Approximated compact Gaussian density functiona
This computes a compactly supported approximation of a Gaussian density function by convolving a box filter with itself n times in each dimension and multiplying, as described in "Poisson Surface Reconstruction" by Kazhdan et al. 2006.
Parameters:
| Name |
Type |
Description |
Default |
x |
(nx,dim) numpy double array
|
Coordinates where the function is to be evaluated |
required
|
n |
int, optional (default 2)
|
Number of convolutions (between 1 and 4), a higher number will more closely resemble a Gaussian but have broader support |
2
|
sigma |
double, optional (default 1.0)
|
Scale / standard deviation function (bigger leades to broader support) |
1.0
|
center |
|
Coordinates where the function is centered (i.e., distribution mean). If None, center at the origin. |
None
|
second_derivative |
int, optional (default -1)
|
If greater or equal than zero, this is the index of the dimension along which to compute the second partial derivative instead of the pure density. |
-1
|
Returns:
| Name | Type |
Description |
v |
(nx,) numpy double array
|
Values of compacly supported approximate Gaussian function at x |
Examples:
from gpytoolbox import compactly_supported_normal
import matplotlib.pyplot as plt
x = np.reshape(np.linspace(-4,4,1000),(-1,1))
plt.plot(x,compactly_supported_normal(x, n=4,center=np.array([0.5])))
plt.plot(x,compactly_supported_normal(x, n=3,center=np.array([0.5])))
plt.plot(x,compactly_supported_normal(x, n=2,center=np.array([0.5])))
plt.plot(x,compactly_supported_normal(x, n=1,center=np.array([0.5])))
plt.show()
Source code in src/gpytoolbox/compactly_supported_normal.py
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88 | def compactly_supported_normal(x,n=2,sigma=1.,center=None,second_derivative=-1):
"""Approximated compact Gaussian density functiona
This computes a compactly supported approximation of a Gaussian density function by convolving a box filter with itself n times in each dimension and multiplying, as described in "Poisson Surface Reconstruction" by Kazhdan et al. 2006.
Parameters
----------
x : (nx,dim) numpy double array
Coordinates where the function is to be evaluated
n : int, optional (default 2)
Number of convolutions (between 1 and 4), a higher number will more closely resemble a Gaussian but have broader support
sigma : double, optional (default 1.0)
Scale / standard deviation function (bigger leades to broader support)
center: (dim,) numpy double array (default None)
Coordinates where the function is centered (i.e., distribution mean). If None, center at the origin.
second_derivative : int, optional (default -1)
If greater or equal than zero, this is the index of the dimension along which to compute the second partial derivative instead of the pure density.
Returns
-------
v : (nx,) numpy double array
Values of compacly supported approximate Gaussian function at x
Examples
--------
```python
from gpytoolbox import compactly_supported_normal
import matplotlib.pyplot as plt
x = np.reshape(np.linspace(-4,4,1000),(-1,1))
plt.plot(x,compactly_supported_normal(x, n=4,center=np.array([0.5])))
plt.plot(x,compactly_supported_normal(x, n=3,center=np.array([0.5])))
plt.plot(x,compactly_supported_normal(x, n=2,center=np.array([0.5])))
plt.plot(x,compactly_supported_normal(x, n=1,center=np.array([0.5])))
plt.show()
```
"""
ndim = x.ndim
if ndim==0:
x = np.array([[x]])
if ndim==1:
x = np.reshape(x,(-1,1))
# print(ndim)
dim = x.shape[1]
if (center is None):
center = np.zeros(dim)
x_new = (x - np.tile(center[None,:],(x.shape[0],1)))/sigma
# This is the function centered at 0 and with sigma=1
def compactly_supported_normal_centered(x,n,second_derivative):
vals = np.ones(x.shape[0])
# I got this by writing into Wolfram Alpha:
# Convolve[rect(y),rect(y)]
# Convolve[rect(y),triangle(y)]
# Convolve[rect(y),Piecewise[{{1/6 (-abs(y)^3 + 6 y^2 - 12 abs(y) + 8),1<=abs(y)<=2},{1/6 (3abs(y)^3 - 6y^2 + 4 ),abs(y)<1}}]]
for i in range(dim):
xx = x[:,i]
if second_derivative==i:
if n==2:
xx_val = (np.abs(xx)<0.5)*(-2.0) + (np.abs(xx)<1.5)*(np.abs(xx)>0.5)*(1.0)
elif n==1:
xx_val = 0.0*np.abs(xx)
elif n==3:
xx_val = (np.abs(xx)<1)*(3*3*2*(np.abs(xx)) - 12)*(1/6) + (np.abs(xx)<=2)*(np.abs(xx)>=1)*( -3*2*(np.abs(xx)) + 12)*(1/6.)
elif n==4:
xx_val = (np.abs(xx)<=2.5)*(np.abs(xx)>=1.5)*( 16*4*3*(np.abs(xx)**2.0) - 160*3*2*(np.abs(xx)) + 600*2)*(1/384) + (np.abs(xx)<=1.5)*(np.abs(xx)>=0.5)*( -16*4*3*(np.abs(xx)**2.0) + 80*3*2*(np.abs(xx)) - 120*2)*(1/96) + (np.abs(xx)<=0.5)*( 48*4*3*(np.abs(xx)**2.0) - 120*2)*(1/192)
else:
if n==2:
xx_val = (np.abs(xx)<0.5)*(-(xx**2.0)+0.75) + (np.abs(xx)<1.5)*(np.abs(xx)>0.5)*(0.5*(xx**2.0) - 1.5*np.abs(xx) + 1.125)
elif n==1:
xx_val = (np.abs(xx)<1.0)*(1 - np.abs(xx))
elif n==3:
xx_val = (np.abs(xx)<1)*(3*(np.abs(xx)**3.0) - 6*(np.abs(xx)**2.0) + 4)*(1/6) + (np.abs(xx)<=2)*(np.abs(xx)>=1)*( -(np.abs(xx)**3.0) + 6*(np.abs(xx)**2.0) - 12*np.abs(xx) + 8 )*(1/6.)
elif n==4:
xx_val = (np.abs(xx)<=2.5)*(np.abs(xx)>=1.5)*( 16*(np.abs(xx)**4.0) - 160*(np.abs(xx)**3.0) + 600*(np.abs(xx)**2.0) - 1000*(np.abs(xx)) + 625)*(1/384) + (np.abs(xx)<=1.5)*(np.abs(xx)>=0.5)*( -16*(np.abs(xx)**4.0) + 80*(np.abs(xx)**3.0) - 120*(np.abs(xx)**2.0) + 20*(np.abs(xx)) + 55)*(1/96) + (np.abs(xx)<=0.5)*( 48*(np.abs(xx)**4.0) - 120*(np.abs(xx)**2.0) + 115)*(1/192)
vals = vals * xx_val
return vals
val = compactly_supported_normal_centered(x_new,n,second_derivative)/(sigma**dim)
if second_derivative>=0:
val = val/(sigma**2.0)
# We call it with the transformed x and rescale by sigma
return val
|