Surface Curvature Matlab equivalent in Python

问题

I was trying to calculate the curvature of a surface given by array of points (x,y,z). Initially I was trying to fit a polynomial equation z=a + bx + cx^2 + dy + exy + fy^2) and then calculate the gaussian curvature

$ K = \frac{F_{xx}\cdot F_{yy}-{F_{xy}}^2}{(1+{F_x}^2+{F_y}^2)^2} $

However the problem is fitting if the surface is complex. I found this Matlab code to numerically calculate curvature. I wonder how to do the same in Python.

function [K,H,Pmax,Pmin] = surfature(X,Y,Z),
% SURFATURE -  COMPUTE GAUSSIAN AND MEAN CURVATURES OF A SURFACE
%   [K,H] = SURFATURE(X,Y,Z), WHERE X,Y,Z ARE 2D ARRAYS OF POINTS ON THE
%   SURFACE.  K AND H ARE THE GAUSSIAN AND MEAN CURVATURES, RESPECTIVELY.
%   SURFATURE RETURNS 2 ADDITIONAL ARGUEMENTS,
%   [K,H,Pmax,Pmin] = SURFATURE(...), WHERE Pmax AND Pmin ARE THE MINIMUM
%   AND MAXIMUM CURVATURES AT EACH POINT, RESPECTIVELY.


% First Derivatives
[Xu,Xv] = gradient(X);
[Yu,Yv] = gradient(Y);
[Zu,Zv] = gradient(Z);

% Second Derivatives
[Xuu,Xuv] = gradient(Xu);
[Yuu,Yuv] = gradient(Yu);
[Zuu,Zuv] = gradient(Zu);

[Xuv,Xvv] = gradient(Xv);
[Yuv,Yvv] = gradient(Yv);
[Zuv,Zvv] = gradient(Zv);

% Reshape 2D Arrays into Vectors
Xu = Xu(:);   Yu = Yu(:);   Zu = Zu(:); 
Xv = Xv(:);   Yv = Yv(:);   Zv = Zv(:); 
Xuu = Xuu(:); Yuu = Yuu(:); Zuu = Zuu(:); 
Xuv = Xuv(:); Yuv = Yuv(:); Zuv = Zuv(:); 
Xvv = Xvv(:); Yvv = Yvv(:); Zvv = Zvv(:); 

Xu          =   [Xu Yu Zu];
Xv          =   [Xv Yv Zv];
Xuu         =   [Xuu Yuu Zuu];
Xuv         =   [Xuv Yuv Zuv];
Xvv         =   [Xvv Yvv Zvv];

% First fundamental Coeffecients of the surface (E,F,G)
E           =   dot(Xu,Xu,2);
F           =   dot(Xu,Xv,2);
G           =   dot(Xv,Xv,2);

m           =   cross(Xu,Xv,2);
p           =   sqrt(dot(m,m,2));
n           =   m./[p p p]; 

% Second fundamental Coeffecients of the surface (L,M,N)
L           =   dot(Xuu,n,2);
M           =   dot(Xuv,n,2);
N           =   dot(Xvv,n,2);

[s,t] = size(Z);

% Gaussian Curvature
K = (L.*N - M.^2)./(E.*G - F.^2);
K = reshape(K,s,t);

% Mean Curvature
H = (E.*N + G.*L - 2.*F.*M)./(2*(E.*G - F.^2));
H = reshape(H,s,t);

% Principal Curvatures
Pmax = H + sqrt(H.^2 - K);
Pmin = H - sqrt(H.^2 - K);

回答1:

I hope I'm not too late here. I work with exactely the same problem (a product for the company I work to).

The first thing you must consider is that the points must represent a rectangular mesh. X is a 2D array, Y is a 2D array, and Z is a 2D array. If you have an unstructured cloudpoint, with a single matrix shaped Nx3 (the first column being X, the second being Y and the third being Z) then you can't apply this matlab function.

I have developed a Python equivalent of this Matlab script, where I only calculate Mean curvature (I guess you can get inspired by the script and adapt it to get all your desired curvatures) for the Z matrix, ignoring the X and Y by assuming the grid is square. I think you can "grasp" what and how I am doing, and adapt it for your needs:

def mean_curvature(Z):
    Zy, Zx  = numpy.gradient(Z)
    Zxy, Zxx = numpy.gradient(Zx)
    Zyy, _ = numpy.gradient(Zy)

    H = (Zx**2 + 1)*Zyy - 2*Zx*Zy*Zxy + (Zy**2 + 1)*Zxx
    H = -H/(2*(Zx**2 + Zy**2 + 1)**(1.5))

    return

回答2:

In case others stumble across this question, for completeness I offer the following code, inspired by heltonbiker.

Here is some python code to calculate Gaussian curvature as described by equation (3) in "Computation of Surface Curvature from Range Images Using Geometrically Intrinsic Weights"*, T. Kurita and P. Boulanger, 1992.

import numpy as np

def gaussian_curvature(Z):
    Zy, Zx = np.gradient(Z)                                                     
    Zxy, Zxx = np.gradient(Zx)                                                  
    Zyy, _ = np.gradient(Zy)                                                    
    K = (Zxx * Zyy - (Zxy ** 2)) /  (1 + (Zx ** 2) + (Zy **2)) ** 2             
    return K

Note:

heltonbiker's method is essentially equation (4) from the paper
heltonbiker's method is also the same on "Surfaces in 3D space, Mean Curvature" on Wikipedia: http://en.wikipedia.org/wiki/Mean_curvature)
If you need both K and H then include the calculation of "K" (Gaussian curvature) in heltonbiker code and return K and H. Saves a little processing time.
I assume the surface is defined as a function of two coordinates, e.g. z = Z(x, y). In my case Z is a range image.

回答3:

Although very late, but no harm in posting. I modified the "surfature" function for use in Python. Disclaimer: I'm not the author original code. Credits wherever they are due.

def surfature(X,Y,Z):
    # where X, Y, Z matrices have a shape (lr+1,lb+1)

    #First Derivatives
    Xv,Xu=np.gradient(X)
    Yv,Yu=np.gradient(Y)
    Zv,Zu=np.gradient(Z)

    #Second Derivatives
    Xuv,Xuu=np.gradient(Xu)
    Yuv,Yuu=np.gradient(Yu)
    Zuv,Zuu=np.gradient(Zu)   

    Xvv,Xuv=np.gradient(Xv)
    Yvv,Yuv=np.gradient(Yv)
    Zvv,Zuv=np.gradient(Zv) 

    #Reshape to 1D vectors
    nrow=(lr+1)*(lb+1) #total number of rows after reshaping
    Xu=Xu.reshape(nrow,1)
    Yu=Yu.reshape(nrow,1)
    Zu=Zu.reshape(nrow,1)
    Xv=Xv.reshape(nrow,1)
    Yv=Yv.reshape(nrow,1)
    Zv=Zv.reshape(nrow,1)
    Xuu=Xuu.reshape(nrow,1)
    Yuu=Yuu.reshape(nrow,1)
    Zuu=Zuu.reshape(nrow,1)
    Xuv=Xuv.reshape(nrow,1)
    Yuv=Yuv.reshape(nrow,1)
    Zuv=Zuv.reshape(nrow,1)
    Xvv=Xvv.reshape(nrow,1)
    Yvv=Yvv.reshape(nrow,1)
    Zvv=Zvv.reshape(nrow,1)

    Xu=np.c_[Xu, Yu, Zu]
    Xv=np.c_[Xv, Yv, Zv]
    Xuu=np.c_[Xuu, Yuu, Zuu]
    Xuv=np.c_[Xuv, Yuv, Zuv]
    Xvv=np.c_[Xvv, Yvv, Zvv]

    #% First fundamental Coeffecients of the surface (E,F,G)
    E=np.einsum('ij,ij->i', Xu, Xu) 
    F=np.einsum('ij,ij->i', Xu, Xv) 
    G=np.einsum('ij,ij->i', Xv, Xv) 

    m=np.cross(Xu,Xv,axisa=1, axisb=1)
    p=sqrt(np.einsum('ij,ij->i', m, m))
    n=m/np.c_[p,p,p]

    #% Second fundamental Coeffecients of the surface (L,M,N)
    L= np.einsum('ij,ij->i', Xuu, n) 
    M= np.einsum('ij,ij->i', Xuv, n) 
    N= np.einsum('ij,ij->i', Xvv, n) 

    #% Gaussian Curvature
    K=(L*N-M**2)/(E*G-L**2)
    K=K.reshape(lr+1,lb+1)

    #% Mean Curvature
    H = (E*N + G*L - 2*F*M)/(2*(E*G - F**2))
    H = H.reshape(lr+1,lb+1)

    #% Principle Curvatures
    Pmax = H + sqrt(H**2 - K)
    Pmin = H - sqrt(H**2 - K)

    return Pmax,Pmin

回答4:

Dot Product in Python

Derivates in Python

Reshaping in Python

Oddly enough all of these are SO questions. Take a gander around next time and you can likely find an answer. Also note that you'll want to be using NumPy for Python to do this. It's fairly intuitive to use. Matlibplot (or something like that) might be helpful for you too!