Python source code: tv_denoise.py

def _tv_denoise_3d(im, weight=100, eps=2.e-4, keep_type=False, n_iter_max=200):
"""
Perform total-variation denoising on 3-D arrays
Parameters
----------
im: ndarray
3-D input data to be denoised
weight: float, optional
denoising weight. The greater ``weight``, the more denoising (at
the expense of fidelity to ``input``)
eps: float, optional
relative difference of the value of the cost function that determines
the stop criterion. The algorithm stops when:
(E_(n-1) - E_n) < eps * E_0
keep_type: bool, optional (False)
whether the output has the same dtype as the input array.
keep_type is False by default, and the dtype of the output
is np.float
n_iter_max: int, optional
maximal number of iterations used for the optimization.
Returns
-------
out: ndarray
denoised array
Notes
-----
Rudin, Osher and Fatemi algorithm
Examples
---------
First build synthetic noisy data
>>> x, y, z = np.ogrid[0:40, 0:40, 0:40]
>>> mask = (x -22)**2 + (y - 20)**2 + (z - 17)**2 < 8**2
>>> mask = mask.astype(np.float)
>>> mask += 0.2*np.random.randn(*mask.shape)
>>> res = tv_denoise_3d(mask, weight=100)
"""
im_type = im.dtype
if im_type is not np.float:
im = im.astype(np.float)
px = np.zeros_like(im)
py = np.zeros_like(im)
pz = np.zeros_like(im)
gx = np.zeros_like(im)
gy = np.zeros_like(im)
gz = np.zeros_like(im)
d = np.zeros_like(im)
i = 0
while i < n_iter_max:
d = - px - py - pz
d[1:] += px[:-1]
d[:, 1:] += py[:, :-1]
d[:, :, 1:] += pz[:, :, :-1]
out = im + d
E = (d**2).sum()
gx[:-1] = np.diff(out, axis=0)
gy[:, :-1] = np.diff(out, axis=1)
gz[:, :, :-1] = np.diff(out, axis=2)
norm = np.sqrt(gx**2 + gy**2 + gz**2)
E += weight * norm.sum()
norm *= 0.5 / weight
norm += 1.
px -= 1./6.*gx
px /= norm
py -= 1./6.*gy
py /= norm
pz -= 1/6.*gz
pz /= norm
E /= float(im.size)
if i == 0:
E_init = E
E_previous = E
else:
if np.abs(E_previous - E) < eps * E_init:
break
else:
E_previous = E
i += 1
if keep_type:
return out.astype(im_type)
else:
return out
def _tv_denoise_2d(im, weight=50, eps=2.e-4, keep_type=False, n_iter_max=200):
"""
Perform total-variation denoising
Parameters
----------
im: ndarray
input data to be denoised
weight: float, optional
denoising weight. The greater ``weight``, the more denoising (at
the expense of fidelity to ``input``)
eps: float, optional
relative difference of the value of the cost function that determines
the stop criterion. The algorithm stops when:
(E_(n-1) - E_n) < eps * E_0
keep_type: bool, optional (False)
whether the output has the same dtype as the input array.
keep_type is False by default, and the dtype of the output
is np.float
n_iter_max: int, optional
maximal number of iterations used for the optimization.
Returns
-------
out: ndarray
denoised array
Notes
-----
The principle of total variation denoising is explained in
http://en.wikipedia.org/wiki/Total_variation_denoising
This code is an implementation of the algorithm of Rudin, Fatemi and Osher
that was proposed by Chambolle in [1]_.
References
----------
.. [1] A. Chambolle, An algorithm for total variation minimization and
applications, Journal of Mathematical Imaging and Vision,
Springer, 2004, 20, 89-97.
Examples
---------
>>> import scipy
>>> lena = scipy.misc.lena().astype(np.float)
>>> lena += 0.5 * lena.std()*np.random.randn(*lena.shape)
>>> denoised_lena = tv_denoise(lena, weight=60.0)
"""
im_type = im.dtype
if im_type is not np.float:
im = im.astype(np.float)
px = np.zeros_like(im)
py = np.zeros_like(im)
gx = np.zeros_like(im)
gy = np.zeros_like(im)
d = np.zeros_like(im)
i = 0
while i < n_iter_max:
d = -px -py
d[1:] += px[:-1]
d[:, 1:] += py[:, :-1]
out = im + d
E = (d**2).sum()
gx[:-1] = np.diff(out, axis=0)
gy[:, :-1] = np.diff(out, axis=1)
norm = np.sqrt(gx**2 + gy**2)
E += weight * norm.sum()
norm *= 0.5 / weight
norm += 1
px -= 0.25*gx
px /= norm
py -= 0.25*gy
py /= norm
E /= float(im.size)
if i == 0:
E_init = E
E_previous = E
else:
if np.abs(E_previous - E) < eps * E_init:
break
else:
E_previous = E
i += 1
if keep_type:
return out.astype(im_type)
else:
return out
def tv_denoise(im, weight=50, eps=2.e-4, keep_type=False, n_iter_max=200):
"""
Perform total-variation denoising on 2-d and 3-d images
Parameters
----------
im: ndarray (2d or 3d) of ints, uints or floats
input data to be denoised. `im` can be of any numeric type,
but it is cast into an ndarray of floats for the computation
of the denoised image.
weight: float, optional
denoising weight. The greater ``weight``, the more denoising (at
the expense of fidelity to ``input``)
eps: float, optional
relative difference of the value of the cost function that
determines the stop criterion. The algorithm stops when:
(E_(n-1) - E_n) < eps * E_0
keep_type: bool, optional (False)
whether the output has the same dtype as the input array.
keep_type is False by default, and the dtype of the output
is np.float
n_iter_max: int, optional
maximal number of iterations used for the optimization.
Returns
-------
out: ndarray
denoised array
Notes
-----
The principle of total variation denoising is explained in
http://en.wikipedia.org/wiki/Total_variation_denoising
The principle of total variation denoising is to minimize the
total variation of the image, which can be roughly described as
the integral of the norm of the image gradient. Total variation
denoising tends to produce "cartoon-like" images, that is,
piecewise-constant images.
This code is an implementation of the algorithm of Rudin, Fatemi and Osher
that was proposed by Chambolle in [1]_.
References
----------
.. [1] A. Chambolle, An algorithm for total variation minimization and
applications, Journal of Mathematical Imaging and Vision,
Springer, 2004, 20, 89-97.
Examples
---------
>>> import scipy
>>> # 2D example using lena
>>> lena = scipy.misc.lena().astype(np.float)
>>> lena += 0.5 * lena.std()*np.random.randn(*lena.shape)
>>> denoised_lena = tv_denoise(lena, weight=60)
>>> # 3D example on synthetic data
>>> x, y, z = np.ogrid[0:40, 0:40, 0:40]
>>> mask = (x -22)**2 + (y - 20)**2 + (z - 17)**2 < 8**2
>>> mask = mask.astype(np.float)
>>> mask += 0.2*np.random.randn(*mask.shape)
>>> res = tv_denoise_3d(mask, weight=100)
"""
if im.ndim == 2:
return _tv_denoise_2d(im, weight, eps, keep_type, n_iter_max)
elif im.ndim == 3:
return _tv_denoise_3d(im, weight, eps, keep_type, n_iter_max)
else:
raise ValueError('only 2-d and 3-d images may be denoised with this function')