariace matrix A = X ix T i. We may use a umerical eigesolver to obtai W, ad the solve the aforemetioed equatios for the cylider ceter ad squared radius. The distributio ca be such that the estimated directio W from the covariace matrix is ot good, as is show i the experimets of the ext sectio. 5 Pseudocode ad Experimets The simplest algorithm to implemet ivolves the miimizatio of the fuctio G i Equatio (28. A implemetio is show i Listig 1. Listig 1. Pseudocode for evaluatig the fuctio G(W ad geeratig the correspodig P C ad r 2. Real G( i t, Vector3 X [ ], Vector3 W, Vector3& PC, Real& rsqr M a t rix3x3 P = M atrix3x3 : : I d e t i t y ( OuterProduct (W, W ; // P = I W WˆT // S = 0, w2, w1}, w2, 0, w0}, w1, w0, 0}}, i e r b r a c e s a r e rows M a t rix3x3 S ( 0, W[ 2 ], W[ 1 ], W[ 2 ], 0, W[ 0 ], W[ 1 ], W[ 0 ], 0 ; M a t rix3x3 A = M a t r ix3x3 : : Zero ( ; Vector3 B = Vector3 : : Zero (, Y [ ] ; Real averagesqrlegth = 0, sqrlegth [ ] ; f o r ( i t i = 0 ; i < ; ++i Y [ i ] = P X [ i ] ; s q r L e g t h [ i ] = Dot (Y [ i ], Y [ i ] ; A += OuterProduct (Y [ i ], Y [ i ] ; B += s q r L e g t h [ i ] Y [ i ] ; averagesqrlegth += sqrlegth [ i ] ; } A /= ; // To keep e l e m e t s s m a l l whe i s l a r g e. B /= ; // To keep e l e m e t s s m a l l whe i s l a r g e. averagesqrlegth /= ; Matrix3x3 Ahat = S A S ; PC = ( Ahat B/ Trace ( Ahat A ; Real e r r o r = 0 ; r S q r = 0 ; f o r ( i t i = 0 ; i < ; ++i 8

9 } Real term = sqrlegth [ i ] averagesqrlegth 2 Dot (Y [ i ], PC ; e r r o r += term term ; Vector3 d i f f = PC Y [ i ] ; r S q r += Dot ( d i f f, d i f f ; } e r r o r /= ; // For root mea s q u a r e e r r o r. r S q r /= ; r e t u r e r r o r ; The fittig is performed by searchig a large umber of directios W, as show i Listig 2. Listig 2. Fittig a cylider to a set of poits. // The X [ ] a r e t h e p o i t s to be f i t. The o u t p u t s rsqr, C, ad W a r e t h e // c y l i d e r p a r a m e t e r s. The f u c t i o r e t u r v a l u e i s t h e e r r o r f u c t i o // e v a l u a t e d a t t h e c y l i d e r p a r a m e t e r s. Real F i t C y l i d e r ( i t, Vector3 X [ ], R e a l& rsqr, V e c t o r 3& C, V e c t o r 3& W // Compute t h e a v e r a g e ad s u b t r a c t from t h e p o i t s. This i s eeded // f o r u m e r i c a l r o b u s t e s s o f t h e i m p l e m e t a t i o. Vector3 average = Vector3 : : Zero ( ; f o r ( i t i = 0 ; i < ; ++i a v e r a g e += X [ i ] ; } average /= ; f o r ( i t i = 0 ; i < ; ++i X [ i ] = a v e r a g e ; } // Choose imax ad jmax as d e s i r e d f o r t h e l e v e l o f g r a u l a r i t y you // wat f o r s a m p l i g W v e c t o r s o t h e h e m i s p h e r e. Real m i E r r o r = i f i i t y ; W = Vector3 : : Zero ( ; C = Vector3 : : Zero ( ; r S q r = 0 ; f o r ( i t j = 0 ; j <= jmax ; ++j Real p h i = h a l f P i j / jmax ; // i [ 0, p i / 2 ] Real c s p h i = cos ( p h i, s p h i = s i ( p h i ; f o r ( i t i = 0 ; i < imax ; ++i Real t h e t a = twopi i / imax ; // i [ 0, 2 p i Real c s t h e t a = cos ( t h e t a, s t h e t a = s i ( t h e t a ; Vector3 curretw ( c s t h e t a s p h i, s t h e t a s p h i, c s p h i ; Vector3 curretc ; Real curretrsqr ; Real e r r o r = G(, X, curretw, c u r r e t C, c u r r e t R S q r ; i f ( e r r o r < m i E r r o r m i E r r o r = e r r o r ; W = curretw ; C = curretc ; rsqr = curretrsqr ; } } } // T r a s l a t e t h e c e t e r to t h e o r i g i a l c o o r d i a t e system. C += average ; 9

10 } r e t u r m i E r r o r ; The followig experimets show how well the cylider fittig works. A regular lattice of samples were chose for a cylider cetered at the origi, with axis directio (0, 0, 1, with radius 1, ad height 4. The samples are (cos θ j, si θ j, z ij, where θ j = 2πj/64 ad z ij = 2 + 4i/64 for 0 i 64 ad 0 j < 64. The fitted ceter, radius, ad axis directio are the actual oes (modulo umerical roud-off errors. Figure 1 shows a rederig of the poits (i black ad a wire frame of the fitted cylider (i blue. Figure 1. Fittig of samples o a cylider rig that is cut perpedicular to the cylider axis. A lattice of samples was chose for the same cylider but the samples are skewed to lie o a cut that is ot perpedicular to the cylider axis. The samples are (cos θ j, si θ j, z ij, where θ j = 2πj/64 for 0 j < 64 ad z ij = b + cos θ j + 2bi/64 for b = 1/4 ad 0 i 64. The fitted ceter, radius, ad axis directio are 10

11 the actual oes (modulo umerical roud-off errors. Figure 2 shows a rederig of the poits (i black ad a wire frame of the fitted cylider (i blue. Figure 2. Fittig of samples o a cylider rig that is cut skewed relative to the cylider axis. I this example, if you were to compute the covariace matrix of the samples ad choose the cylider axis directio to be the eigevector i the directio of maximum variace, that directio is skewed relative to the origial cylider axis. The fitted parameters are (approximately W = (0.699, 0, 0.715, P C = (0, 0, 0, ad r 2 = Figure 3 shows a rederig of the poits (i black ad a wire frame of the fitted cylider (i blue. 11

12 Figure 3. Fittig of samples o a cylider rig that is cut skewed relative to the cylider axis. The directio W was chose to be a eigevector correspodig to the maximum eigevalue of the covariace matrix of the samples. You ca see that the fitted cylider is ot a good approximatio to the poits. Figure 4 shows a poit cloud geerated from a DIC file (data set courtesy of Carolia Lavecchia ad the fitted cylider. The data set has early poits. 12

13 Figure 4. A view of a fitted poit cloud. Figure 5 shows a differet view of the poit cloud ad cylider. 13

14 Figure 5. Aother view of a fitted poit cloud, lookig alog the top of the cylier to get a idea of how well the cylider fits the data. 14