def __init__(self, Vm, parameters=[]):
""" Vm = FunctionSpace for the parameters m1, and m2 """
self.parameters = {}
self.parameters['k'] = 1.0
self.parameters['eps'] = 1e-2
self.parameters['amg'] = 'default'
self.parameters['nb_param'] = 2
self.parameters['use_i'] = False
self.parameters['print'] = False
self.parameters.update(parameters)
n = self.parameters['nb_param']
use_i = self.parameters['use_i']
assert not ((not use_i) * (n > 2))
if not use_i:
VmVm = createMixedFS(Vm, Vm)
else:
if self.parameters['print']:
print '[V_TV] Using createMixedFSi'
Vms = []
for ii in range(n):
Vms.append(Vm)
VmVm = createMixedFSi(Vms)
self.parameters['Vm'] = VmVm
self.regTV = TV(self.parameters)
if not use_i:
self.m1, self.m2 = Function(Vm), Function(Vm)
self.m = Function(VmVm)
def __init__(self, mesh, k, regularization='tikhonov'):
"""
Inputs:
pbtype = 'denoising' or 'deblurring'
mesh = Fenics mesh
k = Fenics Expression of the blurring kernel; must have parameter t
f = target image
"""
self.mesh = mesh
self.V = dl.FunctionSpace(self.mesh, 'Lagrange', 1)
self.dimV = self.V.dim()
self.xx = self.V.dofmap().tabulate_all_coordinates(self.mesh)
self.test, self.trial = dl.TestFunction(self.V), dl.TrialFunction(
self.V)
# Target data:
self.f_true = 0.75 * (self.xx >= .1) * (self.xx <= .25)
self.f_true += (self.xx >= 0.28) * (self.xx <= 0.3) * (15 * self.xx -
15 * 0.28)
self.f_true += (self.xx > 0.3) * (self.xx < 0.33) * 0.3
self.f_true += (self.xx >= 0.33) * (self.xx <= 0.35) * (-15 * self.xx +
15 * 0.35)
self.f_true += (self.xx >= .4) * (self.xx <= .9) * (
self.xx - .4)**2 * (self.xx - 0.9)**2 / .25**4
self.g = None # current iterate
# kernel operator
self.k = k
self.Kweak = dl.inner(self.k, self.test) * dl.dx
self.assembleK()
# mass matrix
self.Mweak = dl.inner(self.test, self.trial) * dl.dx
self.M = dl.assemble(self.Mweak)
# regularization
self.parameters['regularization'] = regularization
if regularization == 'tikhonov':
self.RegTikh = LaplacianPrior({
'gamma': 1.0,
'beta': 0.0,
'Vm': self.V
})
self.R = self.RegTikh.Minvprior.array()
elif regularization == 'TV':
self.RegTV = TV({'eps': 1e-2, 'Vm': self.V})
# line search parameters
self.parameters['alpha0'] = 1.0
self.parameters['rho'] = 0.5
self.parameters['c'] = 5e-5
self.parameters['max_backtrack'] = 12
def define_regularization(self, parameters=None):
if not parameters == None:
self.parameters.update(parameters)
regularization = self.parameters["regularization"]
if regularization == "tikhonov":
gamma = self.parameters["gamma"]
beta = self.parameters["beta"]
self.Reg = LaplacianPrior({"gamma": gamma, "beta": beta, "Vm": self.V})
self.inexact = False
elif regularization == "TV":
eps = self.parameters["eps"]
k = self.parameters["k"]
mode = self.parameters["mode"]
if mode == "primaldual":
self.Reg = self.Reg = TVPD({"eps": eps, "k": k, "Vm": self.V, "GNhessian": False})
elif mode == "full":
self.Reg = TV({"eps": eps, "k": k, "Vm": self.V, "GNhessian": False})
else:
self.Reg = TV({"eps": eps, "k": k, "Vm": self.V, "GNhessian": True})
self.inexact = False
def __init__(self, mesh, k, regularization="tikhonov"):
"""
Inputs:
pbtype = 'denoising' or 'deblurring'
mesh = Fenics mesh
k = Fenics Expression of the blurring kernel; must have parameter t
f = target image
"""
self.mesh = mesh
self.V = dl.FunctionSpace(self.mesh, "Lagrange", 1)
self.dimV = self.V.dim()
self.xx = self.V.dofmap().tabulate_all_coordinates(self.mesh)
self.test, self.trial = dl.TestFunction(self.V), dl.TrialFunction(self.V)
# Target data:
self.f_true = 0.75 * (self.xx >= 0.1) * (self.xx <= 0.25)
self.f_true += (self.xx >= 0.28) * (self.xx <= 0.3) * (15 * self.xx - 15 * 0.28)
self.f_true += (self.xx > 0.3) * (self.xx < 0.33) * 0.3
self.f_true += (self.xx >= 0.33) * (self.xx <= 0.35) * (-15 * self.xx + 15 * 0.35)
self.f_true += (self.xx >= 0.4) * (self.xx <= 0.9) * (self.xx - 0.4) ** 2 * (self.xx - 0.9) ** 2 / 0.25 ** 4
self.g = None # current iterate
# kernel operator
self.k = k
self.Kweak = dl.inner(self.k, self.test) * dl.dx
self.assembleK()
# mass matrix
self.Mweak = dl.inner(self.test, self.trial) * dl.dx
self.M = dl.assemble(self.Mweak)
# regularization
self.parameters["regularization"] = regularization
if regularization == "tikhonov":
self.RegTikh = LaplacianPrior({"gamma": 1.0, "beta": 0.0, "Vm": self.V})
self.R = self.RegTikh.Minvprior.array()
elif regularization == "TV":
self.RegTV = TV({"eps": 1e-2, "Vm": self.V})
# line search parameters
self.parameters["alpha0"] = 1.0
self.parameters["rho"] = 0.5
self.parameters["c"] = 5e-5
self.parameters["max_backtrack"] = 12
class ObjectiveImageDeblurring1D():
"""
Class for linear 1D image denoising problem, built on an integral (blurring) kernel
"""
def __init__(self, mesh, k, regularization='tikhonov'):
"""
Inputs:
pbtype = 'denoising' or 'deblurring'
mesh = Fenics mesh
k = Fenics Expression of the blurring kernel; must have parameter t
f = target image
"""
self.mesh = mesh
self.V = dl.FunctionSpace(self.mesh, 'Lagrange', 1)
self.dimV = self.V.dim()
self.xx = self.V.dofmap().tabulate_all_coordinates(self.mesh)
self.test, self.trial = dl.TestFunction(self.V), dl.TrialFunction(
self.V)
# Target data:
self.f_true = 0.75 * (self.xx >= .1) * (self.xx <= .25)
self.f_true += (self.xx >= 0.28) * (self.xx <= 0.3) * (15 * self.xx -
15 * 0.28)
self.f_true += (self.xx > 0.3) * (self.xx < 0.33) * 0.3
self.f_true += (self.xx >= 0.33) * (self.xx <= 0.35) * (-15 * self.xx +
15 * 0.35)
self.f_true += (self.xx >= .4) * (self.xx <= .9) * (
self.xx - .4)**2 * (self.xx - 0.9)**2 / .25**4
self.g = None # current iterate
# kernel operator
self.k = k
self.Kweak = dl.inner(self.k, self.test) * dl.dx
self.assembleK()
# mass matrix
self.Mweak = dl.inner(self.test, self.trial) * dl.dx
self.M = dl.assemble(self.Mweak)
# regularization
self.parameters['regularization'] = regularization
if regularization == 'tikhonov':
self.RegTikh = LaplacianPrior({
'gamma': 1.0,
'beta': 0.0,
'Vm': self.V
})
self.R = self.RegTikh.Minvprior.array()
elif regularization == 'TV':
self.RegTV = TV({'eps': 1e-2, 'Vm': self.V})
# line search parameters
self.parameters['alpha0'] = 1.0
self.parameters['rho'] = 0.5
self.parameters['c'] = 5e-5
self.parameters['max_backtrack'] = 12
def assembleK(self):
self.K = np.zeros((self.dimV, self.dimV))
for ii, tt in enumerate(self.xx):
self.k.t = tt
self.K[ii, :] = dl.assemble(self.Kweak).array()
def generatedata(self, noisepercent):
""" compute data and add noisepercent (%) of noise """
pbtype = self.parameters['pbtype']
self.d = self.K.dot(self.f_true)
sigma = noisepercent * np.linalg.norm(self.d) / np.sqrt(self.dimV)
eta = sigma * np.random.randn(self.dimV)
print 'noise residual={}'.format(.5 * np.linalg.norm(eta)**2)
self.dn = self.d + eta
def update_reg(self, gamma):
regularization = self.parameters['regularization']
if regularization == 'tikhonov':
self.gamma = gamma
self.R = self.RegTikh.Minvprior.array() * self.gamma
elif regularization == 'TV':
self.RegTV.update({'k': gamma})
### COST and DERIVATIVES
def computecost(self, f=None):
""" Compute cost functional at f """
regularization = self.parameters['regularization']
if f == None: f = self.g
#
if pbtype == 'deblurring':
self.misfit = .5 * np.linalg.norm(self.K.dot(f) - self.dn)**2
if regularization == 'tikhonov': self.reg = .5 * (self.R.dot(f)).dot(f)
elif regularization == 'TV': self.reg = self.RegTV.cost(f)
self.cost = self.misfit + self.reg
return self.cost
def gradient(self, f=None):
""" Compute M.g (discrete gradient) at a given point f """
regularization = self.parameters['regularization']
if f == None: f = self.g
#
self.MGk = self.K.T.dot(self.K.dot(f) - self.dn)
if regularization == 'tikhonov': self.MGr = self.R.dot(f)
elif regularization == 'TV': self.MGr = self.RegTV.grad(f).array()
self.MG = self.MGk + self.MGr
def Hessian(self, f=None):
""" Assemble Hessian at f """
regularization = self.parameters['regularization']
if f == None: f = self.g
#
self.Hessk = self.K.T.dot(self.K)
if regularization == 'tikhonov': self.Hessr = self.R
elif regularization == 'TV':
self.RegTV.assemble_hessian(f)
self.Hessr = self.RegTV.H.array()
self.Hess = self.Hessk + self.Hessr
### SOLVER
def searchdirection(self):
""" Compute search direction """
self.gradient()
self.Hessian()
self.df = np.linalg.solve(self.Hess, -self.MG)
assert self.df.dot(self.MG) < 0.0, "not a descent direction"
def linesearch(self):
""" Perform inexact backtracking line search """
self.alpha = self.parameters['alpha0']
rho = self.parameters['rho']
c = self.parameters['c']
costref = self.cost
cdJdf = c * self.MG.dot(self.df)
self.LS = False
for ii in xrange(self.parameters['max_backtrack']):
if self.computecost(self.g + self.alpha*self.df) < \
costref + self.alpha*cdJdf:
self.g = self.g + self.alpha * self.df
self.LS = True
break
else:
self.alpha *= rho
def solve(self, plot=False):
""" Solve image denoising pb """
regularization = self.parameters['regularization']
#
if regularization == 'tikhonov':
self.Hessian(None)
self.g = np.linalg.solve(self.Hess, self.K.T.dot(self.dn))
self.computecost()
self.alpha = 1.0
elif regularization == 'TV':
self.computecost()
self.alpha = 1.0
self.printout()
cost = self.cost
self.COST = [cost]
self.MEDMIS = [self.relmedmisfit]
for ii in xrange(500):
self.searchdirection()
self.linesearch(1.0)
print ii,
self.printout()
if plot and ii % 50 == 0:
self.plot()
plt.show()
if not self.LS:
print 'Line search failed'
break
if np.abs(cost - self.cost) / cost < 1e-10:
print 'optimization converged'
break
cost = self.cost
self.COST.append(cost)
self.MEDMIS.append(self.relmedmisfit)
### TESTS
def test_gradient(self, f=None, n=5):
""" test gradient with FD approx around point f """
if f == None: f = self.f_true.copy()
pm = [1.0, -1.0]
eps = 1e-5
self.gradient(f)
for nn in xrange(1, n + 1):
df = np.sin(np.pi * nn * self.xx)
cost = []
for sign in pm:
self.g = f + sign * eps * df
cost.append(self.computecost(self.g))
MGFD = (cost[0] - cost[1]) / (2 * eps)
MGdf = self.MG.dot(df)
print 'n={}:\tMGFD={:.5e}, MGdf={:.5e}, error={:.2e}'.format(\
nn, MGFD, MGdf, np.abs(MGdf-MGFD)/np.abs(MGdf))
def test_hessian(self, f=None, n=5):
""" test Hessian with FD approx around point f """
if f == None: f = self.f_true.copy()
pm = [1.0, -1.0]
eps = 1e-5
self.Hessian(f)
for nn in xrange(1, n + 1):
df = np.sin(np.pi * nn * self.xx)
MG = []
for sign in pm:
self.g = f + sign * eps * df
self.gradient(self.g)
MG.append(self.MG)
HFD = (MG[0] - MG[1]) / (2 * eps)
Hdf = self.Hess.dot(df)
print 'n={}:\tHFD={:.5e}, Hdf={:.5e}, error={:.2e}'.format(\
nn, np.linalg.norm(HFD), np.linalg.norm(Hdf), \
np.linalg.norm(Hdf-HFD)/np.linalg.norm(Hdf))
### OUTPUT
def printout(self):
""" Print results """
self.medmisfit = np.linalg.norm(self.g - self.f_true)
self.relmedmisfit = self.medmisfit / np.linalg.norm(self.f_true)
print 'cost={:.2e}, misfit={:.2e}, reg={:.2e}, alpha={:.2e}, medmisfit={:.2e} ({:.3f})'.format(\
self.cost, self.misfit, self.reg, self.alpha, self.medmisfit, self.relmedmisfit)
def plot(self, u=None):
""" Plot data and target """
fig = plt.figure()
ax = fig.add_subplot(111)
#ax.plot(self.xx, self.dn, label='noisy data')
ax.plot(self.xx, self.f_true, label='target')
if not u == None:
ax.plot(self.xx, u, label='u')
elif not self.g == None:
ax.plot(self.xx, self.g, label='sol')
ax.legend(loc='best')
return fig
def __init__(self,
CGdeg,
regularizationtype,
h=1.0,
parameters=[],
image='image.dat'):
class Image(dl.Expression):
def __init__(self, Lx, Ly, data):
self.data = data
self.hx = Lx / float(self.data.shape[1] - 1)
self.hy = Ly / float(self.data.shape[0] - 1)
def eval(self, values, x):
j = math.floor(x[0] / self.hx)
i = math.floor(x[1] / self.hy)
values[0] = self.data[i, j]
data = np.loadtxt(image, delimiter=',')
#Lx, Ly = float(data.shape[1])/float(data.shape[0]), 1.
Lx, Ly = 2., 1.
scaling = 100. * h # =1.0 => h~0.01
Lx, Ly = scaling * Lx, scaling * Ly
np.random.seed(seed=1)
noise_std_dev = 0.3
noise = noise_std_dev * np.random.randn(data.shape[0], data.shape[1])
print '||noise||={}'.format(np.linalg.norm(noise))
mesh = dl.RectangleMesh(dl.Point(0, 0), dl.Point(Lx, Ly), 200, 100)
mcoord = mesh.coordinates()
print 'hx={}, hy={}'.format((mcoord[-1][0] - mcoord[0][0]) / 200.,
(mcoord[-1][1] - mcoord[0][1]) / 100.)
V = dl.FunctionSpace(mesh, 'Lagrange', CGdeg)
trueImage = Image(Lx, Ly, data)
noisyImage = Image(Lx, Ly, data + noise)
print 'min(data)={}, max(data)={}'.format(np.amin(data), np.amax(data))
print 'min(data+noise)={}, max(data+noise)={}'.format(
np.amin(data + noise), np.amax(data + noise))
self.u_true = dl.interpolate(trueImage, V)
self.u_0 = dl.interpolate(noisyImage, V)
self.u = dl.Function(V)
self.ucopy = dl.Function(V)
self.G = dl.Function(V)
self.du = dl.Function(V)
u_test = dl.TestFunction(V)
u_trial = dl.TrialFunction(V)
Mweak = dl.inner(u_test, u_trial) * dl.dx
self.M = dl.assemble(Mweak)
self.solverM = dl.LUSolver('petsc')
self.solverM.parameters['symmetric'] = True
self.solverM.parameters['reuse_factorization'] = True
self.solverM.set_operator(self.M)
self.regul = regularizationtype
if self.regul == 'tikhonov':
self.Regul = LaplacianPrior({'Vm': V, 'gamma': 1.0, 'beta': 0.0})
elif self.regul == 'TV':
paramTV = {'Vm': V, 'k': 1.0, 'eps': 1e-4, 'GNhessian': True}
paramTV.update(parameters)
self.Regul = TV(paramTV)
self.inexact = False
elif self.regul == 'TVPD':
paramTV = {'Vm': V, 'k': 1.0, 'eps': 1e-4, 'exact': False}
paramTV.update(parameters)
self.Regul = TVPD(paramTV)
self.inexact = False
self.alpha = 1.0
self.Hess = self.M
self.parametersLS = {'alpha0':1.0, 'rho':0.5, 'c':5e-5, \
'max_backtrack':12, 'cgtol':0.5, 'maxiter':50000}
filename, ext = os.path.splitext(sys.argv[0])
#if os.path.isdir(filename + '/'): shutil.rmtree(filename + '/')
self.myplot = PlotFenics(filename)
try:
solver = PETScKrylovSolver('cg', 'ml_amg')
self.precond = 'ml_amg'
except:
print '*** WARNING: ML not installed -- using petsc_amg instead'
self.precond = 'petsc_amg'
class V_TV():
""" Definite Vectorial Total Variation regularization from Total Variation class """
def __init__(self, Vm, parameters=[]):
""" Vm = FunctionSpace for the parameters m1, and m2 """
self.parameters = {}
self.parameters['k'] = 1.0
self.parameters['eps'] = 1e-2
self.parameters['amg'] = 'default'
self.parameters['nb_param'] = 2
self.parameters['use_i'] = False
self.parameters['print'] = False
self.parameters.update(parameters)
n = self.parameters['nb_param']
use_i = self.parameters['use_i']
assert not ((not use_i) * (n > 2))
if not use_i:
VmVm = createMixedFS(Vm, Vm)
else:
if self.parameters['print']:
print '[V_TV] Using createMixedFSi'
Vms = []
for ii in range(n):
Vms.append(Vm)
VmVm = createMixedFSi(Vms)
self.parameters['Vm'] = VmVm
self.regTV = TV(self.parameters)
if not use_i:
self.m1, self.m2 = Function(Vm), Function(Vm)
self.m = Function(VmVm)
def isTV(self):
return True
def isPD(self):
return False
def costab(self, m1, m2):
assign(self.m.sub(0), m1)
assign(self.m.sub(1), m2)
return self.regTV.cost(self.m)
def costabvect(self, m1, m2):
setfct(self.m1, m1)
setfct(self.m2, m2)
return self.costab(self.m1, self.m2)
def costabvecti(self, m):
return self.regTV.cost(m)
def gradab(self, m1, m2):
assign(self.m.sub(0), m1)
assign(self.m.sub(1), m2)
return self.regTV.grad(self.m)
def gradabvect(self, m1, m2):
setfct(self.m1, m1)
setfct(self.m2, m2)
return self.gradab(self.m1, self.m2)
def gradabvecti(self, m):
return self.regTV.grad(m)
def assemble_hessianab(self, m1, m2):
setfct(self.m1, m1)
setfct(self.m2, m2)
assign(self.m.sub(0), self.m1)
assign(self.m.sub(1), self.m2)
self.regTV.assemble_hessian(self.m)
def assemble_hessianabi(self, m):
self.regTV.assemble_hessian(m)
def hessianab(self, m1h, m2h):
""" m1h, m2h = Vector(V) """
setfct(self.m1, m1h)
setfct(self.m2, m2h)
assign(self.m.sub(0), self.m1)
assign(self.m.sub(1), self.m2)
return self.regTV.hessian(self.m.vector())
def hessianabi(self, mh):
return self.regTV.hessian(mh)
def getprecond(self):
return self.regTV.getprecond()
class ObjectiveImageDeblurring1D:
"""
Class for linear 1D image denoising problem, built on an integral (blurring) kernel
"""
def __init__(self, mesh, k, regularization="tikhonov"):
"""
Inputs:
pbtype = 'denoising' or 'deblurring'
mesh = Fenics mesh
k = Fenics Expression of the blurring kernel; must have parameter t
f = target image
"""
self.mesh = mesh
self.V = dl.FunctionSpace(self.mesh, "Lagrange", 1)
self.dimV = self.V.dim()
self.xx = self.V.dofmap().tabulate_all_coordinates(self.mesh)
self.test, self.trial = dl.TestFunction(self.V), dl.TrialFunction(self.V)
# Target data:
self.f_true = 0.75 * (self.xx >= 0.1) * (self.xx <= 0.25)
self.f_true += (self.xx >= 0.28) * (self.xx <= 0.3) * (15 * self.xx - 15 * 0.28)
self.f_true += (self.xx > 0.3) * (self.xx < 0.33) * 0.3
self.f_true += (self.xx >= 0.33) * (self.xx <= 0.35) * (-15 * self.xx + 15 * 0.35)
self.f_true += (self.xx >= 0.4) * (self.xx <= 0.9) * (self.xx - 0.4) ** 2 * (self.xx - 0.9) ** 2 / 0.25 ** 4
self.g = None # current iterate
# kernel operator
self.k = k
self.Kweak = dl.inner(self.k, self.test) * dl.dx
self.assembleK()
# mass matrix
self.Mweak = dl.inner(self.test, self.trial) * dl.dx
self.M = dl.assemble(self.Mweak)
# regularization
self.parameters["regularization"] = regularization
if regularization == "tikhonov":
self.RegTikh = LaplacianPrior({"gamma": 1.0, "beta": 0.0, "Vm": self.V})
self.R = self.RegTikh.Minvprior.array()
elif regularization == "TV":
self.RegTV = TV({"eps": 1e-2, "Vm": self.V})
# line search parameters
self.parameters["alpha0"] = 1.0
self.parameters["rho"] = 0.5
self.parameters["c"] = 5e-5
self.parameters["max_backtrack"] = 12
def assembleK(self):
self.K = np.zeros((self.dimV, self.dimV))
for ii, tt in enumerate(self.xx):
self.k.t = tt
self.K[ii, :] = dl.assemble(self.Kweak).array()
def generatedata(self, noisepercent):
""" compute data and add noisepercent (%) of noise """
pbtype = self.parameters["pbtype"]
self.d = self.K.dot(self.f_true)
sigma = noisepercent * np.linalg.norm(self.d) / np.sqrt(self.dimV)
eta = sigma * np.random.randn(self.dimV)
print "noise residual={}".format(0.5 * np.linalg.norm(eta) ** 2)
self.dn = self.d + eta
def update_reg(self, gamma):
regularization = self.parameters["regularization"]
if regularization == "tikhonov":
self.gamma = gamma
self.R = self.RegTikh.Minvprior.array() * self.gamma
elif regularization == "TV":
self.RegTV.update({"k": gamma})
### COST and DERIVATIVES
def computecost(self, f=None):
""" Compute cost functional at f """
regularization = self.parameters["regularization"]
if f == None:
f = self.g
#
if pbtype == "deblurring":
self.misfit = 0.5 * np.linalg.norm(self.K.dot(f) - self.dn) ** 2
if regularization == "tikhonov":
self.reg = 0.5 * (self.R.dot(f)).dot(f)
elif regularization == "TV":
self.reg = self.RegTV.cost(f)
self.cost = self.misfit + self.reg
return self.cost
def gradient(self, f=None):
""" Compute M.g (discrete gradient) at a given point f """
regularization = self.parameters["regularization"]
if f == None:
f = self.g
#
self.MGk = self.K.T.dot(self.K.dot(f) - self.dn)
if regularization == "tikhonov":
self.MGr = self.R.dot(f)
elif regularization == "TV":
self.MGr = self.RegTV.grad(f).array()
self.MG = self.MGk + self.MGr
def Hessian(self, f=None):
""" Assemble Hessian at f """
regularization = self.parameters["regularization"]
if f == None:
f = self.g
#
self.Hessk = self.K.T.dot(self.K)
if regularization == "tikhonov":
self.Hessr = self.R
elif regularization == "TV":
self.RegTV.assemble_hessian(f)
self.Hessr = self.RegTV.H.array()
self.Hess = self.Hessk + self.Hessr
### SOLVER
def searchdirection(self):
""" Compute search direction """
self.gradient()
self.Hessian()
self.df = np.linalg.solve(self.Hess, -self.MG)
assert self.df.dot(self.MG) < 0.0, "not a descent direction"
def linesearch(self):
""" Perform inexact backtracking line search """
self.alpha = self.parameters["alpha0"]
rho = self.parameters["rho"]
c = self.parameters["c"]
costref = self.cost
cdJdf = c * self.MG.dot(self.df)
self.LS = False
for ii in xrange(self.parameters["max_backtrack"]):
if self.computecost(self.g + self.alpha * self.df) < costref + self.alpha * cdJdf:
self.g = self.g + self.alpha * self.df
self.LS = True
break
else:
self.alpha *= rho
def solve(self, plot=False):
""" Solve image denoising pb """
regularization = self.parameters["regularization"]
#
if regularization == "tikhonov":
self.Hessian(None)
self.g = np.linalg.solve(self.Hess, self.K.T.dot(self.dn))
self.computecost()
self.alpha = 1.0
elif regularization == "TV":
self.computecost()
self.alpha = 1.0
self.printout()
cost = self.cost
self.COST = [cost]
self.MEDMIS = [self.relmedmisfit]
for ii in xrange(500):
self.searchdirection()
self.linesearch(1.0)
print ii,
self.printout()
if plot and ii % 50 == 0:
self.plot()
plt.show()
if not self.LS:
print "Line search failed"
break
if np.abs(cost - self.cost) / cost < 1e-10:
print "optimization converged"
break
cost = self.cost
self.COST.append(cost)
self.MEDMIS.append(self.relmedmisfit)
### TESTS
def test_gradient(self, f=None, n=5):
""" test gradient with FD approx around point f """
if f == None:
f = self.f_true.copy()
pm = [1.0, -1.0]
eps = 1e-5
self.gradient(f)
for nn in xrange(1, n + 1):
df = np.sin(np.pi * nn * self.xx)
cost = []
for sign in pm:
self.g = f + sign * eps * df
cost.append(self.computecost(self.g))
MGFD = (cost[0] - cost[1]) / (2 * eps)
MGdf = self.MG.dot(df)
print "n={}:\tMGFD={:.5e}, MGdf={:.5e}, error={:.2e}".format(
nn, MGFD, MGdf, np.abs(MGdf - MGFD) / np.abs(MGdf)
)
def test_hessian(self, f=None, n=5):
""" test Hessian with FD approx around point f """
if f == None:
f = self.f_true.copy()
pm = [1.0, -1.0]
eps = 1e-5
self.Hessian(f)
for nn in xrange(1, n + 1):
df = np.sin(np.pi * nn * self.xx)
MG = []
for sign in pm:
self.g = f + sign * eps * df
self.gradient(self.g)
MG.append(self.MG)
HFD = (MG[0] - MG[1]) / (2 * eps)
Hdf = self.Hess.dot(df)
print "n={}:\tHFD={:.5e}, Hdf={:.5e}, error={:.2e}".format(
nn, np.linalg.norm(HFD), np.linalg.norm(Hdf), np.linalg.norm(Hdf - HFD) / np.linalg.norm(Hdf)
)
### OUTPUT
def printout(self):
""" Print results """
self.medmisfit = np.linalg.norm(self.g - self.f_true)
self.relmedmisfit = self.medmisfit / np.linalg.norm(self.f_true)
print "cost={:.2e}, misfit={:.2e}, reg={:.2e}, alpha={:.2e}, medmisfit={:.2e} ({:.3f})".format(
self.cost, self.misfit, self.reg, self.alpha, self.medmisfit, self.relmedmisfit
)
def plot(self, u=None):
""" Plot data and target """
fig = plt.figure()
ax = fig.add_subplot(111)
# ax.plot(self.xx, self.dn, label='noisy data')
ax.plot(self.xx, self.f_true, label="target")
if not u == None:
ax.plot(self.xx, u, label="u")
elif not self.g == None:
ax.plot(self.xx, self.g, label="sol")
ax.legend(loc="best")
return fig
class ObjectiveImageDenoising:
"""
Class to do image denoising
"""
def __init__(self, mesh, trueImage, parameters=[]):
"""
Inputs:
mesh = Fenics mesh
trueImage = object from class Image
parameters = dict
"""
# Mesh
self.mesh = mesh
self.V = dl.FunctionSpace(self.mesh, "Lagrange", 1)
self.xx = self.V.dofmap().tabulate_all_coordinates(self.mesh)
self.dimV = self.V.dim()
self.test, self.trial = dl.TestFunction(self.V), dl.TrialFunction(self.V)
self.f_true = dl.interpolate(trueImage, self.V)
self.g, self.dg, self.gtmp = dl.Function(self.V), dl.Function(self.V), dl.Function(self.V)
self.Grad = dl.Function(self.V)
self.Gradnorm0 = None
# mass matrix
self.Mweak = dl.inner(self.test, self.trial) * dl.dx
self.M = dl.assemble(self.Mweak)
self.solverM = dl.LUSolver("petsc")
self.solverM.parameters["symmetric"] = True
self.solverM.parameters["reuse_factorization"] = True
self.solverM.set_operator(self.M)
# identity matrix
self.I = dl.assemble(self.Mweak)
self.I.zero()
self.I.set_diagonal(dl.interpolate(dl.Constant(1), self.V).vector())
# self.targetnorm = np.sqrt((self.M*self.f_true.vector()).inner(self.f_true.vector()))
self.targetnorm = np.sqrt((self.f_true.vector()).inner(self.f_true.vector()))
# line search parameters
self.parameters = {"alpha0": 1.0, "rho": 0.5, "c": 5e-5, "max_backtrack": 12}
# regularization
self.parameters.update({"eps": 1e-4, "k": 1.0, "regularization": "TV", "mode": "primaldual"})
self.parameters.update(parameters)
self.define_regularization()
self.regparam = 1.0
# plots:
filename, ext = os.path.splitext(sys.argv[0])
if os.path.isdir(filename + "/"):
shutil.rmtree(filename + "/")
self.myplot = PlotFenics(filename)
def generatedata(self, noisepercent):
""" compute data and add noisepercent (%) of noise """
sigma = noisepercent * np.linalg.norm(self.f_true.vector().array()) / np.sqrt(self.dimV)
print "sigma_noise = ", sigma
np.random.seed(11) # TODO: tmp
eta = sigma * np.random.randn(self.dimV)
self.dn = dl.Function(self.V)
setfct(self.dn, eta)
self.dn.vector().axpy(1.0, self.f_true.vector())
print "min(true)={}, max(true)={}".format(
np.amin(self.f_true.vector().array()), np.amax(self.f_true.vector().array())
)
print "min(noisy)={}, max(noisy)={}".format(
np.amin(self.dn.vector().array()), np.amax(self.dn.vector().array())
)
def define_regularization(self, parameters=None):
if not parameters == None:
self.parameters.update(parameters)
regularization = self.parameters["regularization"]
if regularization == "tikhonov":
gamma = self.parameters["gamma"]
beta = self.parameters["beta"]
self.Reg = LaplacianPrior({"gamma": gamma, "beta": beta, "Vm": self.V})
self.inexact = False
elif regularization == "TV":
eps = self.parameters["eps"]
k = self.parameters["k"]
mode = self.parameters["mode"]
if mode == "primaldual":
self.Reg = self.Reg = TVPD({"eps": eps, "k": k, "Vm": self.V, "GNhessian": False})
elif mode == "full":
self.Reg = TV({"eps": eps, "k": k, "Vm": self.V, "GNhessian": False})
else:
self.Reg = TV({"eps": eps, "k": k, "Vm": self.V, "GNhessian": True})
self.inexact = False
### COST and DERIVATIVES
def computecost(self, f=None):
""" Compute cost functional at f """
if f == None:
f = self.g
df = f.vector() - self.dn.vector()
# self.misfit = 0.5 * (self.M*df).inner(df)
self.misfit = 0.5 * df.inner(df)
self.reg = self.Reg.cost(f)
self.cost = self.misfit + self.regparam * self.reg
return self.cost
def gradient(self, f=None):
""" Compute M.g (discrete gradient) at a given point f """
if f == None:
f = self.g
df = f.vector() - self.dn.vector()
# self.MGk = self.M*df
self.MGk = df
self.MGr = self.Reg.grad(f)
self.MG = self.MGk + self.MGr * self.regparam
self.solverM.solve(self.Grad.vector(), self.MG)
self.Gradnorm = np.sqrt((self.MG).inner(self.Grad.vector()))
if self.Gradnorm0 == None:
self.Gradnorm0 = self.Gradnorm
def Hessian(self, f=None):
""" Assemble Hessian at f """
if f == None:
f = self.g
regularization = self.parameters["regularization"]
if regularization == "TV":
self.Reg.assemble_hessian(f)
self.Hess = self.I + self.Reg.H * self.regparam
# self.Hess = self.M + self.Reg.H*self.regparam
elif regularization == "tikhonov":
self.Hess = self.M + self.Reg.Minvprior * self.regparam
### SOLVER
def searchdirection(self):
""" Compute search direction """
self.gradient()
self.Hessian()
solver = dl.PETScKrylovSolver("cg", "petsc_amg")
solver.parameters["nonzero_initial_guess"] = False
# Inexact CG:
if self.inexact:
self.cgtol = min(0.5, np.sqrt(self.Gradnorm / self.Gradnorm0))
else:
self.cgtol = 1e-8
solver.parameters["relative_tolerance"] = self.cgtol
solver.set_operator(self.Hess)
self.cgiter = solver.solve(self.dg.vector(), -1.0 * self.MG)
if (self.MG).inner(self.dg.vector()) > 0.0:
print "*** WARNING: NOT a descent direction"
def linesearch(self):
""" Perform inexact backtracking line search """
regularization = self.parameters["regularization"]
# compute new direction for dual variables
if regularization == "TV" and self.Reg.isPD():
self.Reg.compute_dw(self.dg)
# line search for primal variable
self.alpha = self.parameters["alpha0"]
rho = self.parameters["rho"]
c = self.parameters["c"]
self.computecost()
costref = self.cost
cdJdf = ((self.MG).inner(self.dg.vector())) * c
self.LS = False
for ii in xrange(self.parameters["max_backtrack"]):
setfct(self.gtmp, self.g.vector() + self.dg.vector() * self.alpha)
if self.computecost(self.gtmp) < costref + self.alpha * cdJdf:
self.g.vector().axpy(self.alpha, self.dg.vector())
self.LS = True
break
else:
self.alpha *= rho
# update dual variable
if regularization == "TV" and self.Reg.isPD():
self.Reg.update_w(self.alpha)
def solve(self, plot=False):
""" Solve image denoising pb """
regularization = self.parameters["regularization"]
print "\t{:12s} {:12s} {:12s} {:12s} {:12s} {:12s} {:12s}\t{:12s} {:12s}".format(
"a_reg", "cost", "misfit", "reg", "||G||", "a_LS", "medmisfit", "tol_cg", "n_cg"
)
#
if regularization == "tikhonov":
# pb is linear with tikhonov regularization
self.searchdirection()
self.g.vector().axpy(1.0, self.dg.vector())
self.computecost()
self.alpha = 1.0
self.printout()
else:
self.computecost()
cost = self.cost
# initial printout
df = self.f_true.vector() - self.g.vector()
self.medmisfit = np.sqrt(df.inner(df))
# self.medmisfit = np.sqrt((self.M*df).inner(df))
self.relmedmisfit = self.medmisfit / self.targetnorm
print ("{:12.1e} {:12.4e} {:12.4e} {:12.4e} {:12s} {:12s} {:12.2e}" + " ({:.3f})").format(
self.regparam, self.cost, self.misfit, self.reg, "", "", self.medmisfit ** 2, self.relmedmisfit
)
# iterate
for ii in xrange(1000):
self.searchdirection()
self.linesearch()
print ii + 1,
self.printout()
# Check termination conditions:
if not self.LS:
print "Line search failed"
break
if self.Gradnorm < min(1e-12, 1e-10 * self.Gradnorm0):
print "gradient sufficiently reduced -- optimization converged"
break
elif np.abs(cost - self.cost) / cost < 1e-12:
print "cost functional stagnates -- optimization converged"
break
cost = self.cost
### OUTPUT
def printout(self):
""" Print results """
df = self.f_true.vector() - self.g.vector()
self.medmisfit = np.sqrt((self.M * df).inner(df))
self.relmedmisfit = self.medmisfit / self.targetnorm
print ("{:12.1e} {:12.4e} {:12.4e} {:12.4e} {:12.4e} {:12.2e} {:12.2e}" + " ({:.3f}) {:12.2e} {:6d}").format(
self.regparam,
self.cost,
self.misfit,
self.reg,
self.Gradnorm,
self.alpha,
self.medmisfit ** 2,
self.relmedmisfit,
self.cgtol,
self.cgiter,
)
def plot(self, index=0, add=""):
""" Plot target (w/ noise 0, or w/o noise 1) or current iterate (2) """
if index == 0:
self.myplot.set_varname("target" + add)
self.myplot.plot_vtk(self.f_true)
elif index == 1:
self.myplot.set_varname("data" + add)
self.myplot.plot_vtk(self.dn)
elif index == 2:
self.myplot.set_varname("solution" + add)
self.myplot.plot_vtk(self.g)
### TESTS
def test_gradient(self, f=None, n=5):
""" test gradient with FD approx around point f """
if f == None:
f = self.f_true
pm = [1.0, -1.0]
eps = 1e-5
self.gradient(f)
for nn in xrange(1, n + 1):
expr = dl.Expression("sin(n*pi*x[0]/200)*sin(n*pi*x[1]/100)", n=nn)
df = dl.interpolate(expr, self.V)
MGdf = self.MG.inner(df.vector())
cost = []
for sign in pm:
setfct(self.g, f)
self.g.vector().axpy(sign * eps, df.vector())
cost.append(self.computecost(self.g))
MGFD = (cost[0] - cost[1]) / (2 * eps)
print "n={}:\tMGFD={:.5e}, MGdf={:.5e}, error={:.2e}".format(
nn, MGFD, MGdf, np.abs(MGdf - MGFD) / np.abs(MGdf)
)
def test_hessian(self, f=None, n=5):
""" test Hessian with FD approx around point f """
if f == None:
f = self.f_true
pm = [1.0, -1.0]
eps = 1e-5
self.Hessian(f)
for nn in xrange(1, n + 1):
expr = dl.Expression("sin(n*pi*x[0]/200)*sin(n*pi*x[1]/100)", n=nn)
df = dl.interpolate(expr, self.V)
Hdf = (self.Hess * df.vector()).array()
MG = []
for sign in pm:
setfct(self.g, f)
self.g.vector().axpy(sign * eps, df.vector())
self.gradient(self.g)
MG.append(self.MG.array())
HFD = (MG[0] - MG[1]) / (2 * eps)
print "n={}:\tHFD={:.5e}, Hdf={:.5e}, error={:.2e}".format(
nn, np.linalg.norm(HFD), np.linalg.norm(Hdf), np.linalg.norm(Hdf - HFD) / np.linalg.norm(Hdf)
)
