Question

我有一组(x, y, z)点，我需要找到最适合它们的平面。平面由其系数定义为：

a*x + b*y + c*z + d = 0

或等效地：

A*X +B*y + C = z

第二个等式只是对第一个等式的重写。

我使用了in this gist开发的方法，这是从this answer中给出的Matlab代码到Python的翻译。该方法找到系数来定义最适合点集的平面方程。

问题在于我能够提出一组系数，使更适合这组点。

为了更好地定义＆＃34;，我按照给定的here数学计算每个点到给定平面的绝对距离之和。较小的值意味着更好的＆＃34;因为这些点平均距离飞机很近。

MWE如下。可以看出，手工挑选的系数导致较小的绝对距离值之和（~155.89），而不是使用＆＃34; best＆＃34;通过上述方法找到的系数（~158.78）。

我在这里缺少什么？

MWE

import numpy as np
import scipy.linalg


def sum_dist_2_plane(x, y, z, a, b, c, d):
    """
    Sum of the absolute values of the distances to a plane, given by the
    a,b,c,d coefficients, for the set of points defined by x,y,z.
    """
    return np.sum(abs(a*x + b*y + c*z + d)/np.sqrt(a**2+b**2+c**2), axis=0)


# Some xyz points.
xyz = np.array([[1.1724546888698482, 0.67037911349217505, 1.6014525241637045], [2.0029440384631063, 1.2163076402918147, -1.1082409593302032], [-0.87863180025363918, 1.261853987259635, 1.1598532675831237], [0.42789396045777467, 0.67325845732274703, 1.1421266649135475], [1.366142552248496, 1.0959456367043121, -1.6046393305927751], [-2.1595534005011485, -2.2582441035518794, -1.0663372184011806], [2.1104543583371633, -2.3711560770628917, 0.33077589412150843], [1.1974640975387107, 1.2100068141421523, 0.71395322259985505], [0.44492797840962123, 0.51098686422493145, 0.23383900276620295], [-2.0810094204638281, -2.11327958929372, -1.0758230448163033], [1.1655230345226737, 2.3777304002844968, -1.5663228128649394], [0.90952208156596781, 0.84978064084217519, 1.5986081506274985], [1.2951624720758836, 1.2231899029278033, 1.6154291293114866], [0.97545563477882025, 1.1844143994262264, 0.25292733170194026], [2.0281659385206012, 1.3370146330231019, 1.1961575550766028], [-1.9843445684092424, -0.012247402159192651, -2.0732736152121092], [1.0852175044560746, 1.8083916604163963, 0.27402181385868829], [-0.97983337631837208, 1.1032503818628847, 1.1579341604311182], [2.5033961310304029, 1.5628354191569325, -0.60785250636200061], [0.84123393662217383, 1.6169587554844618, -0.66116704633280676], [-1.8572657771039134, 0.043103553120073364, -2.0779545355975415], [2.6979128603518787, 1.70987170366249, -0.59306759275995091], [1.898614831265683, -2.9411794973775129, 1.7095862940118209], [0.81052668401212824, 0.89107411631439926, 1.597589407046101], [-2.0466083174114331, 0.14841369250699468, -1.120794708199135], [2.7004384737959648, 1.3616954868011328, 1.2294957766312749], [2.5373220833750385, 1.7067484497548233, 0.32345763726774379], [0.42025310188487158, 0.25762913945011717, -2.5899822318304473], [1.0425582222020597, 1.2902156453507225, 1.1638276333984123], [1.8492329386150801, 1.369745208770941, -1.1101559957041474], [-1.9685282554587256, -0.053725287173628226, 0.26827797508054374], [2.1798881190450285, 1.2454661605758286, -1.5732113885771071], [2.097212096433736, -2.9271738140601462, -0.56568133063870363], [-4.0108387171254396, -0.95559594599890008, 1.7588521192455815], [1.1558287640906737, 0.84330421357278096, 1.1565989504480143], [-2.9571643443632118, -2.847346163285049, 1.3087401683271338], [1.8592900784537116, 1.3952561066549967, 0.28365423946831214], [-3.4841441062982867, -3.0501496018162109, -0.48161393173162992], [2.5524429115550777, 0.62723764313314334, 0.29882336571990464], [2.2267279436912251, -3.8561674586606758, 1.3393813829669483], [2.1214758016437449, -0.20203416631090113, -1.5903243997743601], [0.14882165322179747, 0.4127883227210779, 0.23115527212661391], [1.2042041122995621, 1.2013226392201846, -0.2014020012510187], [-0.91807770884292583, 1.1176994160488214, -2.5723612427329385], [1.910565457302241, 1.1857852625952567, -1.5853233609652335], [1.0660312416826301, 1.3594393638452948, 0.71483235729161265], [0.65109075860726373, 0.58395151990229632, 1.590486638605114], [2.0967121651174518, 3.5121496638531586, 0.85481080660772335], [1.1484000297535542, 0.93256813649663772, 0.25125672956252743], [-1.7670514601312102, 0.17479726844255272, 0.26097336908379276], [-0.38814151285133675, -1.36837872393391, -2.0916940966530149], [1.5825758742579219, -0.34854211056693962, 0.2556641250097158], [2.586881293405797, -4.371974479474976, -2.3458559556297445], [0.22496107684878977, 0.26917053206799602, -0.69280100767942088], [-0.92198332953292639, 5.3103622894708327, 1.4344469946544294], [1.5669967464035819, -0.13527817891479368, 1.6081806927677107], [-0.56872000311273319, -1.9823395333139691, -2.5517609300755879], [-3.7708737466313824, -3.2863308845331081, 1.3928734104180975], [0.26086111146896701, 0.91063726352187491, -2.1025221562973897], [4.3490818342473947, 1.7969605233982313, -0.94470942930075807], [0.8202509554992351, 1.6178074457637883, -0.66148472916848533], [-1.5947972211483237, 0.18933818654144918, -0.20453683465790107], [0.9736103155058905, 1.4905334895713331, -2.0806647444063202], [1.2838541958241105, 2.0842224244281931, -0.17045822168000058], [3.7985716232291624, 2.5292902540646183, -0.022070946178700979], [1.175697191763003, 0.70063646974704663, 0.24808027552254686], [1.7834118390535998, 1.2937296781793448, -0.1818232448888395], [1.1281441478154344, 0.89641394438231292, 1.6040641573676311], [-2.0118889302553362, 2.7916846393274373, -0.57683324778643197], [-0.5995803308341846, -2.2434949940054554, 0.2835440401850704], [0.32077033536702831, -0.95844872063257081, -1.6245015133016167], [0.81357199339193753, 1.5540883407880133, -0.19956720143058249], [0.62611590692268004, 2.5129849486626958, -0.62767513959140331], [1.3018663649626585, 0.92514176013041427, 0.71042211390030729], [-0.72715254964437737, -2.3705643250823436, -0.63320562968051775], [1.9172742234794142, -2.8680592171367834, -1.9965843559235594], [-0.7108415762295921, -2.2783943434144658, -0.63767826146936812], [1.968546542650037, -2.8305910089272146, -0.11154135958968681], [-3.1492524087194655, -2.8503098024243823, -0.049957063615551078], [-4.0600431110777313, -0.97891479243488955, -0.055962425569617835], [-3.3752702254780629, 5.7587998072406652, 2.0459797674238658], [-1.9855135921592455, 2.7466682542750638, -0.58034791274582886], [2.033073141968945, 1.5208650449610079, -0.16592183863411947], [-1.0379089220195949, -4.7336396164389383, 0.0045652508195388464], [0.059579198580756186, 0.50654688886459498, -0.69144595015375643], [2.1785293390435458, -2.67576518666927, -2.4787451249989232], [2.1096278381494935, -0.41668256763302775, -2.5482230530414327], [2.898772426390924, 1.9762337520130302, 1.2619960149795091], [0.95620776766155502, 1.4639884373148864, -0.19976180368861662], [0.78751831482788348, 1.6888070662998231, -1.1280318812973462], [0.75574071441925506, -0.89893698883953688, -0.21651308186821439], [-0.26825101547751962, -3.4496728096007274, 1.7066486428460195], [1.6690385240329706, -0.49893224975237227, -0.66401176702524367], [-0.28877792353045606, 1.5139628395303639, 0.25314013342428154], [0.33435105972001761, 0.72567663189581422, -2.5862147225048417], [-0.29757422904759573, 1.5866751937867298, -0.6682501010682671], [2.7581055173587461, -3.973585217996157, 0.0036824743223959899], [-3.4344275379769509, -3.089933175898083, 0.44457796620464052], [-2.9394415977285413, -2.6122275577950083, 1.2944549102942418], [2.0038460695984823, 1.515512638618338, -1.5731231727332897], [2.206216953170296, 1.4688891052013793, -1.5661966567970254], [-1.035208468220836, 4.4666436487176657, 0.89858770640569929], [-2.0039938640838546, 0.24894412179006209, -1.1220951191237916], [-3.9104727661324539, -0.70689702779279451, 1.2978242803460915], [1.7290487193475563, 1.2850859351795931, -0.18395259620439219], [1.1198244545179541, 1.7335817969585154, -0.18776435816536718], [0.32239533364835676, 0.2896168073626299, -1.1602117002106667], [0.36649393980823192, 0.28244286109766281, -0.69190114531475189], [0.71629324271161154, 0.62574841994964003, 1.1448784055936088], [-0.65109499789331204, -1.3933343864454197, -2.0884024350786063], [0.97046822380567643, 1.5321191441287463, -0.19744980702830617], [-0.9585141324426697, 1.3494884330155692, 1.610936445675776], [0.9615111008482673, 2.4535668843530907, -1.0939899554364985], [-1.0667872216702354, 0.9585914740866075, 1.6038639420443772], [1.8021244106955299, 1.1320598433704154, 1.1820726259869971], [-0.060098920604716666, 0.46839599864404674, 2.0277692055269654], [0.1721690681247055, 0.33837718694053642, 1.137078044079125], [-1.5964760388322969, 0.29775223476696611, 1.1626558382504655], [2.233093222044507, -2.8349614127699461, 0.36052101139762271], [1.9257633093026034, -2.5325763598899247, -1.5360887301240496], [1.116293873468281, 0.82698434754975214, -2.5739062165349651], [1.1781306304855363, 0.67917370389645249, 1.6017135739225736], [-1.8600651472693519, 0.078727875114422086, 1.6184578422253679], [-1.43994317003447, 0.13431327308359137, 2.0472930703748276], [0.84521838040660946, 0.63970047924770745, -2.100345751420285], [1.7661749989776647, -0.37651847162651875, -2.0797840873592222], [0.83547092354865804, 1.7219104152802622, 0.2661115369175846], [1.8300570222025725, -0.28592323411250137, 1.6180934388285593], [-0.62076647836845089, -0.99191053757063119, -1.1486388713745725], [-1.6239006006253158, 0.41366361326031414, 0.2574990624750626], [0.89195815704237569, 2.2004172385784, -0.17400231396826626], [0.36791088305589931, 0.36096348396301231, -2.5897662606427687], [0.073648763901347059, 0.19675260582587464, -2.1107265203482299], [2.161140531872539, -2.842373820387067, 0.35775402140617274], [-2.0416416353442859, -4.4051625504298446, 0.0054589213454931951], [-2.0525396585901774, 3.6758248479033888, -2.4231570023949089], [-0.96441167578601306, -4.6667609706070516, -0.0032107139968431397], [-1.8689820843196163, 0.021432805852950151, 0.26440433366338567], [-0.15613351765730205, -1.0964152703770347, 1.5952653951331826], [-0.91084152695600051, 1.2388514346844914, 1.1598544561959656], [0.94699177145572266, 1.2276340276860185, 2.0505581774713733], [-0.8929399989505632, 1.2806485400811793, -0.20595242802870217], [1.2023125342023806, 2.3477287603163717, -1.5668539565738087], [1.1651535046949058, 1.3836371788871575, 0.26217241277176129], [-1.0929407572158512, 1.3887078738113698, -0.19910861560325088], [-0.76452840903206265, 1.4237410113821392, -1.6090659495628117], [-1.5594385646555604, 0.1455415355638911, 1.1607640518832483], [-0.59734981961340872, -1.2800366176149909, 1.6032259368271653], [1.2325774703556955, 0.80804053623212702, 0.25109224401040819], [1.177240124012167, 0.90163100927998241, -1.1405108476689563]])
x, y, z = xyz[:, 0], xyz[:, 1], xyz[:, 2]

# Best-fit linear plane, for the Eq: z = a*x + b*y + c.
# See: https://gist.github.com/amroamroamro/1db8d69b4b65e8bc66a6
A = np.c_[x, y, np.ones(xyz.shape[0])]
C, _, _, _ = scipy.linalg.lstsq(A, z)

# Coefficients in the form: a*x + b*y + c*z + d = 0.
a, b, c, d = C[0], C[1], -1., C[2]

# Sum of absolute distances of each point to this plane.
print sum_dist_2_plane(x, y, z, a, b, c, d)

# Hand-picked coefficients.
a, b, c, d = 0.28, -0.14, 0.95, 0.

# Sum of absolute distances of each point to this plane.
print sum_dist_2_plane(x, y, z, a, b, c, d)

Answer 1

飞机的公式可以写成

z_plane = a*x + b*y + d

从点到平面的垂直z距离由

给出

|z_plane - z| = |a*x + b*y + d - z|

scipy.linalg.lstsq最小化这些距离之和的平方。

def zerror(x, y, z, a, b, d):
    return (((a*x + b*y + d) - z)**2).sum()

实际上，scipy.linalg.lstsq返回的参数产生的误差小于手工挑选的值：

In [113]: zerror(x, y, z, C[0], C[1], C[2])
Out[113]: 245.03516402045813

In [114]: zerror(x, y, z, 0.28, -0.14, 0.)
Out[114]: 323.81785779708787

formula

给出点(x_0, y_0, z_0)和平面ax + by + cz + d = 0之间的垂直距离。

您可以使用scipy.optimize.minimize最小化与飞机的垂直距离（请参阅下面的minimize_perp_distance）。

import math
import numpy as np
import scipy.optimize as optimize
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d.axes3d as axes3d
np.random.seed(2016)

mean = np.array([0.0,0.0,0.0])
cov = np.array([[1.0,-0.5,0.8], [-0.5,1.1,0.0], [0.8,0.0,1.0]])
xyz = np.random.multivariate_normal(mean, cov, 50)
x, y, z = xyz[:, 0], xyz[:, 1], xyz[:, 2]

def minimize_z_error(x, y, z):
    # Best-fit linear plane, for the Eq: z = a*x + b*y + c.
    # See: https://gist.github.com/amroamroamro/1db8d69b4b65e8bc66a6
    A = np.c_[x, y, np.ones(x.shape)]
    C, resid, rank, singular_values = np.linalg.lstsq(A, z)

    # Coefficients in the form: a*x + b*y + c*z + d = 0.
    return C[0], C[1], -1., C[2]

def minimize_perp_distance(x, y, z):
    def model(params, xyz):
        a, b, c, d = params
        x, y, z = xyz
        length_squared = a**2 + b**2 + c**2
        return ((a * x + b * y + c * z + d) ** 2 / length_squared).sum() 

    def unit_length(params):
        a, b, c, d = params
        return a**2 + b**2 + c**2 - 1

    # constrain the vector perpendicular to the plane be of unit length
    cons = ({'type':'eq', 'fun': unit_length})
    sol = optimize.minimize(model, initial_guess, args=[x, y, z], constraints=cons)
    return tuple(sol.x)

initial_guess = 0.28, -0.14, 0.95, 0.
vert_params = minimize_z_error(x, y, z)
perp_params = minimize_perp_distance(x, y, z)

def z_error(x, y, z, a, b, d):
    return math.sqrt((((a*x + b*y + d) - z)**2).sum())

def perp_error(x, y, z, a, b, c, d):
    length_squared = a**2 + b**2 + c**2
    return ((a * x + b * y + c * z + d) ** 2 / length_squared).sum() 

def report(kind, params):
    a, b, c, d = params
    paramstr = ','.join(['{:.2f}'.format(p) for p in params])
    print('{:7}: params: ({}), z_error: {:>5.2f}, perp_error: {:>5.2f}'.format(
        kind, paramstr, z_error(x, y, z, a, b, d), perp_error(x, y, z, a, b, c, d)))

report('vert', vert_params)
report('perp', perp_params)
report('guess', initial_guess)

X, Y = np.meshgrid(np.arange(-3.0, 3.0, 0.5), np.arange(-3.0, 3.0, 0.5))
fig = plt.figure()
ax = fig.gca(projection='3d')

def Z(X, Y, params):
    a, b, c, d = params
    return -(a*X + b*Y + d)/c

ax.plot_surface(X, Y, Z(X, Y, initial_guess), rstride=1, cstride=1, alpha=0.3, color='magenta')
ax.plot_surface(X, Y, Z(X, Y, vert_params), rstride=1, cstride=1, alpha=0.3, color='yellow')
ax.plot_surface(X, Y, Z(X, Y, perp_params), rstride=1, cstride=1, alpha=0.3, color='green')
ax.scatter(x, y, z, c='r', s=50)
plt.xlabel('X')
plt.ylabel('Y')
ax.set_zlabel('Z')
ax.axis('equal')
ax.axis('tight')
plt.show()

上面的代码计算最小化与平面的垂直距离和与平面的垂直距离的参数。然后我们可以计算总误差：

vert   : params: (0.94,0.52,-1.00,0.10), z_error:  2.63, perp_error:  3.21
perp   : params: (-0.68,-0.39,0.63,-0.06), z_error:  9.50, perp_error:  2.96
guess  : params: (0.28,-0.14,0.95,0.00), z_error:  5.22, perp_error: 52.31

请注意，vert_params最小化z_error，但perp_params最小化perp_error。

洋红色平面对应initial_guess，黄色平面对应vert_params，绿色平面对应perp_params。

Answer 2

unutbu的回答非常出色，是我在网上看到的最好的答案。我已经测试了很多。

为了稍微估计一下它，以便估计一个平面的初始猜测，我发现这样做非常有用，仅使用较大集合中的3个点即可。我们可以像这样估算参数a，b，c，d：

    def estimate_initial_parameters_with_first_three_points(points):
        # points is a list of lists [[x,y,z], [x,y,z], ...]
        p1 = points[0] 
        p2 = points[1]
        p3 = points[2]
        a1 = p2[0] - p1[0] 
        b1 = p2[1] - p1[1]
        c1 = p2[2] - p1[2]
        a2 = p3[0] - p1[0]
        b2 = p3[1] - p1[1] 
        c2 = p3[2] - p1[2]
        a = b1 * c2 - b2 * c1 
        b = a2 * c1 - a1 * c2 
        c = a1 * b2 - b1 * a2 
        d = (- a * p1[0] - b * p1[1] - c * p1[2]) 
        return [a,b,c,d]

这当然是从平面方程得出的：

a(x-x0) + b*(y-y0) + c*(z-z0) = d

使用scipy.linalg.lstsq获得最佳拟合平面到一组点的错误结果？

2 个答案: