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2571f24 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 | """Compute spline interpolating coefficients
These functions are ported from the C routines in SPM's bsplines.c
by John Ashburner, which are themselves ports from Philippe Thevenaz's
code. JA furthermore derived the initial conditions for the DFT ("wrap around")
boundary conditions.
Note that similar routines are available in scipy with boundary conditions
DCT1 ("mirror"), DCT2 ("reflect") and DFT ("wrap"); all derived by P. Thevenaz,
according to the comments. Our DCT2 boundary conditions are ported from
scipy.
Only boundary conditions DCT1, DCT2 and DFT are implemented.
References
----------
..[1] M. Unser, A. Aldroubi and M. Eden.
"B-Spline Signal Processing: Part I-Theory,"
IEEE Transactions on Signal Processing 41(2):821-832 (1993).
..[2] M. Unser, A. Aldroubi and M. Eden.
"B-Spline Signal Processing: Part II-Efficient Design and Applications,"
IEEE Transactions on Signal Processing 41(2):834-848 (1993).
..[3] M. Unser.
"Splines: A Perfect Fit for Signal and Image Processing,"
IEEE Signal Processing Magazine 16(6):22-38 (1999).
"""
import torch
import math
from typing import List, Optional
from .jit_utils import movedim1
from .pushpull import pad_list_int
@torch.jit.script
def get_poles(order: int) -> List[float]:
empty: List[float] = []
if order in (0, 1):
return empty
if order == 2:
return [math.sqrt(8.) - 3.]
if order == 3:
return [math.sqrt(3.) - 2.]
if order == 4:
return [math.sqrt(664. - math.sqrt(438976.)) + math.sqrt(304.) - 19.,
math.sqrt(664. + math.sqrt(438976.)) - math.sqrt(304.) - 19.]
if order == 5:
return [math.sqrt(67.5 - math.sqrt(4436.25)) + math.sqrt(26.25) - 6.5,
math.sqrt(67.5 + math.sqrt(4436.25)) - math.sqrt(26.25) - 6.5]
if order == 6:
return [-0.488294589303044755130118038883789062112279161239377608394,
-0.081679271076237512597937765737059080653379610398148178525368,
-0.00141415180832581775108724397655859252786416905534669851652709]
if order == 7:
return [-0.5352804307964381655424037816816460718339231523426924148812,
-0.122554615192326690515272264359357343605486549427295558490763,
-0.0091486948096082769285930216516478534156925639545994482648003]
raise NotImplementedError
@torch.jit.script
def get_gain(poles: List[float]) -> float:
lam: float = 1.
for pole in poles:
lam *= (1. - pole) * (1. - 1./pole)
return lam
@torch.jit.script
def dft_initial(inp, pole: float, dim: int = -1, keepdim: bool = False):
assert inp.shape[dim] > 1
max_iter: int = int(math.ceil(-30./math.log(abs(pole))))
max_iter = min(max_iter, inp.shape[dim])
poles = torch.as_tensor(pole, dtype=inp.dtype, device=inp.device)
poles = poles.pow(torch.arange(1, max_iter, dtype=inp.dtype, device=inp.device))
poles = poles.flip(0)
inp = movedim1(inp, dim, 0)
inp0 = inp[0]
inp = inp[1-max_iter:]
inp = movedim1(inp, 0, -1)
out = torch.matmul(inp.unsqueeze(-2), poles.unsqueeze(-1)).squeeze(-1)
out = out + inp0.unsqueeze(-1)
if keepdim:
out = movedim1(out, -1, dim)
else:
out = out.squeeze(-1)
pole = pole ** max_iter
out = out / (1 - pole)
return out
@torch.jit.script
def dct1_initial(inp, pole: float, dim: int = -1, keepdim: bool = False):
n = inp.shape[dim]
max_iter: int = int(math.ceil(-30./math.log(abs(pole))))
if max_iter < n:
poles = torch.as_tensor(pole, dtype=inp.dtype, device=inp.device)
poles = poles.pow(torch.arange(1, max_iter, dtype=inp.dtype, device=inp.device))
inp = movedim1(inp, dim, 0)
inp0 = inp[0]
inp = inp[1:max_iter]
inp = movedim1(inp, 0, -1)
out = torch.matmul(inp.unsqueeze(-2), poles.unsqueeze(-1)).squeeze(-1)
out = out + inp0.unsqueeze(-1)
if keepdim:
out = movedim1(out, -1, dim)
else:
out = out.squeeze(-1)
else:
max_iter = n
polen = pole ** (n - 1)
inp0 = inp[0] + polen * inp[-1]
inp = inp[1:-1]
inp = movedim1(inp, 0, -1)
poles = torch.as_tensor(pole, dtype=inp.dtype, device=inp.device)
poles = poles.pow(torch.arange(1, n-1, dtype=inp.dtype, device=inp.device))
poles = poles + (polen * polen) / poles
out = torch.matmul(inp.unsqueeze(-2), poles.unsqueeze(-1)).squeeze(-1)
out = out + inp0.unsqueeze(-1)
if keepdim:
out = movedim1(out, -1, dim)
else:
out = out.squeeze(-1)
pole = pole ** (max_iter - 1)
out = out / (1 - pole * pole)
return out
@torch.jit.script
def dct2_initial(inp, pole: float, dim: int = -1, keepdim: bool = False):
# Ported from scipy:
# https://github.com/scipy/scipy/blob/master/scipy/ndimage/src/ni_splines.c
#
# I (YB) unwarped and simplied the terms so that I could use a dot
# product instead of a loop.
# It should certainly be possible to derive a version for max_iter < n,
# as JA did for DCT1, to avoid long recursions when `n` is large. But
# I think it would require a more complicated anticausal/final condition.
n = inp.shape[dim]
polen = pole ** n
pole_last = polen * (1 + 1/(pole + polen * polen))
inp00 = inp[0]
inp0 = inp[0] + pole_last * inp[-1]
inp = inp[1:-1]
inp = movedim1(inp, 0, -1)
poles = torch.as_tensor(pole, dtype=inp.dtype, device=inp.device)
poles = (poles.pow(torch.arange(1, n-1, dtype=inp.dtype, device=inp.device)) +
poles.pow(torch.arange(2*n-2, n, -1, dtype=inp.dtype, device=inp.device)))
out = torch.matmul(inp.unsqueeze(-2), poles.unsqueeze(-1)).squeeze(-1)
out = out + inp0.unsqueeze(-1)
out = out * (pole / (1 - polen * polen))
out = out + inp00.unsqueeze(-1)
if keepdim:
out = movedim1(out, -1, dim)
else:
out = out.squeeze(-1)
return out
@torch.jit.script
def dft_final(inp, pole: float, dim: int = -1, keepdim: bool = False):
assert inp.shape[dim] > 1
max_iter: int = int(math.ceil(-30./math.log(abs(pole))))
max_iter = min(max_iter, inp.shape[dim])
poles = torch.as_tensor(pole, dtype=inp.dtype, device=inp.device)
poles = poles.pow(torch.arange(2, max_iter+1, dtype=inp.dtype, device=inp.device))
inp = movedim1(inp, dim, 0)
inp0 = inp[-1]
inp = inp[:max_iter-1]
inp = movedim1(inp, 0, -1)
out = torch.matmul(inp.unsqueeze(-2), poles.unsqueeze(-1)).squeeze(-1)
out = out.add(inp0.unsqueeze(-1), alpha=pole)
if keepdim:
out = movedim1(out, -1, dim)
else:
out = out.squeeze(-1)
pole = pole ** max_iter
out = out / (pole - 1)
return out
@torch.jit.script
def dct1_final(inp, pole: float, dim: int = -1, keepdim: bool = False):
inp = movedim1(inp, dim, 0)
out = pole * inp[-2] + inp[-1]
out = out * (pole / (pole*pole - 1))
if keepdim:
out = movedim1(out.unsqueeze(0), 0, dim)
return out
@torch.jit.script
def dct2_final(inp, pole: float, dim: int = -1, keepdim: bool = False):
# Ported from scipy:
# https://github.com/scipy/scipy/blob/master/scipy/ndimage/src/ni_splines.c
inp = movedim1(inp, dim, 0)
out = inp[-1] * (pole / (pole - 1))
if keepdim:
out = movedim1(out.unsqueeze(0), 0, dim)
return out
@torch.jit.script
class CoeffBound:
def __init__(self, bound: int):
self.bound = bound
def initial(self, inp, pole: float, dim: int = -1, keepdim: bool = False):
if self.bound in (0, 2): # zero, dct1
return dct1_initial(inp, pole, dim, keepdim)
elif self.bound in (1, 3): # nearest, dct2
return dct2_initial(inp, pole, dim, keepdim)
elif self.bound == 6: # dft
return dft_initial(inp, pole, dim, keepdim)
else:
raise NotImplementedError
def final(self, inp, pole: float, dim: int = -1, keepdim: bool = False):
if self.bound in (0, 2): # zero, dct1
return dct1_final(inp, pole, dim, keepdim)
elif self.bound in (1, 3): # nearest, dct2
return dct2_final(inp, pole, dim, keepdim)
elif self.bound == 6: # dft
return dft_final(inp, pole, dim, keepdim)
else:
raise NotImplementedError
@torch.jit.script
def filter(inp, bound: CoeffBound, poles: List[float],
dim: int = -1, inplace: bool = False):
if not inplace:
inp = inp.clone()
if inp.shape[dim] == 1:
return inp
gain = get_gain(poles)
inp *= gain
inp = movedim1(inp, dim, 0)
n = inp.shape[0]
for pole in poles:
inp[0] = bound.initial(inp, pole, dim=0, keepdim=False)
for i in range(1, n):
inp[i].add_(inp[i-1], alpha=pole)
inp[-1] = bound.final(inp, pole, dim=0, keepdim=False)
for i in range(n-2, -1, -1):
inp[i].neg_().add_(inp[i+1]).mul_(pole)
inp = movedim1(inp, 0, dim)
return inp
@torch.jit.script
def spline_coeff(inp, bound: int, order: int, dim: int = -1,
inplace: bool = False):
"""Compute the interpolating spline coefficients, for a given spline order
and boundary conditions, along a single dimension.
Parameters
----------
inp : tensor
bound : {2: dct1, 6: dft}
order : {0..7}
dim : int, default=-1
inplace : bool, default=False
Returns
-------
out : tensor
"""
if not inplace:
inp = inp.clone()
if order in (0, 1):
return inp
poles = get_poles(order)
return filter(inp, CoeffBound(bound), poles, dim=dim, inplace=True)
@torch.jit.script
def spline_coeff_nd(inp, bound: List[int], order: List[int],
dim: Optional[int] = None, inplace: bool = False):
"""Compute the interpolating spline coefficients, for a given spline order
and boundary condition, along the last `dim` dimensions.
Parameters
----------
inp : (..., *spatial) tensor
bound : List[{2: dct1, 6: dft}]
order : List[{0..7}]
dim : int, default=`inp.dim()`
inplace : bool, default=False
Returns
-------
out : (..., *spatial) tensor
"""
if not inplace:
inp = inp.clone()
if dim is None:
dim = inp.dim()
bound = pad_list_int(bound, dim)
order = pad_list_int(order, dim)
for d, b, o in zip(range(dim), bound, order):
inp = spline_coeff(inp, b, o, dim=-dim + d, inplace=True)
return inp
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