"""
File originally part of the Topographica project. Provides
SheetCoordinateSystem, allowing conversion between continuous 'sheet
coordinates' and integer matrix coordinates.
'Sheet coordinates' allow simulation parameters to be specified in
units that are density-independent, whereas 'matrix coordinates'
provide a means of realizing the continuous sheets.
Hence we can have a pair of 'sheet coordinates' (x,y); floating-point
Cartesian coordinates indicating an arbitrary point on the sheet's plane.
We can also have a pair of 'matrix coordinates' (r,c), which are used
to address an underlying matrix. These matrix coordinates are also
floating-point numbers to allow precise conversion between the two
schemes. Where it is necessary to address a specific element of the
matrix (as is often the case in calculations), we also have the usual
matrix index coordinates (r_idx, c_idx). We refer to these as
matrixidx coordinates. SheetCoordinateSystem proxies methods for converting
between sheet and matrix coordinates, as well as sheet and matrixidx
coordinates.
Everyone should use these facilities for conversions between the two
coordinate systems to guarantee consistency.
Example of how the matrix stores the representation of the Sheet
================================================================
For the purposes of this example, assume the goal is a Sheet with
density=3 that has a 1 at (-1/2,-1/2), a 5 at (0.0,0.0), and a 9 at
(1/2,1/2). More precisely, for this Sheet,
the continuous area from -1/2,-1/2 to -1/6,-1/6 has value 1,
the continuous area from -1/6,-1/6 to 1/6,1/6 has value 5, and
the continuous area from 1/6,1/6 to 1/2,1/2 has value 9.
With the rest of the elements filled in, the Sheet would look like::
(-1/2,1/2) -+-----+-----+-----+- (1/2,1/2)
| | | |
| 7 | 8 | 9 |
| | | |
(-1/2,1/6) -+-----+-----+-----+- (1/2,1/6)
| | | |
| 4 | 5 | 6 |
| | | |
(-1/2,-1/6) -+-----+-----+-----+- (1/2,-1/6)
| | | |
| 1 | 2 | 3 |
| | | |
(-1/2,-1/2) -+-----+-----+-----+- (1/2,-1/2)
where element 5 is centered on 0.0,0.0. A matrix that would match
these Sheet coordinates is::
[[7 8 9]
[4 5 6]
[1 2 3]]
If we have such a matrix, we can access it in one of two ways: Sheet
or matrix/matrixidx coordinates. In matrixidx coordinates, the matrix is indexed
by rows and columns, and it is possible to ask for the element at
location [0,2] (which returns 9 as in any normal row-major matrix).
But the values can additionally be accessed in Sheet coordinates,
where the matrix is indexed by a point in the Cartesian plane. In
Sheet coordinates, it is possible to ask for the element at location
(0.3,0.02), which returns floating-point matrix coordinates that can
be cropped to give the nearest matrix element, namely the one with
value 6.
Of course, it would be an error to try to pass matrix coordinates like
[0,2] to the sheet2matrix calls; the result would be a value far
outside of the actual matrix.
"""
import numpy as np
from .boundingregion import BoundingBox
from .util import datetime_types
# Note about the 'bounds-master' approach we have adopted
# =======================================================
#
# Our current approach is a "bounds-master" approach, where we trust
# the user's specified x width, and choose the nearest density and y
# height to make that possible. The advantage of this is that when we
# change the density (which is often), each such simulation is the
# best possible approximation to the given area. Generally, the area
# is much more meaningful than the density, so this approach makes
# sense. Plus, the y height is usually the same as the x width, so
# it's not usually a problem that the y height is not respected. The
# disadvantage is that the user's area can only be trusted in one
# dimension, because of wanting to avoid the complication of separate
# xdensity and ydensity, which makes this approach very difficult to
# explain to the user.
#
# The other approach is density-master: trust the user's specified
# density as-is, and simply choose the nearest area that fits that
# density. The advantages are that (a) it's very simple to describe
# and reason about, and (b) if a user asks for a different area, they
# get a true subsection of the same simulation they would have gotten
# at the larger area. The disadvantage is that the simulation isn't
# the best approximation of the given area that it could be -- e.g. at
# low densities, the sheet area could be quite significantly different
# than the one the user requested. Plus, if we took this approach
# seriously, then we'd let the density specify the matrix coordinate
# system entirely, including the origin, which would mean that the
# actual area would often be offset from the intended one, which is
# even worse. Differences between the area and the offset could cause
# severe problems in the alignment of projections between sheets with
# different densities, which would make low-density versions of
# hierarchical models behave very strangely.
[docs]class SheetCoordinateSystem(object):
"""
Provides methods to allow conversion between sheet and matrix
coordinates.
"""
def __get_xdensity(self):
return self.__xdensity
def __get_ydensity(self):
return self.__ydensity
def __get_shape(self):
return self.__shape
xdensity = property(__get_xdensity, doc="""
The spacing between elements in an underlying matrix
representation, in the x direction.""")
ydensity = property(__get_ydensity, doc="""
The spacing between elements in an underlying matrix
representation, in the y direction.""")
shape = property(__get_shape)
# Determines the unit of time densities are defined relative to
# when one or both axes are datetime types
_time_unit = 'us'
def __init__(self,bounds,xdensity,ydensity=None):
"""
Store the bounds (as l,b,r,t in an array), xdensity, and
ydensity.
If ydensity is not specified, it is assumed that the specified
xdensity is nominal and that the true xdensity should be
calculated. The top and bottom bounds are adjusted so that the
ydensity is equal to the xdensity.
If both xdensity and ydensity are specified, these and the
bounds are taken to be exact and are not adjusted.
"""
if not ydensity:
bounds,xdensity = self.__equalize_densities(bounds,xdensity)
self.bounds = bounds
self.__set_xdensity(xdensity)
self.__set_ydensity(ydensity or xdensity)
self.lbrt = np.array(bounds.lbrt())
r1,r2,c1,c2 = Slice._boundsspec2slicespec(self.lbrt,self)
self.__shape = (r2-r1,c2-c1)
# xstep and ystep allow division to be avoid for speed reasons
def __set_xdensity(self,density):
self.__xdensity=density
self.__xstep = 1.0/density
def __set_ydensity(self,density):
self.__ydensity=density
self.__ystep = 1.0/density
def __equalize_densities(self,nominal_bounds,nominal_density):
"""
Calculate the true density along x, and adjust the top and
bottom bounds so that the density along y will be equal.
Returns (adjusted_bounds, true_density)
"""
left,bottom,right,top = nominal_bounds.lbrt()
width = right-left; height = top-bottom
center_y = bottom + height/2.0
# True density is not equal to the nominal_density when
# nominal_density*(right-left) is not an integer.
true_density = int(nominal_density*(width))/float(width)
n_cells = round(height*true_density,0)
adjusted_half_height = n_cells/true_density/2.0
return (BoundingBox(points=((left, center_y-adjusted_half_height),
(right, center_y+adjusted_half_height))),
true_density)
[docs] def sheet2matrix(self,x,y):
"""
Convert a point (x,y) in Sheet coordinates to continuous
matrix coordinates.
Returns (float_row,float_col), where float_row corresponds to
y, and float_col to x.
Valid for scalar or array x and y.
Note about Bounds For a Sheet with
BoundingBox(points=((-0.5,-0.5),(0.5,0.5))) and density=3,
x=-0.5 corresponds to float_col=0.0 and x=0.5 corresponds to
float_col=3.0. float_col=3.0 is not inside the matrix
representing this Sheet, which has the three columns
(0,1,2). That is, x=-0.5 is inside the BoundingBox but x=0.5
is outside. Similarly, y=0.5 is inside (at row 0) but y=-0.5
is outside (at row 3) (it's the other way round for y because
the matrix row index increases as y decreases).
"""
# First translate to (left,top), which is [0,0] in the matrix,
# then scale to the size of the matrix. The y coordinate needs
# to be flipped, because the points are moving down in the
# sheet as the y index increases in the matrix.
xdensity = self.__xdensity
if ((isinstance(x, np.ndarray) and x.dtype.kind == 'M') or
isinstance(x, datetime_types)):
xdensity = np.timedelta64(int(round(1./xdensity)), self._time_unit)
float_col = (x-self.lbrt[0]) / xdensity
else:
float_col = (x-self.lbrt[0]) * xdensity
ydensity = self.__ydensity
if ((isinstance(y, np.ndarray) and y.dtype.kind == 'M') or
isinstance(y, datetime_types)):
ydensity = np.timedelta64(int(round(1./ydensity)), self._time_unit)
float_row = (self.lbrt[3]-y) / ydensity
else:
float_row = (self.lbrt[3]-y) * ydensity
return float_row, float_col
[docs] def sheet2matrixidx(self,x,y):
"""
Convert a point (x,y) in sheet coordinates to the integer row
and column index of the matrix cell in which that point falls,
given a bounds and density. Returns (row,column).
Note that if coordinates along the right or bottom boundary
are passed into this function, the returned matrix coordinate
of the boundary will be just outside the matrix, because the
right and bottom boundaries are exclusive.
Valid for scalar or array x and y.
"""
r,c = self.sheet2matrix(x,y)
r = np.floor(r)
c = np.floor(c)
if hasattr(r,'astype'):
return r.astype(int), c.astype(int)
else:
return int(r),int(c)
[docs] def matrix2sheet(self,float_row,float_col):
"""
Convert a floating-point location (float_row,float_col) in
matrix coordinates to its corresponding location (x,y) in
sheet coordinates.
Valid for scalar or array float_row and float_col.
Inverse of sheet2matrix().
"""
xoffset = float_col*self.__xstep
if isinstance(self.lbrt[0], datetime_types):
xoffset = np.timedelta64(int(round(xoffset)), self._time_unit)
x = self.lbrt[0] + xoffset
yoffset = float_row*self.__ystep
if isinstance(self.lbrt[3], datetime_types):
yoffset = np.timedelta64(int(round(yoffset)), self._time_unit)
y = self.lbrt[3] - yoffset
return x, y
[docs] def matrixidx2sheet(self,row,col):
"""
Return (x,y) where x and y are the floating point coordinates
of the *center* of the given matrix cell (row,col). If the
matrix cell represents a 0.2 by 0.2 region, then the center
location returned would be 0.1,0.1.
NOTE: This is NOT the strict mathematical inverse of
sheet2matrixidx(), because sheet2matrixidx() discards all but
the integer portion of the continuous matrix coordinate.
Valid only for scalar or array row and col.
"""
x,y = self.matrix2sheet((row+0.5), (col+0.5))
# Rounding allows easier comparison with user specified values
if not isinstance(x, datetime_types):
x = np.around(x,10)
if not isinstance(y, datetime_types):
y = np.around(y,10)
return x, y
[docs] def closest_cell_center(self,x,y):
"""
Given arbitrary sheet coordinates, return the sheet coordinates
of the center of the closest unit.
"""
return self.matrixidx2sheet(*self.sheet2matrixidx(x,y))
[docs] def sheetcoordinates_of_matrixidx(self):
"""
Return x,y where x is a vector of sheet coordinates
representing the x-center of each matrix cell, and y
represents the corresponding y-center of the cell.
"""
rows,cols = self.shape
return self.matrixidx2sheet(np.arange(rows), np.arange(cols))
[docs]class Slice(np.ndarray):
"""
Represents a slice of a SheetCoordinateSystem; i.e., an array
specifying the row and column start and end points for a submatrix
of the SheetCoordinateSystem.
The slice is created from the supplied bounds by calculating the
slice that corresponds most closely to the specified bounds.
Therefore, the slice does not necessarily correspond exactly to
the specified bounds. The bounds that do exactly correspond to the
slice are available via the 'bounds' attribute.
Note that the slice does not respect the bounds of the
SheetCoordinateSystem, and that actions such as translate() also
do not respect the bounds. To ensure that the slice is within the
SheetCoordinateSystem's bounds, use crop_to_sheet().
"""
__slots__ = []
def compute_bounds(self,scs):
spec = self._slicespec2boundsspec(self,scs)
return BoundingBox(points=spec)
def __new__(cls, bounds, sheet_coordinate_system, force_odd=False,
min_matrix_radius=1):
"""
Create a slice of the given sheet_coordinate_system from the
specified bounds.
"""
if force_odd:
slicespec=Slice._createoddslicespec(bounds,sheet_coordinate_system,
min_matrix_radius)
else:
slicespec=Slice._boundsspec2slicespec(bounds.lbrt(),sheet_coordinate_system)
# Using numpy.int32 for legacy reasons
a = np.array(slicespec, dtype=np.int32, copy=False).view(cls)
return a
[docs] def submatrix(self,matrix):
"""
Return the submatrix of the given matrix specified by this
slice.
Equivalent to computing the intersection between the
SheetCoordinateSystem's bounds and the bounds, and returning
the corresponding submatrix of the given matrix.
The submatrix is just a view into the sheet_matrix; it is not
an independent copy.
"""
return matrix[self[0]:self[1],self[2]:self[3]]
[docs] def positionlesscrop(self,x,y,sheet_coord_system):
"""
Return the correct slice for a weights/mask matrix at this
ConnectionField's location on the sheet (i.e. for getting the
correct submatrix of the weights or mask in case the unit is
near the edge of the sheet).
"""
slice_inds = self.findinputslice(
sheet_coord_system.sheet2matrixidx(x,y),
self.shape_on_sheet(), sheet_coord_system.shape)
self.set(slice_inds)
[docs] def positionedcrop(self,x,y,sheet_coord_system):
"""
Offset the bounds_template to this cf's location and store the
result in the 'bounds' attribute.
Also stores the input_sheet_slice for access by C.
"""
cf_row,cf_col = sheet_coord_system.sheet2matrixidx(x,y)
bounds_x,bounds_y=self.compute_bounds(sheet_coord_system).centroid()
b_row,b_col=sheet_coord_system.sheet2matrixidx(bounds_x,bounds_y)
row_offset = cf_row-b_row
col_offset = cf_col-b_col
self.translate(row_offset,col_offset)
[docs] def translate(self, r, c):
"Translate the slice by the given number of rows and columns."
self+=[r,r,c,c]
[docs] def set(self,slice_specification):
"Set this slice from some iterable that specifies (r1,r2,c1,c2)."
self.put([0,1,2,3],slice_specification) # pylint: disable-msg=E1101
[docs] def shape_on_sheet(self):
"Return the shape of the array of the Slice on its sheet."
return self[1]-self[0],self[3]-self[2]
[docs] def crop_to_sheet(self,sheet_coord_system):
"Crop the slice to the SheetCoordinateSystem's bounds."
maxrow,maxcol = sheet_coord_system.shape
self[0] = max(0,self[0])
self[1] = min(maxrow,self[1])
self[2] = max(0,self[2])
self[3] = min(maxcol,self[3])
@staticmethod
def _createoddslicespec(bounds,scs,min_matrix_radius):
"""
Create the 'odd' Slice that best approximates the specified
sheet-coordinate bounds.
The supplied bounds are translated to have a center at the
center of one of the sheet's units (we arbitrarily use the
center unit), and then these bounds are converted to a slice
in such a way that the slice exactly includes all units whose
centers are within the bounds (see boundsspec2slicespec()).
However, to ensure that the bounds are treated symmetrically,
we take the right and bottom bounds and reflect these about
the center of the slice (i.e. we take the 'xradius' to be
right_col-center_col and the 'yradius' to be
bottom_col-center_row). Hence, if the bounds happen to go
through units, if the units are included on the right and
bottom bounds, they will be included on the left and top
bounds. This ensures that the slice has odd dimensions.
"""
bounds_xcenter,bounds_ycenter=bounds.centroid()
sheet_rows,sheet_cols = scs.shape
center_row,center_col = sheet_rows/2,sheet_cols/2
unit_xcenter,unit_ycenter=scs.matrixidx2sheet(center_row,
center_col)
bounds.translate(unit_xcenter-bounds_xcenter,
unit_ycenter-bounds_ycenter)
r1,r2,c1,c2 = Slice._boundsspec2slicespec(bounds.lbrt(),scs)
xrad=max(c2-center_col-1,min_matrix_radius)
yrad=max(r2-center_row-1,min_matrix_radius)
r2=center_row+yrad+1
c2=center_col+xrad+1
r1=center_row-yrad
c1=center_col-xrad
return (r1,r2,c1,c2)
@staticmethod
def _boundsspec2slicespec(boundsspec,scs):
"""
Convert an iterable boundsspec (supplying l,b,r,t of a
BoundingRegion) into a Slice specification.
Includes all units whose centers are within the specified
sheet-coordinate bounds specified by boundsspec.
Exact inverse of _slicespec2boundsspec().
"""
l,b,r,t = boundsspec
t_m,l_m = scs.sheet2matrix(l,t)
b_m,r_m = scs.sheet2matrix(r,b)
l_idx = int(np.ceil(l_m-0.5))
t_idx = int(np.ceil(t_m-0.5))
# CBENHANCEMENT: Python 2.6's math.trunc()?
r_idx = int(np.floor(r_m+0.5))
b_idx = int(np.floor(b_m+0.5))
return t_idx,b_idx,l_idx,r_idx
@staticmethod
def _slicespec2boundsspec(slicespec,scs):
"""
Convert an iterable slicespec (supplying r1,r2,c1,c2 of a
Slice) into a BoundingRegion specification.
Exact inverse of _boundsspec2slicespec().
"""
r1,r2,c1,c2 = slicespec
left,bottom = scs.matrix2sheet(r2,c1)
right, top = scs.matrix2sheet(r1,c2)
return ((left,bottom),(right,top))
