def BP_parallel_update_direction(h_m_to_px, h_px_to_m, y, proj_operator,
J=.1, use_mask=True, use_micro=False, hext=None):
"""
One iteration of BP (belief propagation), with messages updated
after all messages corresponding to one angle have been computed.
The directions are processed so that closeby directions are not processed
together. If there are ndir directions, they are processed in the order
i + j * ndir / 8 for j in range(0, 8), for i in range(0, ndir/8)
Parameters
----------
h_m_to_px: ndarray of shape (ndir, l_x, l_x) where the original image
is (l_x, l_x)
local magnetizations (measure to field). The frame is the one of the
non-rotated image (ie, Ising chains are taken at an angle
corresponding to the projection)
h_px_to_m: ndarray of shape (ndir, l_x, l_x) where the original image
is (l_x, l_x)
local magnetic field, determined by the other measures. The frame
is the one of the non-rotated image (ie, Ising chains are taken at
an angle corresponding to the projection)
y: ndarray of shape ndir*L (1-D)
Array that contains all measures (sum of spins along rays)
proj_operator: sparse matrix, in lil format
The projection operator (with only 1s in non-zero entries)
J: float
Amplitude of the coupling between spins
use_mask: bool
If True, there are no spins outside the central circle
use_micro: bool
If True, a microcanonical computation is performed for short chains,
or chains with few non-blocked spins. If False, only canonical
computations are performed.
hext : ndarray of same shape as y
Initial guess for the external fields.
Returns
-------
h_m_to_px, h_px_to_m, h_sum
h_m_to_px and h_px_to_m are the new iterates
h_sum is the sum over all directions of h_m_to_px, from which the marginal
of a spin to be +1 or -1 can be computed.
Notes
-----
The steps performed here are as follows:
1. Loop over the projection angles
2. Loop over measures for a given projection angle
3. Compute effective couplings Js
4. Solve the line
Both types of fields (h_px_to_m and h_m_to_px) are in the image frame,
and indices corresponding to the different light-rays are extracted
from the projection operator.
"""
if not joblib_import:
raise ImportError("""joblib could not be imported, use the
function BP_step instead""")
ndir = len(h_m_to_px)
if hext is None:
hext = np.zeros_like(y)
hext_new = np.zeros_like(y)
l_x = np.sqrt(h_m_to_px.shape[1])
h_tmp = np.copy(h_m_to_px)
if use_mask:
X, Y = np.ogrid[:l_x, :l_x]
mask = ((X - l_x/2)**2 + (Y - l_x/2)**2 <= (l_x/2)**2).ravel()
order = _order_dir(ndir)
for direction in order:
h_dir_tmp = np.zeros_like(h_px_to_m[direction])
for pixel_ind in range(int(l_x)):
measure_index = l_x * direction + pixel_ind
proj_value = y[measure_index]
inds = np.array(proj_operator.rows[measure_index])
if measure_index > len(y)/2:
inds = _reorder(inds, l_x)
mask_inds = mask[inds]
inds = inds[mask_inds]
if len(inds) == 0:
continue
Js = _calc_Jeff(inds, l_x, J)
h_dir_tmp[inds], hext_new[measure_index] = \
solve_line(h_px_to_m[direction][inds], Js,
proj_value, use_micro=use_micro, hext=hext[measure_index])
damping = 0.9
h_old = np.copy(h_m_to_px[direction])
h_dir_tmp = (1 - damping) * h_dir_tmp + damping * h_old
h_m_to_px[direction] = h_dir_tmp[:]
delta_h = h_dir_tmp - h_old
h_px_to_m = _calc_hatf_partial_dir(h_px_to_m, delta_h, direction)
h_sum = h_m_to_px.sum(axis=0)
h_sum[~mask] = 0
return h_m_to_px, h_px_to_m, h_sum, hext_new
def BP_step(h_m_to_px, h_px_to_m, y, proj_operator, J=.1,
use_mask=True, use_micro=False, hext=None):
"""
One iteration of BP (belief propagation), with messages updated
after all new messages have been computed. A strong damping is needed
with this update method.
Parameters
----------
h_m_to_px: ndarray of shape (ndir, l_x, l_x) where the original image
is (l_x, l_x)
local magnetizations (measure to field). The frame is the one of the
non-rotated image (ie, Ising chains are taken at an angle
corresponding to the projection)
h_px_to_m: ndarray of shape (ndir, l_x, l_x) where the original image
is (l_x, l_x)
local magnetic field, determined by the other measures. The frame
is the one of the non-rotated image (ie, Ising chains are taken at
an angle corresponding to the projection)
y: ndarray of shape ndir*L (1-D)
Array that contains all measures (sum of spins along rays)
proj_operator: sparse matrix, in lil format
The projection operator (with only 1s in non-zero entries)
J: float
Amplitude of the coupling between spins
use_mask: bool
If True, there are no spins outside the central circle
use_micro: bool
If True, a microcanonical computation is performed for short chains,
or chains with few non-blocked spins. If False, only canonical
computations are performed.
Returns
-------
h_m_to_px, h_px_to_m, h_sum
h_m_to_px and h_px_to_m are the new iterates
h_sum is the sum over all directions of h_m_to_px, from which the marginal
of a spin to be +1 or -1 can be computed.
"""
ndir = len(h_m_to_px)
if hext is None:
hext = np.zeros_like(y)
hext_new = np.zeros_like(y)
# Heuristic value that works well for the damping factor
# The more projections, the more damping we need
damping = 1 - 1.6/ndir
l_x = np.sqrt(h_m_to_px.shape[1])
h_tmp = np.copy(h_m_to_px)
if use_mask:
X, Y = np.ogrid[:l_x, :l_x]
mask = ((X - l_x/2)**2 + (Y - l_x/2)**2 <= (l_x/2)**2).ravel()
for i, proj_value in enumerate(y):
# Which pixels are in this measurement?
inds = np.array(proj_operator.rows[i])
# How are they ordered?
if i > len(y)/2:
inds = _reorder(inds, l_x)
# Handle pixels outside circle -- remove them
mask_inds = mask[inds]
inds = inds[mask_inds]
if len(inds) == 0:
continue
# effective couplings
Js = _calc_Jeff(inds, l_x, J)
mu = i / int(l_x) # angle number
# Solve the chain
h_m_to_px[mu][inds], hext_new[i] = solve_line(h_px_to_m[mu][inds], Js,
proj_value, hext=hext[i], use_micro=use_micro)
h_m_to_px = (1 - damping) * h_m_to_px + damping * h_tmp
# Then we update h_px_to_m
h_px_to_m, h_sum = _calc_hatf(h_m_to_px)
h_sum[~mask] = 0
return h_m_to_px, h_px_to_m, h_sum, hext_new
def BP_step_always_update(h_m_to_px, h_px_to_m, y, proj_operator,
J=.1, use_mask=True, use_micro=False, hext=None):
"""
One iteration of BP (belief propagation), with messages updated
after each resolution of an Ising chain.
A new random permutation of the chains is computed to determine the
order in which the chains are processed.
Parameters
----------
h_m_to_px: ndarray of shape (ndir, l_x, l_x) where the original image
is (l_x, l_x)
local magnetizations (measure to field). The frame is the one of the
non-rotated image (ie, Ising chains are taken at an angle
corresponding to the projection)
h_px_to_m: ndarray of shape (ndir, l_x, l_x) where the original image
is (l_x, l_x)
local magnetic field, determined by the other measures. The frame
is the one of the non-rotated image (ie, Ising chains are taken at
an angle corresponding to the projection)
y: ndarray of shape ndir*L (1-D)
Array that contains all measures (sum of spins along rays)
proj_operator: sparse matrix, in lil format
The projection operator (with only 1s in non-zero entries)
J: float
Amplitude of the coupling between spins
use_mask: bool
If True, there are no spins outside the central circle
use_micro: bool
If True, a microcanonical computation is performed for short chains,
or chains with few non-blocked spins. If False, only canonical
computations are performed.
Returns
-------
h_m_to_px, h_px_to_m, h_sum
h_m_to_px and h_px_to_m are the new iterates
h_sum is the sum over all directions of h_m_to_px, from which the marginal
of a spin to be +1 or -1 can be computed.
"""
ndir = len(h_m_to_px)
if hext is None:
hext = np.zeros_like(y)
hext_new = np.zeros_like(y)
l_x = np.sqrt(h_m_to_px.shape[1])
h_tmp = np.copy(h_m_to_px)
if use_mask:
X, Y = np.ogrid[:l_x, :l_x]
mask = ((X - l_x/2)**2 + (Y - l_x/2)**2 <= (l_x/2)**2).ravel()
order = np.random.permutation(range(len(y)))
for measure_index in order:
proj_value = y[measure_index]
inds = np.array(proj_operator.rows[measure_index])
if measure_index > len(y)/2:
inds = _reorder(inds, l_x)
mask_inds = mask[inds]
inds = inds[mask_inds]
if len(inds) == 0:
continue
Js = _calc_Jeff(inds, l_x, J)
mu = measure_index / int(l_x)
h_m_to_px_line, hext_new[measure_index] = \
solve_line(h_px_to_m[mu][inds], Js,
proj_value, use_micro=use_micro, hext=hext[measure_index])
damping = 0.3#5
h_m_to_px_line = (1 - damping) * h_m_to_px_line \
+ damping * h_tmp[mu][inds]
h_m_to_px[mu][inds] = h_m_to_px_line[:]
delta_h = h_m_to_px_line - h_tmp[mu][inds]
h_px_to_m = _calc_hatf_partial(h_px_to_m, delta_h, mu, inds)
# Then we update h_px_to_m
h_sum = h_m_to_px.sum(axis=0)
h_sum[~mask] = 0
return h_m_to_px, h_px_to_m, h_sum, hext_new