From b5b28e6717ead89b6880fd3581038dbc68d6b5e1 Mon Sep 17 00:00:00 2001
From: Jan Caron <j.caron@fz-juelich.de>
Date: Sun, 15 Mar 2015 17:01:54 +0100
Subject: [PATCH] Implemented RotTiltProjector for arbitrary tilts using
quaternions! projector: RotTiltProjector implemented! quaternion: New module
providing quaternions!
---
pyramid/__init__.py | 2 +
pyramid/projector.py | 233 +++++++++++++++++++++++++++++++++++-----
pyramid/quaternion.py | 72 +++++++++++++
tests/test_projector.py | 9 +-
4 files changed, 286 insertions(+), 30 deletions(-)
create mode 100644 pyramid/quaternion.py
diff --git a/pyramid/__init__.py b/pyramid/__init__.py
index 5a4b786..a5e055d 100644
--- a/pyramid/__init__.py
+++ b/pyramid/__init__.py
@@ -57,6 +57,7 @@ from .phasemap import * # analysis:ignore
from .phasemapper import * # analysis:ignore
from .projector import * # analysis:ignore
from .regularisator import * # analysis:ignore
+from .quaternion import * # analysis:ignore
from .config import * # analysis:ignore
from .version import version as __version__
from .version import hg_revision as __hg_revision__
@@ -78,3 +79,4 @@ __all__.extend(phasemap.__all__)
__all__.extend(phasemapper.__all__)
__all__.extend(projector.__all__)
__all__.extend(regularisator.__all__)
+__all__.extend(quaternion.__all__)
diff --git a/pyramid/projector.py b/pyramid/projector.py
index ef845e9..f7bedb0 100644
--- a/pyramid/projector.py
+++ b/pyramid/projector.py
@@ -6,6 +6,8 @@
projections of vector and scalar fields."""
+from __future__ import division
+
import numpy as np
from numpy import pi
@@ -16,12 +18,13 @@ import itertools
from scipy.sparse import coo_matrix, csr_matrix
from pyramid.magdata import MagData
+from pyramid.quaternion import Quaternion
import pyramid.fft as fft
import logging
-__all__ = ['XTiltProjector', 'YTiltProjector', 'SimpleProjector']
+__all__ = ['RotTiltProjector', 'XTiltProjector', 'YTiltProjector', 'SimpleProjector']
class Projector(object):
@@ -91,40 +94,49 @@ class Projector(object):
return mag_proj
def _vector_field_projection(self, vector):
- size_2d, size_3d = self.size_2d, self.size_3d
- result = fft.zeros(2*size_2d, dtype=fft.FLOAT)
+ result = fft.zeros(2*self.size_2d, dtype=fft.FLOAT)
# Go over all possible component projections (z, y, x) to (u, v):
- # TODO: save self.weight.dot(...)
+ vec_x, vec_y, vec_z = np.split(vector, 3)
+ vec_x_weighted = self.weight.dot(vec_x)
+ vec_y_weighted = self.weight.dot(vec_y)
+ vec_z_weighted = self.weight.dot(vec_z)
+ slice_u = slice(0, self.size_2d)
+ slice_v = slice(self.size_2d, 2*self.size_2d)
if self.coeff[0][0] != 0: # x to u
- result[:size_2d] += self.coeff[0][0] * self.weight.dot(vector[:size_3d])
+ result[slice_u] += self.coeff[0][0] * vec_x_weighted
if self.coeff[0][1] != 0: # y to u
- result[:size_2d] += self.coeff[0][1] * self.weight.dot(vector[size_3d:2*size_3d])
+ result[slice_u] += self.coeff[0][1] * vec_y_weighted
if self.coeff[0][2] != 0: # z to u
- result[:size_2d] += self.coeff[0][2] * self.weight.dot(vector[2*size_3d:])
+ result[slice_u] += self.coeff[0][2] * vec_z_weighted
if self.coeff[1][0] != 0: # x to v
- result[size_2d:] += self.coeff[1][0] * self.weight.dot(vector[:size_3d])
+ result[slice_v] += self.coeff[1][0] * vec_x_weighted
if self.coeff[1][1] != 0: # y to v
- result[size_2d:] += self.coeff[1][1] * self.weight.dot(vector[size_3d:2*size_3d])
+ result[slice_v] += self.coeff[1][1] * vec_y_weighted
if self.coeff[1][2] != 0: # z to v
- result[size_2d:] += self.coeff[1][2] * self.weight.dot(vector[2*size_3d:])
+ result[slice_v] += self.coeff[1][2] * vec_z_weighted
return result
def _vector_field_projection_T(self, vector):
- size_2d, size_3d = self.size_2d, self.size_3d
- result = np.zeros(3*size_3d, dtype=fft.FLOAT)
+ result = np.zeros(3*self.size_3d, dtype=fft.FLOAT)
# Go over all possible component projections (u, v) to (z, y, x):
+ vec_u, vec_v = np.split(vector, 2)
+ vec_u_weighted = self.weight.T.dot(vec_u)
+ vec_v_weighted = self.weight.T.dot(vec_v)
+ slice_x = slice(0, self.size_3d)
+ slice_y = slice(self.size_3d, 2*self.size_3d)
+ slice_z = slice(2*self.size_3d, 3*self.size_3d)
if self.coeff[0][0] != 0: # u to x
- result[:size_3d] += self.coeff[0][0] * self.weight.T.dot(vector[:size_2d])
+ result[slice_x] += self.coeff[0][0] * vec_u_weighted
if self.coeff[0][1] != 0: # u to y
- result[size_3d:2*size_3d] += self.coeff[0][1] * self.weight.T.dot(vector[:size_2d])
+ result[slice_y] += self.coeff[0][1] * vec_u_weighted
if self.coeff[0][2] != 0: # u to z
- result[2*size_3d:] += self.coeff[0][2] * self.weight.T.dot(vector[:size_2d])
+ result[slice_z] += self.coeff[0][2] * vec_u_weighted
if self.coeff[1][0] != 0: # v to x
- result[:size_3d] += self.coeff[1][0] * self.weight.T.dot(vector[size_2d:])
+ result[slice_x] += self.coeff[1][0] * vec_v_weighted
if self.coeff[1][1] != 0: # v to y
- result[size_3d:2*size_3d] += self.coeff[1][1] * self.weight.T.dot(vector[size_2d:])
+ result[slice_y] += self.coeff[1][1] * vec_v_weighted
if self.coeff[1][2] != 0: # v to z
- result[2*size_3d:] += self.coeff[1][2] * self.weight.T.dot(vector[size_2d:])
+ result[slice_z] += self.coeff[1][2] * vec_v_weighted
return result
def _scalar_field_projection(self, vector):
@@ -202,6 +214,173 @@ class Projector(object):
raise NotImplementedError()
+class RotTiltProjector(Projector):
+
+ '''Class representing a projection function with a rotation around z followed by tilt around x.
+
+ The :class:`~.XTiltProjector` class represents a projection function for a 3-dimensional
+ vector- or scalar field onto a 2-dimensional grid, which is a concrete subclass of
+ :class:`~.Projector`.
+
+ Attributes
+ ----------
+ dim : tuple (N=3)
+ Dimensions (z, y, x) of the magnetization distribution.
+ rotation : float
+ Angle in `rad` describing the rotation around the z-axis before the tilt is happening.
+ tilt : float
+ Angle in `rad` describing the tilt of the beam direction relative to the x-axis.
+ dim_uv : tuple (N=2), optional
+ Dimensions (v, u) of the projection. If not set defaults to the (y, x)-dimensions.
+ subcount : int (optional)
+ Number of subpixels along one axis. This is used to create the lookup table which uses
+ a discrete subgrid to estimate the impact point of a voxel onto a pixel and the weight on
+ all surrounding pixels. Default is 11 (odd numbers provide a symmetric center).
+ sparsity : float
+ Measures the sparsity of the weighting (not the complete one!), 1 means completely sparse!
+
+ '''
+
+ _log = logging.getLogger(__name__+'.RotTiltProjector')
+
+ def __init__(self, dim, rotation, tilt, dim_uv=None, subcount=11):
+ self._log.debug('Calling __init__')
+ self.rotation = rotation
+ self.tilt = tilt
+ # Determine dimensions:
+ dim_z, dim_y, dim_x = dim
+ center = (dim_z/2., dim_y/2., dim_x/2.)
+ if dim_uv is None:
+ dim_v = max(dim_x, dim_y) # first rotate around z-axis (take x and y into account)
+ dim_u = max(dim_v, dim_z) # then tilt around x-axis (now z matters, too)
+ dim_uv = (dim_v, dim_u)
+ dim_v, dim_u = dim_uv
+ # Creating coordinate list of all voxels:
+ voxels = list(itertools.product(range(dim_z), range(dim_y), range(dim_x)))
+ # Calculate vectors to voxels relative to rotation center:
+ voxel_vecs = (np.asarray(voxels)+0.5 - np.asarray(center)).T
+ # Create tilt, rotation and combined quaternion, careful: Quaternion(w,x,y,z), not (z,y,x):
+ quat_x = Quaternion.from_axisangle((0, 1, 0), tilt) # Tilt around y-axis
+ quat_z = Quaternion.from_axisangle((0, 0, 1), rotation) # Rotate around z-axis
+ quat = quat_x * quat_z # Combined quaternion (first rotate around z, then tilt around x)
+ # Calculate impact positions on the projected pixel coordinate grid (flip because quat.):
+ impacts = np.flipud(quat.matrix[:2, :].dot(np.flipud(voxel_vecs))) # only care for x/y
+ impacts[1, :] += dim_u/2. # Shift back to normal indices
+ impacts[0, :] += dim_v/2. # Shift back to normal indices
+ # Calculate equivalence radius:
+ R = (3/(4*np.pi))**(1/3.)
+ # Prepare weight matrix calculation:
+ rows = [] # 2D projection
+ columns = [] # 3D distribution
+ data = [] # weights
+ # Create 4D lookup table (1&2: which neighbour weight?, 3&4: which subpixel is hit?)
+ weight_lookup = self._create_weight_lookup(subcount, R)
+ # Go over all voxels:
+ for i, voxel in enumerate(voxels):
+ column_index = voxel[0]*dim_y*dim_x + voxel[1]*dim_x + voxel[2]
+ remainder, impact = np.modf(impacts[:, i]) # split index of impact and remainder!
+ sub_pixel = (remainder * subcount).astype(dtype=np.int) # sub_pixel inside impact px.
+ # Go over all influenced pixels (impact and neighbours, indices are [0, 1, 2]!):
+ for px_ind in list(itertools.product(range(3), range(3))):
+ # Pixel indices influenced by the impact (px_ind-1 to center them around impact):
+ pixel = (impact + np.array(px_ind)-1).astype(dtype=np.int)
+ # Check if pixel is out of bound:
+ if 0 <= pixel[0] < dim_uv[0] and 0 <= pixel[1] < dim_uv[1]:
+ # Lookup weight in 4-dimensional lookup table!
+ weight = weight_lookup[px_ind[0], px_ind[1], sub_pixel[0], sub_pixel[1]]
+ # Only write into sparse matrix if weight is not zero:
+ if weight != 0.:
+ row_index = pixel[0]*dim_u + pixel[1]
+ columns.append(column_index)
+ rows.append(row_index)
+ data.append(weight)
+ # Calculate weight matrix and coefficients for jacobi matrix:
+ shape = (np.prod(dim_uv), np.prod(dim))
+ self.sparsity = 1. - len(data)/np.prod(shape)
+ print self.sparsity
+ weights = csr_matrix(coo_matrix((data, (rows, columns)), shape=shape))
+ # Calculate coefficients by rotating unity matrix (unit vectors, (x,y,z)):
+ coeff = quat.matrix[:2, :].dot(np.eye(3))
+ super(RotTiltProjector, self).__init__(dim, dim_uv, weights, coeff)
+ self._log.debug('Created '+str(self))
+
+ def _create_weight_lookup(self, subcount, R):
+ s = subcount
+ Rz = R * s # Radius in subgrid units
+ dim_zoom = (3*s, 3*s) # Dimensions of the subgrid, (3, 3) because of neighbour count!
+ cent_zoom = (np.asarray(dim_zoom)/2.).astype(dtype=np.int) # Center of the subgrid
+ y, x = np.indices(dim_zoom)
+ y -= cent_zoom[0]
+ x -= cent_zoom[1]
+ # Calculate projected thickness of an equivalence sphere (normed!):
+ d = np.where(np.hypot(x, y) <= Rz, np.sqrt(Rz**2-x**2-y**2), 0)
+ d /= d.sum()
+ # Create lookup table (4D):
+ lookup = np.zeros((3, 3, s, s))
+ # Go over all 9 pixels (center and neighbours):
+ for pixel in list(itertools.product(range(3), range(3))):
+ pixel_lb = np.array(pixel) * s # Convert to subgrid, hit bottom left of the pixel!
+ # Go over all subpixels in the center that can be hit:
+ for sub_pixel in list(itertools.product(range(s), range(s))):
+ shift = np.array(sub_pixel) - np.array((s//2, s//2)) # relative to center!
+ lb = pixel_lb - shift # Shift summing zone according to hit subpixel!
+ # Make sure, that the summing zone is in bounds (otherwise correct accordingly):
+ lb = np.where(lb >= 0, lb, [0, 0])
+ tr = np.where(lb < 3*s, lb+np.array((s, s)), [3*s, 3*s])
+ # Calculate weight by summing over the summing zone:
+ weight = d[lb[0]:tr[0], lb[1]:tr[1]].sum()
+ lookup[pixel[0], pixel[1], sub_pixel[0], sub_pixel[1]] = weight
+ return lookup
+
+ def _create_weight_lookupNEW(self, subcount, R):
+ s = subcount
+ Rz = R * s # Radius in subgrid units
+ dim_zoom = (3*s, 3*s) # Dimensions of the subgrid, (3, 3) because of neighbour count!
+ cent_zoom = (np.asarray(dim_zoom)/2.).astype(dtype=np.int) # Center of the subgrid
+ y, x = np.indices(dim_zoom)
+ y -= cent_zoom[0]
+ x -= cent_zoom[1]
+ # Calculate projected thickness of an equivalence sphere (normed!):
+ d = np.where(np.hypot(x, y) <= Rz, np.sqrt(Rz**2-x**2-y**2), 0)
+ d /= d.sum()
+ # Create lookup table (4D):
+ lookup = np.zeros((3, 3, s, s))
+ # Go over all 9 pixels (center and neighbours):
+ for pixel in list(itertools.product(range(3), range(3))):
+ pixel_lb = np.array(pixel) * s # Convert to subgrid, hit bottom left of the pixel!
+ # Go over all subpixels in the center that can be hit:
+ for sub_pixel in list(itertools.product(range(s), range(s))):
+ shift = np.array(sub_pixel) - np.array((s//2, s//2)) # relative to center!
+ pixel_lb -= shift # Shift summing zone according to the hit subpixel!
+ # Make sure, that the summing zone is in bounds (otherwise correct accordingly):
+ pixel_lb = np.where(pixel_lb >= 0, pixel_lb, [0, 0])
+ pixel_tr = np.where(pixel_lb < 3*s, pixel_lb+np.array((s, s)), [3*s, 3*s])
+ # Calculate weight by summing over the summing zone:
+ weight = d[pixel_lb[0]:pixel_tr[0], pixel_lb[1]:pixel_tr[1]].sum()
+ lookup[pixel[0], pixel[1], sub_pixel[0], sub_pixel[1]] = weight
+ return lookup
+
+ def get_info(self, verbose=False):
+ '''Get specific information about the projector as a string.
+
+ Parameters
+ ----------
+ verbose: boolean, optional
+ If this is true, the text looks prettier (maybe using latex). Default is False for the
+ use in file names and such.
+
+ Returns
+ -------
+ info : string
+ Information about the projector as a string, e.g. for the use in plot titles.
+
+ '''
+ if verbose:
+ return u'x-tilt: $\phi = {:d}$°'.format(int(np.round(self.tilt*180/pi)))
+ else:
+ return u'xtilt_phi={:d}°'.format(int(np.round(self.tilt*180/pi)))
+
+
class XTiltProjector(Projector):
'''Class representing a projection function with a tilt around the x-axis.
@@ -230,7 +409,7 @@ class XTiltProjector(Projector):
direc_vec = np.array((np.cos(tilt), -np.sin(tilt))) # vector pointing along projection
distances = -np.cross(point_vecs, direc_vec) # here (special case): divisor is one!
# minus because sign is derived of -sin(angle(point_vec, direc_vec)) neg between 0-180°
- return distances + size/2. # Shift to center the projection
+ return distances + size/2. # Shift to the center of the projection
def get_impact(pos, r, size):
return [x for x in np.arange(np.floor(pos-r), np.floor(pos+r)+1, dtype=int)
@@ -275,11 +454,13 @@ class XTiltProjector(Projector):
# one slice:
for i, voxel in enumerate(voxels):
impacts = get_impact(positions[i], r, dim_v) # impact along projected y-axis
+ voxel_index = voxel[0]*dim_rot*dim_perp + voxel[1]*dim_rot # 0: z, 1: y
for impact in impacts:
+ impact_index = impact*dim_u+int((dim_u-dim_rot)/2)
distance = np.abs(impact+0.5 - positions[i])
delta = distance / r
- col.append(voxel[0]*dim_rot*dim_perp + voxel[1]*dim_rot) # 0: z, 1: y
- row.append(impact*dim_u+int((dim_u-dim_rot)/2))
+ col.append(voxel_index)
+ row.append(impact_index)
data.append(get_weight(delta, rho))
# All other slices (along x):
columns = col
@@ -343,7 +524,7 @@ class YTiltProjector(Projector):
direc_vec = np.array((np.cos(tilt), -np.sin(tilt))) # vector pointing along projection
distances = -np.cross(point_vecs, direc_vec) # here (special case): divisor is one!
# minus because sign is derived of -sin(angle(point_vec, direc_vec)) neg between 0-180°
- return distances + size/2. # Shift to center the projection
+ return distances + size/2. # Shift to the center of the projection
def get_impact(pos, r, size):
return [x for x in np.arange(np.floor(pos-r), np.floor(pos+r)+1, dtype=int)
@@ -388,11 +569,13 @@ class YTiltProjector(Projector):
# one slice:
for i, voxel in enumerate(voxels):
impacts = get_impact(positions[i], r, dim_u) # impact along projected x-axis
+ voxel_index = voxel[0]*dim_perp*dim_rot + voxel[1] # 0: z, 1: x
for impact in impacts:
+ impact_index = impact+int((dim_v-dim_rot)/2)*dim_u
distance = np.abs(impact+0.5 - positions[i])
delta = distance / r
- col.append(voxel[0]*dim_perp*dim_rot + voxel[1]) # 0: z, 1: x
- row.append(impact+int((dim_v-dim_rot)/2)*dim_u)
+ col.append(voxel_index)
+ row.append(impact_index)
data.append(get_weight(delta, rho))
# All other slices (along y):
columns = col
@@ -516,5 +699,3 @@ class SimpleProjector(Projector):
return 'projected along {}-axis'.format(self.axis)
else:
return '{}axis'.format(self.axis)
-
-# TODO: Arbitrary Projections!
diff --git a/pyramid/quaternion.py b/pyramid/quaternion.py
new file mode 100644
index 0000000..f350d91
--- /dev/null
+++ b/pyramid/quaternion.py
@@ -0,0 +1,72 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Tue Mar 10 18:07:53 2015
+
+@author: Jan
+"""
+
+# TODO: Cleanup
+
+import numpy as np
+
+
+__all__ = ['Quaternion']
+
+
+class Quaternion(object):
+
+ NORM_TOLERANCE = 1E-6
+
+ @property
+ def conj(self):
+ w, x, y, z = self.values
+ return Quaternion((w, -x, -y, -z))
+
+ @property
+ def matrix(self):
+ w, x, y, z = self.values
+ return np.array([[1-2*(y**2+z**2), 2*(x*y-w*z), 2*(x*z+w*y)],
+ [2*(x*y+w*z), 1-2*(x**2+z**2), 2*(y*z-w*x)],
+ [2*(x*z-w*y), 2*(y*z+w*x), 1-2*(x**2+y**2)]])
+
+ def __init__(self, values):
+ self.values = values
+ self._normalize()
+
+ def __mul__(self, other): # self * other
+ if isinstance(other, Quaternion): # Quaternion multiplication
+ return self.dot_quat(self, other)
+ elif len(other) == 3: # vector multiplication
+ q_vec = Quaternion((0,)+tuple(other))
+ q = self.dot_quat(self.dot_quat(self, q_vec), self.conj)
+ return q.values[1:]
+
+ def dot_quat(self, q1, q2):
+ w1, x1, y1, z1 = q1.values
+ w2, x2, y2, z2 = q2.values
+ w = w1*w2 - x1*x2 - y1*y2 - z1*z2
+ x = w1*x2 + x1*w2 + y1*z2 - z1*y2
+ y = w1*y2 + y1*w2 + z1*x2 - x1*z2
+ z = w1*z2 + z1*w2 + x1*y2 - y1*x2
+ return Quaternion((w, x, y, z))
+
+ def _normalize(self):
+ mag2 = np.sum(n**2 for n in self.values)
+ if abs(mag2 - 1.0) > self.NORM_TOLERANCE:
+ mag = np.sqrt(mag2)
+ self.values = tuple(n / mag for n in self.values)
+
+ @classmethod
+ def from_axisangle(cls, vector, theta):
+ x, y, z = vector
+ theta /= 2.
+ w = np.cos(theta)
+ x *= np.sin(theta)
+ y *= np.sin(theta)
+ z *= np.sin(theta)
+ return Quaternion((w, x, y, z))
+
+ def to_axisangle(self):
+ w, x, y, z = self.values
+ theta = 2.0 * np.arccos(w)
+ return np.array((x, y, z)), theta
diff --git a/tests/test_projector.py b/tests/test_projector.py
index ca6fa33..b945beb 100644
--- a/tests/test_projector.py
+++ b/tests/test_projector.py
@@ -246,11 +246,12 @@ class TestCaseYTiltProjector(unittest.TestCase):
assert_allclose(jac_T_90, jac_T_90_ref,
err_msg='Unexpected behaviour in the the transp. jacobi matrix! (90°)')
+# TODO: Test RotTiltProjector!!!
if __name__ == '__main__':
suite = unittest.TestLoader().loadTestsFromTestCase(TestCaseSimpleProjector)
unittest.TextTestRunner(verbosity=2).run(suite)
-# suite = unittest.TestLoader().loadTestsFromTestCase(TestCaseXTiltProjector)
-# unittest.TextTestRunner(verbosity=2).run(suite)
-# suite = unittest.TestLoader().loadTestsFromTestCase(TestCaseYTiltProjector)
-# unittest.TextTestRunner(verbosity=2).run(suite)
+ suite = unittest.TestLoader().loadTestsFromTestCase(TestCaseXTiltProjector)
+ unittest.TextTestRunner(verbosity=2).run(suite)
+ suite = unittest.TestLoader().loadTestsFromTestCase(TestCaseYTiltProjector)
+ unittest.TextTestRunner(verbosity=2).run(suite)
--
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