My goal is to interpolate curves in 2D and 3D space on a predefined sequential distance to perform a PCA on multiple curves.
Assume a data frame of multiple 3D-arrays (each of different size):
>>> df.curves
0 [[0.0, 0.0, 0.91452991453, 0.91452991453, 1.0]...
1 [[0.0, 0.0, 0.734693877551, 0.734693877551, 1....
2 [[0.0, 0.0, 1.0, 1.0, 1.0], [0.0, 0.6435643564...
3 [[0.0, 0.0, 0.551020408163, 0.551020408163, 1....
4 [[0.0, 0.0, 1.0, 1.0, 1.0], [0.0, 0.4389027431...
5 [[0.0, 0.0, 0.734693877551, 0.734693877551, 1....
Name: curves, dtype: object
>>> df.curves[0]
array([[ 0. , 0. , 0.73469388, 0.73469388, 1. ],
[ 0. , 0.1097561 , 0.47560976, 0.5 , 1. ],
[ 1. , 0.65036675, 0.08801956, 0.06845966, 0. ]])
Let's name the dimensions x,y,z where all dimensions have the same length and x and y dimensions are monotonically increasing:
3D Plot
I try to sample the data to be equidistant and allow comparability between curves with uniform sample rate.
A simple sampling function for a 2D curve (no y dim) would be per data frame row:
def sample2DCurve(row, res=10, method='linear'):
# coords of interpolation
xnew = np.linspace(0, 1, res)
# call scipy interpolator interp1d
# create interpolation function for 2D data
sample2D = interpolate.interp1d(row[0], row[1], kind=method)
# sample data points based on xnew
znew = sample2D(xnew)
return np.array([xnew, znew])
For 3D data I am using an interpolation along the path:
def sample3DCurves(row, res=10, method='linear'):
#npts = row[0].size
#p = np.zeros(npts, dtype=float)
#for i in range(1, npts):
# dx = row[0][i] - row[0][i-1]
# dy = row[1][i] - row[1][i-1]
# dz = row[2][i] - row[2][i-1]
# v = np.array([dx, dy, dz])
# p[i] = p[i-1] + np.linalg.norm(v)
#==============================================================================
# edit: cleaner algebra
x, *y, z = row
# vecs between subsequently measured points
vecs = np.diff(row)
# path: cum distance along points (norm from first to ith point)
path = np.cumsum(np.linalg.norm(vecs, axis=0))
path = np.insert(path, 0, 0)
#==============================================================================
## coords of interpolation
coords = np.linspace(p[0], p[-1], res) #p[0]=0 p[-1]=max(p)
# interpolation func for each axis with the path
sampleX = interpolate.interp1d(p, row[0], kind=method)
sampleY = interpolate.interp1d(p, row[1], kind=method)
sampleZ = interpolate.interp1d(p, row[2], kind=method)
# sample each dim
xnew = sampleX(coords)
ynew = sampleY(coords)
znew = sampleZ(coords)
return np.array([xnew, ynew, znew])
As another approach in 3D I'd like to perform an interpolation along isolines forming circles in x,y-plane with uniform radius:
Circular isolines around [0,0,0] in x,y-plane with 3D intersection
The z value is then interpolated based on the intersection of the isolines with the (linearly) interpolated curve projected in the x,y-plane.
But I struggle to define the circular lines and intersect with a projection of the curve/path vectors in the x,y-plane.
Any advice is very appreciated! (also in other languages - R/Matlab etc.)