b = np.arctan2(x[:, 0], -2 + x[:, 1])
return (a - b) / np.pi
# define the known discontinuity
bnd_vert = np.array([[0.0, 2.0], [0.0, -2.0]])
bnd_smp = np.array([[0, 1]])
# create synthetic data
pnts_obs = 8 * (rbf.halton.halton(200, 2, start=1) - 0.5)
sigma_obs = 0.2 * np.ones(200)
u_obs = signal(pnts_obs) + np.random.normal(0.0, sigma_obs)
# find the filtered solution
cutoff = 0.5
soln, _ = filter(pnts_obs,
u_obs,
sigma=sigma_obs,
cutoff=cutoff,
vert=bnd_vert,
smp=bnd_smp)
# plot the results and true signal
vals = np.linspace(-4, 4, 200)
grid = np.reshape(np.meshgrid(vals, vals), (2, 200**2)).T
fig, axs = plt.subplots(2, 1, figsize=(6, 10))
axs[0].scatter(pnts_obs[:, 0],
pnts_obs[:, 1],
c=u_obs,
s=50,
vmin=-1.0,
vmax=1.0,
cmap='viridis',
zorder=1)
p = axs[0].tripcolor(grid[:, 0],
x = np.linspace(0.0,1.0,768)
y = np.linspace(1.0,0.0,768)
x,y = np.meshgrid(x,y)
points = np.array([x.flatten(),y.flatten()]).T
values = np.linalg.norm(face(),axis=2)[:,256:].flatten()
#values = lena().flatten().astype(float)
values /= 441.7 # normalize so that the max is 1.0
signal = NearestNDInterpolator(points,values)
# interpolate Lena onto new observation points and add noise
points_obs = np.random.normal(0.5,0.3,(100000,2))
#points_obs = rbf.halton.halton(50000,2)
u_obs = signal(points_obs) + np.random.normal(0.0,0.2,100000)
# find filtered solution
cutoff = 60.0
soln,_ = filter(points_obs,u_obs,cutoff=cutoff,n=20,samples=0)
# plot the observed and filtered results
fig,ax = plt.subplots(2,1,figsize=(6,10))
ax[0].set_aspect('equal')
ax[0].set_axis_bgcolor('blue')
ax[0].set_xlim((0,1))
ax[0].set_ylim((0,1))
ax[0].set_title('noisy data')
ax[1].set_aspect('equal')
ax[1].set_axis_bgcolor('blue')
ax[1].set_xlim((0,1))
ax[1].set_ylim((0,1))
ax[1].set_title(u'filtered solution $\mathregular{(\omega_c = %s)}$' % cutoff)
p1 = ax[0].scatter(points_obs[:,0],points_obs[:,1],s=4,c=u_obs,
edgecolor='none',cmap='Greys_r',vmin=-0.1,vmax=1.1)
'''
This script demonstrates using rbf.filter.filter to denoise and
differentiate a time series
'''
import numpy as np
import matplotlib.pyplot as plt
from rbf.filter import filter
np.random.seed(1)
t = np.linspace(0.0,5.0,200)
u_true,udiff_true = np.sin(t),np.cos(t) # true signal
sigma_obs = 0.5*np.ones(t.shape)
u_obs = u_true + np.random.normal(0.0,sigma_obs) # observed
# find the filtered solution
u_pred,sigma_pred = filter(t[:,None],u_obs,sigma_obs,
samples=1000,n=3,
cutoff=0.5)
# find the derivative of the filtered solution
udiff_pred,sigmadiff_pred = filter(t[:,None],u_obs,sigma_obs,
samples=1000,n=3,
cutoff=0.5,diffs=(1,))
# plot the results
fig,ax = plt.subplots(2,1)
ax[0].plot(t,u_obs,'k.',label='observed',zorder=0)
ax[0].plot(t,u_true,'r-',label='true signal',zorder=2)
ax[0].plot(t,u_pred,'b-',label='filtered',zorder=1)
ax[0].fill_between(t,u_pred-sigma_pred,u_pred+sigma_pred,
color='b',alpha=0.4,edgecolor='none',zorder=1)
ax[0].legend(frameon=False)
ax[1].plot(t,udiff_true,'r-',label='true signal derivative',zorder=2)
this signal has a discontinuity from (0.0,-2.0) to (0.0,2.0)
'''
a = np.arctan2(x[:,0], 2 + x[:,1])
b = np.arctan2(x[:,0],-2 + x[:,1])
return (a - b)/np.pi
# define the known discontinuity
bnd_vert = np.array([[0.0, 2.0],[0.0,-2.0]])
bnd_smp = np.array([[0,1]])
# create synthetic data
pnts_obs = 8*(rbf.halton.halton(200,2,start=1) - 0.5)
sigma_obs = 0.2*np.ones(200)
u_obs = signal(pnts_obs) + np.random.normal(0.0,sigma_obs)
# find the filtered solution
cutoff = 0.5
soln,_ = filter(pnts_obs,u_obs,sigma=sigma_obs,cutoff=cutoff,
vert=bnd_vert,smp=bnd_smp,check_all_edges=True)
# plot the results and true signal
vals = np.linspace(-4,4,200)
grid = np.reshape(np.meshgrid(vals,vals),(2,200**2)).T
fig,axs = plt.subplots(2,1,figsize=(6,10))
axs[0].scatter(pnts_obs[:,0],pnts_obs[:,1],c=u_obs,s=50,
vmin=-1.0,vmax=1.0,cmap='viridis',zorder=1)
p = axs[0].tripcolor(grid[:,0],grid[:,1],signal(grid),
vmin=-1.0,vmax=1.0,cmap='viridis',zorder=0)
axs[0].plot(bnd_vert[:,0],bnd_vert[:,1],'r-',lw=4)
axs[0].set_xlim((-4,4));axs[0].set_ylim((-4,4))
axs[0].set_aspect('equal')
axs[0].set_title(u'observed data and true signal')
plt.colorbar(p,ax=axs[0])
axs[1].scatter(pnts_obs[:,0],pnts_obs[:,1],c=soln,s=50,
vmin=-1.0,vmax=1.0,cmap='viridis')
return (a - b) / np.pi
# define the known discontinuity
bnd_vert = np.array([[0.0, 2.0], [0.0, -2.0]])
bnd_smp = np.array([[0, 1]])
# create synthetic data
pnts_obs = 8 * (rbf.halton.halton(200, 2, start=1) - 0.5)
sigma_obs = 0.2 * np.ones(200)
u_obs = signal(pnts_obs) + np.random.normal(0.0, sigma_obs)
# find the filtered solution
cutoff = 0.5
soln, _ = filter(pnts_obs,
u_obs,
sigma=sigma_obs,
cutoff=cutoff,
vert=bnd_vert,
smp=bnd_smp,
check_all_edges=True)
# plot the results and true signal
vals = np.linspace(-4, 4, 200)
grid = np.reshape(np.meshgrid(vals, vals), (2, 200**2)).T
fig, axs = plt.subplots(2, 1, figsize=(6, 10))
axs[0].scatter(pnts_obs[:, 0],
pnts_obs[:, 1],
c=u_obs,
s=50,
vmin=-1.0,
vmax=1.0,
cmap='viridis',
zorder=1)
x = np.linspace(0.0,1.0,512)
y = np.linspace(1.0,0.0,512)
x,y = np.meshgrid(x,y)
points = np.array([x.flatten(),y.flatten()]).T
values = lena().flatten().astype(float)
values /= 255.0 # normalize so that the max is 1.0
signal = NearestNDInterpolator(points,values)
# interpolate Lena onto new observation points and add noise
N = 10000
points_obs = np.random.normal(0.5,0.25,(N,2))
u_obs = signal(points_obs)
u_obs += np.random.normal(0.0,0.5,N)
# find filtered solution
soln,sigma = filter(points_obs,u_obs,cutoff=40,size=20)
# plot the observed and filtered results
fig,ax = plt.subplots(2,1,figsize=(6,10))
ax[0].set_aspect('equal')
ax[0].set_axis_bgcolor('blue')
ax[0].set_xlim((0,1))
ax[0].set_ylim((0,1))
ax[0].set_title('Noisy Data')
ax[1].set_aspect('equal')
ax[1].set_axis_bgcolor('blue')
ax[1].set_xlim((0,1))
ax[1].set_ylim((0,1))
ax[1].set_title(u'Filtered Solution $(\omega_c = 40)$')
p1 = ax[0].scatter(points_obs[:,0],points_obs[:,1],s=4,c=u_obs,
edgecolor='none',cmap='Greys_r',vmin=-0.2,vmax=1.2)
differentiate a time series
'''
import numpy as np
import matplotlib.pyplot as plt
from rbf.filter import filter
np.random.seed(1)
t = np.linspace(0.0, 5.0, 200)
u_true, udiff_true = np.sin(t), np.cos(t) # true signal
sigma_obs = 0.5 * np.ones(t.shape)
u_obs = u_true + np.random.normal(0.0, sigma_obs) # observed
# find the filtered solution
u_pred, sigma_pred = filter(t[:, None],
u_obs,
sigma_obs,
samples=1000,
n=3,
cutoff=0.5)
# find the derivative of the filtered solution
udiff_pred, sigmadiff_pred = filter(t[:, None],
u_obs,
sigma_obs,
samples=1000,
n=3,
cutoff=0.5,
diffs=(1, ))
# plot the results
fig, ax = plt.subplots(2, 1)
ax[0].plot(t, u_obs, 'k.', label='observed', zorder=0)
ax[0].plot(t, u_true, 'r-', label='true signal', zorder=2)