pl.figure(1)
pl.draw()
# ----------------------------------------------------------------------------
# Code for initialising the online multi-frame deconvolution
# ----------------------------------------------------------------------------
# Initialisation of latent image by averaging the first 20 frames
y0 = 0.
for i in np.arange(1, N0):
y0 += yload(i)
y0 /= N0
y_gpu = cua.to_gpu(y0)
# Pad image since we perform deconvolution with valid boundary conditions
x_gpu = gputools.impad_gpu(y_gpu, sf - 1)
# Create windows for OlaGPU
sx = y0.shape + sf - 1
sf2 = np.floor(sf / 2)
winaux = imagetools.win2winaux(sx, csf, overlap)
# ----------------------------------------------------------------------------
# Loop over all frames and do online blind deconvolution
# ----------------------------------------------------------------------------
import time as t
ti = t.clock()
for i in np.arange(1, N + 1):
print 'Processing frame %d/%d \r' % (i, N)
pl.figure(1)
pl.draw()
# ----------------------------------------------------------------------------
# Code for initialising the online multi-frame deconvolution
# ----------------------------------------------------------------------------
# Initialisation of latent image by averaging the first 20 frames
y0 = 0.
for i in np.arange(1,N0):
y0 += yload(i)
y0 /= N0
y_gpu = cua.to_gpu(y0)
# Pad image since we perform deconvolution with valid boundary conditions
x_gpu = gputools.impad_gpu(y_gpu, sf-1)
# Create windows for OlaGPU
sx = y0.shape + sf - 1
sf2 = np.floor(sf/2)
winaux = imagetools.win2winaux(sx, csf, overlap)
# ----------------------------------------------------------------------------
# Loop over all frames and do online blind deconvolution
# ----------------------------------------------------------------------------
import time as t
ti = t.clock()
for i in np.arange(1,N+1):
print 'Processing frame %d/%d \r' % (i,N)
def process(opts):
# ============================================================================
# Specify some parameter settings
# ----------------------------------------------------------------------------
# Specify data path and file identifier
DATAPATH = '/DATA/LSST/FITS'
RESPATH = '../../../DATA/results';
BASE_N = 141
FILENAME = lambda i: '%s/v88827%03d-fz.R22.S11.fits' % (DATAPATH,(BASE_N+i))
ID = 'LSST'
# ----------------------------------------------------------------------------
# Specify parameter settings
# General
doshow = opts.doShow # put 1 to show intermediate results
backup = opts.backup # put 1 to write intermediate results to disk
N = opts.N # how many frames to process
N0 = opts.N0 # number of averaged frames for initialisation
# OlaGPU parameters
sf = np.array([40,40]) # estimated size of PSF
csf =(3,3) # number of kernels across x and y direction
overlap = 0.5 # overlap of neighboring patches in percent
# Regularization parameters for kernel estimation
f_alpha = opts.f_alpha # promotes smoothness
f_beta = opts.f_beta # Thikhonov regularization
optiter = opts.optiter # number of iterations for minimization
tol = opts.tol # tolerance for when to stop minimization
# ============================================================================
# Create helper functions for file handling
# # # HACK for chunking into available GPU mem # # #
# - loads one 1kx1k block out of the fits image
xOffset=2000
yOffset=0
chunkSize=1000
yload = lambda i: 1. * fitsTools.readFITS(FILENAME(i), use_mask=True, norm=True)[yOffset:yOffset+chunkSize,xOffset:xOffset+chunkSize]
# ----------------------------------------------------------------------------
# Some more code for backuping the results
# ----------------------------------------------------------------------------
# For backup purposes
EXPPATH = '%s/%s_sf%dx%d_csf%dx%d_maxiter%d_alpha%.2f_beta%.2f' % \
(RESPATH,ID,sf[0],sf[1],csf[0],csf[1],optiter,f_alpha,f_beta)
xname = lambda i: '%s/x_%04d.png' % (EXPPATH,i)
yname = lambda i: '%s/y_%04d.png' % (EXPPATH,i)
fname = lambda i: '%s/f_%04d.png' % (EXPPATH,i)
if os.path.exists(EXPPATH) and opts.overwrite:
try:
rmtree(EXPPATH)
except:
print "[ERROR] removing old results dir:",EXPPATH
exit()
elif os.path.exists(EXPPATH):
print "[ERROR] results directory already exists, please remove or use '-o' to overwrite"
exit()
# Create results path if not existing
try:
os.makedirs(EXPPATH)
except:
print "[ERROR] creating results dir:",EXPPATH
exit()
print 'Results are saved to: \n %s \n' % EXPPATH
# ----------------------------------------------------------------------------
# For displaying intermediate results create target figure
# ----------------------------------------------------------------------------
# Create figure for displaying intermediate results
if doshow:
print "showing intermediate results is currently disabled.."
#pl.figure(1)
#pl.draw()
# ----------------------------------------------------------------------------
# Code for initialising the online multi-frame deconvolution
# ----------------------------------------------------------------------------
# Initialisation of latent image by averaging the first 20 frames
y0 = 0.
for i in np.arange(1,N0):
y0 += yload(i)
y0 /= N0
y_gpu = cua.to_gpu(y0)
# Pad image since we perform deconvolution with valid boundary conditions
x_gpu = gputools.impad_gpu(y_gpu, sf-1)
# Create windows for OlaGPU
sx = y0.shape + sf - 1
sf2 = np.floor(sf/2)
winaux = imagetools.win2winaux(sx, csf, overlap)
# ----------------------------------------------------------------------------
# Loop over all frames and do online blind deconvolution
# ----------------------------------------------------------------------------
import time as t
ti = t.clock()
t1 = stopwatch.timer()
t2 = stopwatch.timer()
t3 = stopwatch.timer()
t4 = stopwatch.timer()
t4.start()
for i in np.arange(1,N+1):
print 'Processing frame %d/%d \r' % (i,N)
# Load next observed image
t3.start()
y = yload(i)
print "TIMER load:", t3.elapsed()
# Compute mask for determining saturated regions
mask_gpu = 1. * cua.to_gpu(y < 1.)
y_gpu = cua.to_gpu(y)
# ------------------------------------------------------------------------
# PSF estimation
# ------------------------------------------------------------------------
# Create OlaGPU instance with current estimate of latent image
t2.start()
X = olaGPU.OlaGPU(x_gpu,sf,'valid',winaux=winaux)
print "TIMER GPU: ", t2.elapsed()
t1.start()
# PSF estimation for given estimate of latent image and current observation
f = X.deconv(y_gpu, mode = 'lbfgsb', alpha = f_alpha, beta = f_beta,
maxfun = optiter, verbose = 10)
print "TIMER Optimization: ", t1.elapsed()
fs = f[0]
# Normalize PSF kernels to sum up to one
fs = gputools.normalize(fs)
# ------------------------------------------------------------------------
# Latent image estimation
# ------------------------------------------------------------------------
# Create OlaGPU instance with estimated PSF
t2.start()
F = olaGPU.OlaGPU(fs,sx,'valid',winaux=winaux)
# Latent image estimation by performing one gradient descent step
# multiplicative update is used which preserves positivity
factor_gpu = F.cnvtp(mask_gpu*y_gpu)/(F.cnvtp(mask_gpu*F.cnv(x_gpu))+tol)
gputools.cliplower_GPU(factor_gpu, tol)
x_gpu = x_gpu * factor_gpu
x_max = x_gpu.get()[sf[0]:-sf[0],sf[1]:-sf[1]].max()
gputools.clipupper_GPU(x_gpu, x_max)
print "TIMER GPU: ", t2.elapsed()
# ------------------------------------------------------------------------
# For backup intermediate results
# ------------------------------------------------------------------------
if backup or i == N:
# Write intermediate results to disk incl. input
y_img = y_gpu.get()*1e5
fitsTools.asinhScale(y_img, 450, -50, minCut=0.0, maxCut=40000, fname=yname(i))
# Crop image to input size
xi = (x_gpu.get()[sf2[0]:-sf2[0],sf2[1]:-sf2[1]] / x_max)*1e5
fitsTools.fitsStats(xi)
fitsTools.asinhScale(xi, 450, -50, minCut=0.0, maxCut=40000, fname=xname(i))
# Concatenate PSF kernels for ease of visualisation
f = imagetools.gridF(fs,csf)
f = f*1e5
fitsTools.asinhScale(f, 450, -50, minCut=0.0, maxCut=40000, fname=fname(i))
# ------------------------------------------------------------------------
# For displaying intermediate results
# ------------------------------------------------------------------------
'''
if np.mod(i,1) == 0 and doshow:
pl.figure(1)
pl.subplot(121)
# what is SY?
pl.imshow(imagetools.crop(x_gpu.get(),sy,np.ceil(sf/2)),'gray')
pl.title('x after %d observations' % i)
pl.subplot(122)
pl.imshow(y_gpu.get(),'gray')
pl.title('y(%d)' % i)
pl.draw()
pl.figure(2)
pl.title('PSF(%d)' % i)
imagetools.cellplot(fs, winaux.csf)
tf = t.clock()
print('Time elapsed after %d frames %.3f' % (i,(tf-ti)))
'''
tf = t.clock()
print('Time elapsed for total image sequence %.3f' % (tf-ti))
# ----------------------------------------------------------------------------
print "TOTAL: %.3f" % (t4.elapsed())
print "OptimizeCPUtime %.3f %.3f" % (t1.getTotal(), 100*(t1.getTotal()/t4.getTotal()))
print "GPUtime %.3f %.3f" % (t2.getTotal(), 100*(t2.getTotal()/t4.getTotal()))
print "LoadTime %.3f %.3f" % (t3.getTotal(), 100*(t3.getTotal()/t4.getTotal()))
def deconv(self, y, z0=None, mode='lbfgsb', maxfun=100, alpha=0., beta=0.01,
verbose=10, m=5, edgetapering=1, factor=3, gamma=1e-4):
"""
deconv implements various deconvolution methods. It expects a
blurry image and outputs an estimated blur kernel or a sharp latent
image. Currently, the following algorithms are implemented:
'lbfgsb' uses the lbfgsb optimization code to minimize the following
constrained regularized problem:
|y-Zu|^2 + alpha * |grad(u)|^2 + beta * |u|^2 s.t. u>0
The alpha term promotes smoothness of the solution, while
the beta term is an ordinary Thikhonov regularization
'direct' as above but solves the problem directly, i.e. via
division in Fourier space instead of an iterative
minimization scheme at the cost of the positivity
constraint.
'xdirect' as 'direct' but without corrective term which reduces
artifacts stemming from the windowing
'gdirect' solves the following problem
|grad(y)-grad(Zu)|^2 + alpha * |grad(u)|^2 + beta * |u|^2
This is particularly useful for kernel estimation in the
case of blurred natural images featuring many edges. The
advantage vs. 'direct' is the suppression of noise in the
estimated PSF kernels.
'xdirect' as 'direct' but without corrective term which reduces
artifacts stemming from the windowing
'Fast Image Deconvolution using Hyper-Laplacian Priors'
by Dilip Krishnan and Rob Fergus, NIPS 2009.
It minimizes the following problem
|y-Zu|^2 + gamma * |grad(u)|^(2/3)
via half-quadratic splitting. See paper for details.
----------------------------------------------------------------------
Usage:
Call: Z = OlaGPU(z, sw, mode, winaux)
u = Z.deconv(y)
Input: y blurry image
Ouput: u either image or PSF sized object
"""
from numpy import array
if not all(array(y.shape) == self.sy):
raise IOError ('Sizes incompatible. Expected blurred image!')
# Potential data transfer to GPU
if y.__class__ == cua.GPUArray:
y_gpu = 1. * y
else:
y_gpu = cua.to_gpu(y.astype(np.float32))
# --------------------------------------------------------------------
if mode == 'lbfgsb':
from scipy.optimize import fmin_l_bfgs_b
self.res_gpu = cua.empty_like(y_gpu)
if self.__id__ == 'X':
sz = ((int(np.prod(self.winaux.csf)),
int(self.sz[0]),int(self.sz[1])))
elif self.__id__ == 'F':
sz = self.sz
lf = np.prod(sz)
if z0 == None:
z0_gpu = self.cnvtp(y_gpu)
z0 = z0_gpu.get()
z0 = z0.flatten()
#z0 = np.ones(lf)/(1. * lf) # initialisation with flat kernels
else:
z0 = z0.flatten()
lb = 0. # lower bound
ub = np.infty # upper bound
zhat = fmin_l_bfgs_b(func = self.cnvinv_objfun, x0 = z0, \
fprime = self.cnvinv_gradfun,\
args = [sz, y_gpu, alpha, beta],\
factr = 10., pgtol = 10e-15, \
maxfun = maxfun, bounds = [(lb, ub)] * lf,\
m = m, iprint = verbose)
return np.reshape(zhat[0], sz), zhat[1], zhat[2]
# --------------------------------------------------------------------
elif mode == 'gdirect':
# Use this method only for estimating the PSF
if self.__id__ != 'X':
raise Exception('Use direct mode for image estimation!')
# Compute Laplacian
if alpha > 0.:
gx_gpu = gputools.pad_cpu2gpu(
np.array([[-1,1],[-1,1],[-1,1]]),
self.sfft_gpu, dtype='complex')
gy_gpu = gputools.pad_cpu2gpu(
np.array([[-1,-1,-1],[1,1,1]]),
self.sfft_gpu, dtype='complex')
self.plan.execute(gx_gpu)
self.plan.execute(gy_gpu)
L_gpu = gx_gpu * gx_gpu.conj() + gy_gpu * gy_gpu.conj()
else:
L_gpu = cua.zeros(self.fft_gpu.shape, np.complex64)
if edgetapering == 1:
gputools.edgetaper_gpu(y_gpu, 2*self.sf, 'barthann')
# Transfer to GPU
if self.x.__class__ == cua.GPUArray:
x_gpu = self.x
else:
x_gpu = cua.to_gpu(self.x)
# Compute gradient images
xx_gpu, xy_gpu = gputools.gradient_gpu(x_gpu)
yx_gpu, yy_gpu = gputools.gradient_gpu(y_gpu)
# Chop and pad business
if self.mode == 'valid':
yx_gpu = gputools.chop_pad_GPU(yx_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, self.sf-1,
'complex')
yy_gpu = gputools.chop_pad_GPU(yy_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, self.sf-1,
'complex')
elif self.mode == 'same':
yx_gpu = gputools.chop_pad_GPU(yx_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu,
np.floor(self.sf/2), 'complex')
yy_gpu = gputools.chop_pad_GPU(yy_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu,
np.floor(self.sf/2), 'complex')
else:
raise NotImplementedError('Not a valid mode!')
xx_gpu = gputools.chop_pad_GPU(xx_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, dtype='complex')
xy_gpu = gputools.chop_pad_GPU(xy_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, dtype='complex')
# Here each patch should be windowed to reduce ringing artifacts,
# however since we are working in the gradient domain, the effect
# is negligible
# ws_gpu = gputools.pad_stack_GPU(self.winaux.ws_gpu,
# self.sfft_gpu, self.sf-1,
# dtype='complex')
# xx_gpu = ws_gpu * xx_gpu
# xy_gpu = ws_gpu * xy_gpu
# yx_gpu = ws_gpu * yx_gpu
# yy_gpu = ws_gpu * yy_gpu
# Compute Fourier transform
self.fft(yx_gpu, self.fft_gpu.shape[0])
self.fft(yy_gpu, self.fft_gpu.shape[0])
self.fft(xx_gpu, self.fft_gpu.shape[0])
self.fft(xy_gpu, self.fft_gpu.shape[0])
# Do division in Fourier space
z_gpu = cua.zeros(xy_gpu.shape, np.complex64)
z_gpu = gputools.comp_ola_gdeconv(xx_gpu, xy_gpu,
yx_gpu, yy_gpu,
L_gpu, alpha, beta)
# Computing the inverse FFT
z_gpu = z_gpu.conj()
self.fft(z_gpu, self.fft_gpu.shape[0])
z_gpu = z_gpu.conj()/np.prod(z_gpu.shape[-2::])
# Crop out the kernels
zc_gpu = gputools.crop_stack_GPU(z_gpu, self.sf)
return zc_gpu
# --------------------------------------------------------------------
elif mode == 'direct':
const_gpu = cua.empty_like(y_gpu)
const_gpu.fill(1.)
# First deconvolution without corrective term
y_gpu = self.deconv(y_gpu, mode = 'xdirect', alpha = alpha,
beta = beta, edgetapering = edgetapering)
gputools.cliplower_GPU(y_gpu,0)
# Now same for constant image to get rid of window artifacts
if edgetapering == 1:
gputools.edgetaper_gpu(const_gpu, 2*self.sf, 'barthann')
const_gpu = self.deconv(const_gpu, mode = 'xdirect', alpha = alpha,
beta = beta, edgetapering = edgetapering)
gputools.edgetaper_gpu(const_gpu, 2*self.sf, 'barthann')
gputools.clip_GPU(const_gpu, 0.01, 10.)
# Division of deconvolved latent and constant image to get rid
# of artifacts stemming from windowing
y_gpu = y_gpu / const_gpu
sz = y_gpu.shape
#gputools.clip_GPU(y_gpu, 0., 1.0)
#gputools.edgetaper_gpu(y_gpu, 3*self.sf, 'barthann')
# Do cropping and padding since edges are corrupted by division
y_gpu = gputools.crop_gpu2cpu(y_gpu, sz-factor*self.sf-1,
offset=np.floor((factor*self.sf-1)/2.))
y_gpu = gputools.impad_gpu(y_gpu, tuple(np.array(sz)-y_gpu.shape))
return y_gpu
# --------------------------------------------------------------------
elif mode == 'xdirect':
# Compute Laplacian
if alpha > 0.:
gx_gpu = gputools.pad_cpu2gpu(
np.array([[-1,1]]), self.sfft_gpu, dtype='complex')
gy_gpu = gputools.pad_cpu2gpu(
np.array([[-1],[1]]), self.sfft_gpu, dtype='complex')
self.plan.execute(gx_gpu)
self.plan.execute(gy_gpu)
L_gpu = gx_gpu * gx_gpu.conj() + gy_gpu * gy_gpu.conj()
else:
L_gpu = cua.zeros(self.fft_gpu.shape, np.complex64)
# Edgetapering of blurry input image
if edgetapering == 1:
gputools.edgetaper_gpu(y_gpu, 3*self.sf, 'barthann')
if self.mode == 'valid':
#y_gpu = gputools.pad_cpu2gpu(y_gpu, self.sx, self.sf-1, dtype='real')
offset = self.sf-1
elif self.mode == 'same':
offset = np.floor(self.sf/2)
else:
raise NotImplementedError('Not a valid mode!')
# Chop and pad business
y_gpu = gputools.chop_pad_GPU(y, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, offset, 'complex')
ws_gpu = gputools.pad_stack_GPU(self.winaux.ws_gpu, self.sfft_gpu,
dtype='complex')
# Windowing
y_gpu = ws_gpu * y_gpu
# Compute FFT
self.fft(y_gpu, self.fft_gpu.shape[0])
# Do division in Fourier space
z_gpu = gputools.comp_ola_deconv(self.fft_gpu, y_gpu, L_gpu,
alpha, beta)
# Computing the inverse FFT
z_gpu = z_gpu.conj()
self.fft(z_gpu, self.fft_gpu.shape[0])
z_gpu = z_gpu.conj()/np.prod(z_gpu.shape[-2::])
# Crop the solution to correct output size
if self.__id__ == 'X':
zc_gpu = gputools.crop_stack_GPU(z_gpu, self.sf)
return zc_gpu
elif self.__id__ == 'F':
zs_gpu = gputools.crop_stack_GPU(z_gpu, self.winaux.sw)
#zs_gpu = self.winaux.ws_gpu * zs_gpu
zc_gpu = gputools.ola_GPU_test(zs_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop)
zc_gpu = gputools.crop_gpu2cpu(zc_gpu, self.sx)
return zc_gpu
# --------------------------------------------------------------------
elif mode == 'sparse':
# Compute Laplacian
gx_gpu = gputools.pad_cpu2gpu(np.sqrt(2.)/2. *
np.array([[-1,1]]), self.sfft_gpu, dtype='complex')
gy_gpu = gputools.pad_cpu2gpu(np.sqrt(2.)/2. *
np.array([[-1],[1]]), self.sfft_gpu, dtype='complex')
self.plan.execute(gx_gpu)
self.plan.execute(gy_gpu)
L_gpu = gx_gpu * gx_gpu.conj() + gy_gpu * gy_gpu.conj()
const_gpu = cua.empty_like(y_gpu)
const_gpu.fill(1.)
# Edgetapering
if edgetapering == 1:
gputools.edgetaper_gpu(y_gpu, 2*self.sf, 'barthann')
gputools.edgetaper_gpu(const_gpu, 2*self.sf, 'barthann')
# Parameter settings
beta = 1.
beta_rate = 2. * np.sqrt(2.)
beta_max = 2.**8
# Initialisation of x with padded version of y
x_gpu = 1 * y_gpu
if self.mode == 'valid':
offset = self.sf-1
elif self.mode == 'same':
offset = np.floor(self.sf/2)
else:
raise NotImplementedError('Not a valid mode!')
# Chop and pad business
y_gpu = gputools.chop_pad_GPU(y_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, offset,'complex')
const_gpu = gputools.chop_pad_GPU(const_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, offset,'complex')
ws_gpu = gputools.pad_stack_GPU(self.winaux.ws_gpu, self.sfft_gpu,
offset, dtype='complex')
# Windowing
y_gpu = y_gpu * ws_gpu
# Constant image for corrective weighting term
const_gpu = const_gpu * ws_gpu
del ws_gpu
self.fft(const_gpu, self.fft_gpu.shape[0])
const_gpu = gputools.comp_ola_deconv(self.fft_gpu, const_gpu,
L_gpu, alpha, gamma)
const_gpu = const_gpu.conj()
self.fft(const_gpu, self.fft_gpu.shape[0])
const_gpu = const_gpu.conj()/np.prod(const_gpu.shape[-2::])
const_gpu = gputools.crop_stack_GPU(const_gpu, self.winaux.sw)
const_gpu = const_gpu * self.winaux.ws_gpu
const_gpu = gputools.ola_GPU_test(const_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop)
const_gpu = gputools.crop_gpu2cpu(const_gpu, self.sx)
# For debugging purposes
#scipy.misc.imsave('const1.png', const_gpu.get()/const_gpu.get().max())
gputools.cliplower_GPU(const_gpu, 0.01)
const_gpu = 0.01 / const_gpu
# Precompute F'y
self.fft(y_gpu, self.fft_gpu.shape[0])
y_gpu = y_gpu * self.fft_gpu.conj()
while beta < beta_max:
# Compute gradient images of x
xx_gpu, xy_gpu = gputools.gradient_gpu(x_gpu)
del x_gpu
# w sub-problem for alpha 2/3
gputools.modify_sparse23_gpu(xx_gpu, beta)
gputools.modify_sparse23_gpu(xy_gpu, beta)
#gputools.modify_sparse_gpu(xx_gpu, beta, 0.01)
#gputools.modify_sparse_gpu(xy_gpu, beta, 0.01)
# Chop and pad to size of FFT
xx_gpu = gputools.chop_pad_GPU(xx_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, dtype='complex')
xy_gpu = gputools.chop_pad_GPU(xy_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, dtype='complex')
# Compute Fourier transform
self.fft(xx_gpu, self.fft_gpu.shape[0])
self.fft(xy_gpu, self.fft_gpu.shape[0])
# Do division in Fourier space
x_gpu = gputools.comp_ola_sdeconv(gx_gpu, gy_gpu,
xx_gpu, xy_gpu,
y_gpu, self.fft_gpu,
L_gpu, alpha, beta, gamma)
del xx_gpu, xy_gpu
# Computing the inverse FFT
x_gpu = x_gpu.conj()
self.fft(x_gpu, self.fft_gpu.shape[0])
x_gpu = x_gpu.conj()
x_gpu /= np.prod(x_gpu.shape[-2::])
# Ola and cropping
x_gpu = gputools.crop_stack_GPU(x_gpu, self.winaux.sw)
x_gpu = x_gpu * self.winaux.ws_gpu
x_gpu = gputools.ola_GPU_test(x_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop)
x_gpu = gputools.crop_gpu2cpu(x_gpu, self.sx)
# Enforce positivity
x_gpu = x_gpu * const_gpu
gputools.cliplower_GPU(x_gpu, 0.)
beta *= beta_rate
return x_gpu
else:
raise NotImplementedError('Not a valid deconv mode!')
def deconv(self,
y,
z0=None,
mode='lbfgsb',
maxfun=100,
alpha=0.,
beta=0.01,
verbose=10,
m=5,
edgetapering=1,
factor=3,
gamma=1e-4):
"""
deconv implements various deconvolution methods. It expects a
blurry image and outputs an estimated blur kernel or a sharp latent
image. Currently, the following algorithms are implemented:
'lbfgsb' uses the lbfgsb optimization code to minimize the following
constrained regularized problem:
|y-Zu|^2 + alpha * |grad(u)|^2 + beta * |u|^2 s.t. u>0
The alpha term promotes smoothness of the solution, while
the beta term is an ordinary Thikhonov regularization
'direct' as above but solves the problem directly, i.e. via
division in Fourier space instead of an iterative
minimization scheme at the cost of the positivity
constraint.
'xdirect' as 'direct' but without corrective term which reduces
artifacts stemming from the windowing
'gdirect' solves the following problem
|grad(y)-grad(Zu)|^2 + alpha * |grad(u)|^2 + beta * |u|^2
This is particularly useful for kernel estimation in the
case of blurred natural images featuring many edges. The
advantage vs. 'direct' is the suppression of noise in the
estimated PSF kernels.
'xdirect' as 'direct' but without corrective term which reduces
artifacts stemming from the windowing
'Fast Image Deconvolution using Hyper-Laplacian Priors'
by Dilip Krishnan and Rob Fergus, NIPS 2009.
It minimizes the following problem
|y-Zu|^2 + gamma * |grad(u)|^(2/3)
via half-quadratic splitting. See paper for details.
----------------------------------------------------------------------
Usage:
Call: Z = OlaGPU(z, sw, mode, winaux)
u = Z.deconv(y)
Input: y blurry image
Ouput: u either image or PSF sized object
"""
from numpy import array
if not all(array(y.shape) == self.sy):
raise IOError('Sizes incompatible. Expected blurred image!')
# Potential data transfer to GPU
if y.__class__ == cua.GPUArray:
y_gpu = 1. * y
else:
y_gpu = cua.to_gpu(y.astype(np.float32))
# --------------------------------------------------------------------
if mode == 'lbfgsb':
from scipy.optimize import fmin_l_bfgs_b
self.res_gpu = cua.empty_like(y_gpu)
if self.__id__ == 'X':
sz = ((int(np.prod(self.winaux.csf)), int(self.sz[0]),
int(self.sz[1])))
elif self.__id__ == 'F':
sz = self.sz
lf = np.prod(sz)
if z0 == None:
z0_gpu = self.cnvtp(y_gpu)
z0 = z0_gpu.get()
z0 = z0.flatten()
#z0 = np.zeros(self.sf) # initialisation with flat kernels
#z0[self.sf[0]/2,self.sf[1]/2] = 1.
#z0 = np.tile(z0, [np.prod(self.csf),1,1])
#z0 = z0.flatten()
else:
z0 = z0.flatten()
lb = 0. # lower bound
ub = np.infty # upper bound
zhat = fmin_l_bfgs_b(func = self.cnvinv_objfun, x0 = z0, \
fprime = self.cnvinv_gradfun,\
args = [sz, y_gpu, alpha, beta],\
factr = 10., pgtol = 10e-15, \
maxfun = maxfun, bounds = [(lb, ub)] * lf,\
m = m, iprint = verbose)
return np.reshape(zhat[0], sz), zhat[1], zhat[2]
# --------------------------------------------------------------------
elif mode == 'gdirect':
# Use this method only for estimating the PSF
if self.__id__ != 'X':
raise Exception('Use direct mode for image estimation!')
# Compute Laplacian
if alpha > 0.:
gx_gpu = gputools.pad_cpu2gpu(np.array([[-1, 1], [-1, 1],
[-1, 1]]),
self.sfft_gpu,
dtype='complex')
gy_gpu = gputools.pad_cpu2gpu(np.array([[-1, -1, -1],
[1, 1, 1]]),
self.sfft_gpu,
dtype='complex')
self.plan.execute(gx_gpu)
self.plan.execute(gy_gpu)
L_gpu = gx_gpu * gx_gpu.conj() + gy_gpu * gy_gpu.conj()
else:
L_gpu = cua.zeros(self.fft_gpu.shape, np.complex64)
if edgetapering == 1:
gputools.edgetaper_gpu(y_gpu, 2 * self.sf, 'barthann')
# Transfer to GPU
if self.x.__class__ == cua.GPUArray:
x_gpu = self.x
else:
x_gpu = cua.to_gpu(self.x)
# Compute gradient images
xx_gpu, xy_gpu = gputools.gradient_gpu(x_gpu)
yx_gpu, yy_gpu = gputools.gradient_gpu(y_gpu)
# Chop and pad business
if self.mode == 'valid':
yx_gpu = gputools.chop_pad_GPU(yx_gpu, self.winaux.csf,
self.winaux.sw,
self.winaux.nhop, self.sfft_gpu,
self.sf - 1, 'complex')
yy_gpu = gputools.chop_pad_GPU(yy_gpu, self.winaux.csf,
self.winaux.sw,
self.winaux.nhop, self.sfft_gpu,
self.sf - 1, 'complex')
elif self.mode == 'same':
yx_gpu = gputools.chop_pad_GPU(yx_gpu, self.winaux.csf,
self.winaux.sw,
self.winaux.nhop, self.sfft_gpu,
np.floor(self.sf / 2),
'complex')
yy_gpu = gputools.chop_pad_GPU(yy_gpu, self.winaux.csf,
self.winaux.sw,
self.winaux.nhop, self.sfft_gpu,
np.floor(self.sf / 2),
'complex')
else:
raise NotImplementedError('Not a valid mode!')
xx_gpu = gputools.chop_pad_GPU(xx_gpu,
self.winaux.csf,
self.winaux.sw,
self.winaux.nhop,
self.sfft_gpu,
dtype='complex')
xy_gpu = gputools.chop_pad_GPU(xy_gpu,
self.winaux.csf,
self.winaux.sw,
self.winaux.nhop,
self.sfft_gpu,
dtype='complex')
# Here each patch should be windowed to reduce ringing artifacts,
# however since we are working in the gradient domain, the effect
# is negligible
# ws_gpu = gputools.pad_stack_GPU(self.winaux.ws_gpu,
# self.sfft_gpu, self.sf-1,
# dtype='complex')
# xx_gpu = ws_gpu * xx_gpu
# xy_gpu = ws_gpu * xy_gpu
# yx_gpu = ws_gpu * yx_gpu
# yy_gpu = ws_gpu * yy_gpu
# Compute Fourier transform
self.fft(yx_gpu, self.fft_gpu.shape[0])
self.fft(yy_gpu, self.fft_gpu.shape[0])
self.fft(xx_gpu, self.fft_gpu.shape[0])
self.fft(xy_gpu, self.fft_gpu.shape[0])
# Do division in Fourier space
z_gpu = cua.zeros(xy_gpu.shape, np.complex64)
z_gpu = gputools.comp_ola_gdeconv(xx_gpu, xy_gpu, yx_gpu, yy_gpu,
L_gpu, alpha, beta)
# Computing the inverse FFT
z_gpu = z_gpu.conj()
self.fft(z_gpu, self.fft_gpu.shape[0])
z_gpu = z_gpu.conj() / np.prod(z_gpu.shape[-2::])
# Crop out the kernels
zc_gpu = gputools.crop_stack_GPU(z_gpu, self.sf)
return zc_gpu
# --------------------------------------------------------------------
elif mode == 'direct':
const_gpu = cua.empty_like(y_gpu)
const_gpu.fill(1.)
# First deconvolution without corrective term
y_gpu = self.deconv(y_gpu,
mode='xdirect',
alpha=alpha,
beta=beta,
edgetapering=edgetapering)
gputools.cliplower_GPU(y_gpu, 0)
# Now same for constant image to get rid of window artifacts
if edgetapering == 1:
gputools.edgetaper_gpu(const_gpu, 2 * self.sf, 'barthann')
const_gpu = self.deconv(const_gpu,
mode='xdirect',
alpha=alpha,
beta=beta,
edgetapering=edgetapering)
gputools.edgetaper_gpu(const_gpu, 2 * self.sf, 'barthann')
gputools.clip_GPU(const_gpu, 0.01, 10.)
# Division of deconvolved latent and constant image to get rid
# of artifacts stemming from windowing
y_gpu = y_gpu / const_gpu
sz = y_gpu.shape
#gputools.clip_GPU(y_gpu, 0., 1.0)
#gputools.edgetaper_gpu(y_gpu, 3*self.sf, 'barthann')
# Do cropping and padding since edges are corrupted by division
y_gpu = gputools.crop_gpu2cpu(y_gpu,
sz - factor * self.sf - 1,
offset=np.floor(
(factor * self.sf - 1) / 2.))
y_gpu = gputools.impad_gpu(y_gpu,
tuple(np.array(sz) - y_gpu.shape))
return y_gpu
# --------------------------------------------------------------------
elif mode == 'xdirect':
# Compute Laplacian
if alpha > 0.:
gx_gpu = gputools.pad_cpu2gpu(np.array([[-1, 1]]),
self.sfft_gpu,
dtype='complex')
gy_gpu = gputools.pad_cpu2gpu(np.array([[-1], [1]]),
self.sfft_gpu,
dtype='complex')
self.plan.execute(gx_gpu)
self.plan.execute(gy_gpu)
L_gpu = gx_gpu * gx_gpu.conj() + gy_gpu * gy_gpu.conj()
else:
L_gpu = cua.zeros(self.fft_gpu.shape, np.complex64)
# Edgetapering of blurry input image
if edgetapering == 1:
gputools.edgetaper_gpu(y_gpu, 3 * self.sf, 'barthann')
if self.mode == 'valid':
#y_gpu = gputools.pad_cpu2gpu(y_gpu, self.sx, self.sf-1, dtype='real')
offset = self.sf - 1
elif self.mode == 'same':
offset = np.floor(self.sf / 2)
else:
raise NotImplementedError('Not a valid mode!')
# Chop and pad business
y_gpu = gputools.chop_pad_GPU(y, self.winaux.csf, self.winaux.sw,
self.winaux.nhop, self.sfft_gpu,
offset, 'complex')
ws_gpu = gputools.pad_stack_GPU(self.winaux.ws_gpu,
self.sfft_gpu,
dtype='complex')
# Windowing
y_gpu = ws_gpu * y_gpu
# Compute FFT
self.fft(y_gpu, self.fft_gpu.shape[0])
# Do division in Fourier space
z_gpu = gputools.comp_ola_deconv(self.fft_gpu, y_gpu, L_gpu, alpha,
beta)
# Computing the inverse FFT
z_gpu = z_gpu.conj()
self.fft(z_gpu, self.fft_gpu.shape[0])
z_gpu = z_gpu.conj() / np.prod(z_gpu.shape[-2::])
# Crop the solution to correct output size
if self.__id__ == 'X':
zc_gpu = gputools.crop_stack_GPU(z_gpu, self.sf)
return zc_gpu
elif self.__id__ == 'F':
zs_gpu = gputools.crop_stack_GPU(z_gpu, self.winaux.sw)
#zs_gpu = self.winaux.ws_gpu * zs_gpu
zc_gpu = gputools.ola_GPU_test(zs_gpu, self.winaux.csf,
self.winaux.sw,
self.winaux.nhop)
zc_gpu = gputools.crop_gpu2cpu(zc_gpu, self.sx)
return zc_gpu
# --------------------------------------------------------------------
elif mode == 'sparse':
# Compute Laplacian
gx_gpu = gputools.pad_cpu2gpu(np.sqrt(2.) / 2. *
np.array([[-1, 1]]),
self.sfft_gpu,
dtype='complex')
gy_gpu = gputools.pad_cpu2gpu(np.sqrt(2.) / 2. *
np.array([[-1], [1]]),
self.sfft_gpu,
dtype='complex')
self.plan.execute(gx_gpu)
self.plan.execute(gy_gpu)
L_gpu = gx_gpu * gx_gpu.conj() + gy_gpu * gy_gpu.conj()
const_gpu = cua.empty_like(y_gpu)
const_gpu.fill(1.)
# Edgetapering
if edgetapering == 1:
gputools.edgetaper_gpu(y_gpu, 2 * self.sf, 'barthann')
gputools.edgetaper_gpu(const_gpu, 2 * self.sf, 'barthann')
# Parameter settings
beta = 1.
beta_rate = 2. * np.sqrt(2.)
beta_max = 2.**8
# Initialisation of x with padded version of y
x_gpu = 1 * y_gpu
if self.mode == 'valid':
offset = self.sf - 1
elif self.mode == 'same':
offset = np.floor(self.sf / 2)
else:
raise NotImplementedError('Not a valid mode!')
# Chop and pad business
y_gpu = gputools.chop_pad_GPU(y_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, offset, 'complex')
const_gpu = gputools.chop_pad_GPU(const_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop,
self.sfft_gpu, offset, 'complex')
ws_gpu = gputools.pad_stack_GPU(self.winaux.ws_gpu,
self.sfft_gpu,
offset,
dtype='complex')
# Windowing
y_gpu = y_gpu * ws_gpu
# Constant image for corrective weighting term
const_gpu = const_gpu * ws_gpu
del ws_gpu
self.fft(const_gpu, self.fft_gpu.shape[0])
const_gpu = gputools.comp_ola_deconv(self.fft_gpu, const_gpu,
L_gpu, alpha, gamma)
const_gpu = const_gpu.conj()
self.fft(const_gpu, self.fft_gpu.shape[0])
const_gpu = const_gpu.conj() / np.prod(const_gpu.shape[-2::])
const_gpu = gputools.crop_stack_GPU(const_gpu, self.winaux.sw)
const_gpu = const_gpu * self.winaux.ws_gpu
const_gpu = gputools.ola_GPU_test(const_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop)
const_gpu = gputools.crop_gpu2cpu(const_gpu, self.sx)
# For debugging purposes
#scipy.misc.imsave('const1.png', const_gpu.get()/const_gpu.get().max())
gputools.cliplower_GPU(const_gpu, 0.01)
const_gpu = 0.01 / const_gpu
# Precompute F'y
self.fft(y_gpu, self.fft_gpu.shape[0])
y_gpu = y_gpu * self.fft_gpu.conj()
while beta < beta_max:
# Compute gradient images of x
xx_gpu, xy_gpu = gputools.gradient_gpu(x_gpu)
del x_gpu
# w sub-problem for alpha 2/3
gputools.modify_sparse23_gpu(xx_gpu, beta)
gputools.modify_sparse23_gpu(xy_gpu, beta)
#gputools.modify_sparse_gpu(xx_gpu, beta, 0.01)
#gputools.modify_sparse_gpu(xy_gpu, beta, 0.01)
# Chop and pad to size of FFT
xx_gpu = gputools.chop_pad_GPU(xx_gpu,
self.winaux.csf,
self.winaux.sw,
self.winaux.nhop,
self.sfft_gpu,
dtype='complex')
xy_gpu = gputools.chop_pad_GPU(xy_gpu,
self.winaux.csf,
self.winaux.sw,
self.winaux.nhop,
self.sfft_gpu,
dtype='complex')
# Compute Fourier transform
self.fft(xx_gpu, self.fft_gpu.shape[0])
self.fft(xy_gpu, self.fft_gpu.shape[0])
# Do division in Fourier space
x_gpu = gputools.comp_ola_sdeconv(gx_gpu, gy_gpu, xx_gpu,
xy_gpu, y_gpu, self.fft_gpu,
L_gpu, alpha, beta, gamma)
del xx_gpu, xy_gpu
# Computing the inverse FFT
x_gpu = x_gpu.conj()
self.fft(x_gpu, self.fft_gpu.shape[0])
x_gpu = x_gpu.conj()
x_gpu /= np.prod(x_gpu.shape[-2::])
# Ola and cropping
x_gpu = gputools.crop_stack_GPU(x_gpu, self.winaux.sw)
x_gpu = x_gpu * self.winaux.ws_gpu
x_gpu = gputools.ola_GPU_test(x_gpu, self.winaux.csf,
self.winaux.sw, self.winaux.nhop)
x_gpu = gputools.crop_gpu2cpu(x_gpu, self.sx)
# Enforce positivity
x_gpu = x_gpu * const_gpu
gputools.cliplower_GPU(x_gpu, 0.)
beta *= beta_rate
return x_gpu
else:
raise NotImplementedError('Not a valid deconv mode!')
def process(opts):
# ============================================================================
# Specify some parameter settings
# ----------------------------------------------------------------------------
# Specify data path and file identifier
DATAPATH = '/DATA/LSST/FITS'
RESPATH = '../../../DATA/results'
BASE_N = 141
FILENAME = lambda i: '%s/v88827%03d-fz.R22.S11.fits' % (DATAPATH,
(BASE_N + i))
ID = 'LSST'
# ----------------------------------------------------------------------------
# Specify parameter settings
# General
doshow = opts.doShow # put 1 to show intermediate results
backup = opts.backup # put 1 to write intermediate results to disk
N = opts.N # how many frames to process
N0 = opts.N0 # number of averaged frames for initialisation
# OlaGPU parameters
sf = np.array([40, 40]) # estimated size of PSF
csf = (3, 3) # number of kernels across x and y direction
overlap = 0.5 # overlap of neighboring patches in percent
# Regularization parameters for kernel estimation
f_alpha = opts.f_alpha # promotes smoothness
f_beta = opts.f_beta # Thikhonov regularization
optiter = opts.optiter # number of iterations for minimization
tol = opts.tol # tolerance for when to stop minimization
# ============================================================================
# Create helper functions for file handling
# # # HACK for chunking into available GPU mem # # #
# - loads one 1kx1k block out of the fits image
xOffset = 2000
yOffset = 0
chunkSize = 1000
yload = lambda i: 1. * fitsTools.readFITS(
FILENAME(i), use_mask=True, norm=True)[yOffset:yOffset + chunkSize,
xOffset:xOffset + chunkSize]
# ----------------------------------------------------------------------------
# Some more code for backuping the results
# ----------------------------------------------------------------------------
# For backup purposes
EXPPATH = '%s/%s_sf%dx%d_csf%dx%d_maxiter%d_alpha%.2f_beta%.2f' % \
(RESPATH,ID,sf[0],sf[1],csf[0],csf[1],optiter,f_alpha,f_beta)
xname = lambda i: '%s/x_%04d.png' % (EXPPATH, i)
yname = lambda i: '%s/y_%04d.png' % (EXPPATH, i)
fname = lambda i: '%s/f_%04d.png' % (EXPPATH, i)
if os.path.exists(EXPPATH) and opts.overwrite:
try:
rmtree(EXPPATH)
except:
print "[ERROR] removing old results dir:", EXPPATH
exit()
elif os.path.exists(EXPPATH):
print "[ERROR] results directory already exists, please remove or use '-o' to overwrite"
exit()
# Create results path if not existing
try:
os.makedirs(EXPPATH)
except:
print "[ERROR] creating results dir:", EXPPATH
exit()
print 'Results are saved to: \n %s \n' % EXPPATH
# ----------------------------------------------------------------------------
# For displaying intermediate results create target figure
# ----------------------------------------------------------------------------
# Create figure for displaying intermediate results
if doshow:
print "showing intermediate results is currently disabled.."
#pl.figure(1)
#pl.draw()
# ----------------------------------------------------------------------------
# Code for initialising the online multi-frame deconvolution
# ----------------------------------------------------------------------------
# Initialisation of latent image by averaging the first 20 frames
y0 = 0.
for i in np.arange(1, N0):
y0 += yload(i)
y0 /= N0
y_gpu = cua.to_gpu(y0)
# Pad image since we perform deconvolution with valid boundary conditions
x_gpu = gputools.impad_gpu(y_gpu, sf - 1)
# Create windows for OlaGPU
sx = y0.shape + sf - 1
sf2 = np.floor(sf / 2)
winaux = imagetools.win2winaux(sx, csf, overlap)
# ----------------------------------------------------------------------------
# Loop over all frames and do online blind deconvolution
# ----------------------------------------------------------------------------
import time as t
ti = t.clock()
t1 = stopwatch.timer()
t2 = stopwatch.timer()
t3 = stopwatch.timer()
t4 = stopwatch.timer()
t4.start()
for i in np.arange(1, N + 1):
print 'Processing frame %d/%d \r' % (i, N)
# Load next observed image
t3.start()
y = yload(i)
print "TIMER load:", t3.elapsed()
# Compute mask for determining saturated regions
mask_gpu = 1. * cua.to_gpu(y < 1.)
y_gpu = cua.to_gpu(y)
# ------------------------------------------------------------------------
# PSF estimation
# ------------------------------------------------------------------------
# Create OlaGPU instance with current estimate of latent image
t2.start()
X = olaGPU.OlaGPU(x_gpu, sf, 'valid', winaux=winaux)
print "TIMER GPU: ", t2.elapsed()
t1.start()
# PSF estimation for given estimate of latent image and current observation
f = X.deconv(y_gpu,
mode='lbfgsb',
alpha=f_alpha,
beta=f_beta,
maxfun=optiter,
verbose=10)
print "TIMER Optimization: ", t1.elapsed()
fs = f[0]
# Normalize PSF kernels to sum up to one
fs = gputools.normalize(fs)
# ------------------------------------------------------------------------
# Latent image estimation
# ------------------------------------------------------------------------
# Create OlaGPU instance with estimated PSF
t2.start()
F = olaGPU.OlaGPU(fs, sx, 'valid', winaux=winaux)
# Latent image estimation by performing one gradient descent step
# multiplicative update is used which preserves positivity
factor_gpu = F.cnvtp(
mask_gpu * y_gpu) / (F.cnvtp(mask_gpu * F.cnv(x_gpu)) + tol)
gputools.cliplower_GPU(factor_gpu, tol)
x_gpu = x_gpu * factor_gpu
x_max = x_gpu.get()[sf[0]:-sf[0], sf[1]:-sf[1]].max()
gputools.clipupper_GPU(x_gpu, x_max)
print "TIMER GPU: ", t2.elapsed()
# ------------------------------------------------------------------------
# For backup intermediate results
# ------------------------------------------------------------------------
if backup or i == N:
# Write intermediate results to disk incl. input
y_img = y_gpu.get() * 1e5
fitsTools.asinhScale(y_img,
450,
-50,
minCut=0.0,
maxCut=40000,
fname=yname(i))
# Crop image to input size
xi = (x_gpu.get()[sf2[0]:-sf2[0], sf2[1]:-sf2[1]] / x_max) * 1e5
fitsTools.fitsStats(xi)
fitsTools.asinhScale(xi,
450,
-50,
minCut=0.0,
maxCut=40000,
fname=xname(i))
# Concatenate PSF kernels for ease of visualisation
f = imagetools.gridF(fs, csf)
f = f * 1e5
fitsTools.asinhScale(f,
450,
-50,
minCut=0.0,
maxCut=40000,
fname=fname(i))
# ------------------------------------------------------------------------
# For displaying intermediate results
# ------------------------------------------------------------------------
'''
if np.mod(i,1) == 0 and doshow:
pl.figure(1)
pl.subplot(121)
# what is SY?
pl.imshow(imagetools.crop(x_gpu.get(),sy,np.ceil(sf/2)),'gray')
pl.title('x after %d observations' % i)
pl.subplot(122)
pl.imshow(y_gpu.get(),'gray')
pl.title('y(%d)' % i)
pl.draw()
pl.figure(2)
pl.title('PSF(%d)' % i)
imagetools.cellplot(fs, winaux.csf)
tf = t.clock()
print('Time elapsed after %d frames %.3f' % (i,(tf-ti)))
'''
tf = t.clock()
print('Time elapsed for total image sequence %.3f' % (tf - ti))
# ----------------------------------------------------------------------------
print "TOTAL: %.3f" % (t4.elapsed())
print "OptimizeCPUtime %.3f %.3f" % (t1.getTotal(), 100 *
(t1.getTotal() / t4.getTotal()))
print "GPUtime %.3f %.3f" % (t2.getTotal(), 100 *
(t2.getTotal() / t4.getTotal()))
print "LoadTime %.3f %.3f" % (t3.getTotal(), 100 *
(t3.getTotal() / t4.getTotal()))
