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@@ -5,26 +5,29 @@ import pylab
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import pprint
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from scipy.optimize import nnls
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+import matplotlib.pyplot as plt
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+
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def PhiB(mux, muy, minVal, maxVal, x):
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phib = mux * np.abs( np.sum(np.log( x-minVal)) )
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- #phib += np.abs( np.log(1. - maxVal / np.sum(x)) )
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return phib
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- #phib += np.log std::log(maxVal - x.segment(ib*block, block).sum());
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-
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- #template < typename Scalar >
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- #Scalar PhiB2 (const Scalar& minVal, const Scalar& maxVal, const VectorXr x,
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- # const int& block, const int &nblocks) {
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- # Scalar phib = std::abs((x.array() - minVal).log().sum());
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- # //phib += std::abs((maxVal - x.array()).log().sum()*muy);
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- # for (int ib=0; ib<nblocks; ++ib) {
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- # //HiSegments(ib) = x.segment(ib*block, block).sum();
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- # phib += Scalar(block)*std::log(maxVal - x.segment(ib*block, block).sum());
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- # }
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- # return phib;
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- #}
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-
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-def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False, MAXITER=70, fignum=1000, sigma=1, callback=None):
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+def curvaturefd(x, y, t):
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+ x1 = np.gradient(x,t)
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+ x2 = np.gradient(x1,t)
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+ y1 = np.gradient(y,t)
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+ y2 = np.gradient(y1,t)
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+ return np.abs(x1*y2 - y1*x2) / np.power(x1**2 + y1**2, 3./2)
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+
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+def curvatureg(x, y):
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+ from scipy.ndimage import gaussian_filter1d
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+ #first and second derivative
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+ x1 = gaussian_filter1d(x, sigma=1, order=1)#, mode='constant', cval=x[-1])
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+ x2 = gaussian_filter1d(x1, sigma=1, order=1)#, mode='constant', cval=y[-1])
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+ y1 = gaussian_filter1d(y, sigma=1, order=1)#, mode='constant', cval=x1[-1])
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+ y2 = gaussian_filter1d(y1, sigma=1, order=1)#, mode='constant', cval=y1[-1])
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+ return np.abs(x1*y2 - y1*x2) / np.power(x1**2 + y1**2, 3./2)
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+
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+def logBarrier(A, b, T2Bins, lambdastar, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False, MAXITER=70, fignum=1000, sigma=1, callback=None):
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"""Impliments a log barrier Tikhonov solution to a linear system of equations
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Ax = b s.t. x_min < x < x_max. A log-barrier term is used for the constraint
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"""
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@@ -35,13 +38,11 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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Wd = (np.eye(len(b)) / (sigma)) # Wd = eye( sigma )
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WdTWd = (np.eye(len(b)) / (sigma**2)) # Wd = eye( sigma )
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- #print ("calculating AT WdT.Wd A ") # AT WdTWd A with known data matrix
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- #print (" A.shape" , np.shape(A), np.shape(b))
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ATWdTWdA = np.dot(A.conj().transpose(), np.dot( WdTWd, A )) # TODO, implicit calculation instead?
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N = np.shape(A)[1] # number of model
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M = np.shape(A)[0] # number of data
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- SIGMA = .25 #.25 #.125 # .25 #.01#3e-1
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- EPSILON = 1e-35 # was 35
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+ SIGMA = .25
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+ EPSILON = 1e-35
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# reference model
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if np.size(xr) == 1:
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@@ -54,9 +55,8 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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x = 1e-10 + x_0
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# Construct model constraint base
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- #print ("constructing Phim")
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Phim_base = np.zeros( [N , N] )
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- a1 = 1.0 # smallest
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+ a1 = .05 # smallest too
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# calculate largest term
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D1 = 1./abs(T2Bins[1]-T2Bins[0])
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@@ -64,56 +64,53 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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#a2 = 1. #(1./(2.*D1+D2)) # smooth
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if smooth == "Both":
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- #print ("SMOOTH")
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- # Smooth model
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- print ("Both small and smooth model")
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+ #print ("Both small and smooth model")
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for ip in range(N):
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D1 = 0.
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D2 = 0.
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- DMAX = 2./(T2Bins[1]-T2Bins[0])
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if ip > 0:
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- D1 = 1./abs(T2Bins[ip]-T2Bins[ip-1])
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+ #D1 = np.sqrt(1./abs(T2Bins[ip]-T2Bins[ip-1]))**.5
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+ D1 = (1./abs(T2Bins[ip]-T2Bins[ip-1])) #**2
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if ip < N-1:
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- D2 = 1./abs(T2Bins[ip+1]-T2Bins[ip])
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+ #D2 = np.sqrt(1./abs(T2Bins[ip+1]-T2Bins[ip]))**.5
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+ D2 = (1./abs(T2Bins[ip+1]-T2Bins[ip])) #**2
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if ip > 0:
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- Phim_base[ip,ip-1] = -(D1) # smooth in log space
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+ Phim_base[ip,ip-1] = -(D1)
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if ip == 0:
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- Phim_base[ip,ip ] = (D1+D2) + a1 # Encourage a little low model, no a1
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+ Phim_base[ip,ip ] = 2.*(D1+D2)
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elif ip == N-1:
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- Phim_base[ip,ip ] = (D1+D2) + a1 # Penalize long decays
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+ Phim_base[ip,ip ] = 2.*(D1+D2)
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else:
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- Phim_base[ip,ip ] = (D1+D2) + a1 # Smooth and small
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+ Phim_base[ip,ip ] = 2.*(D1+D2)
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if ip < N-1:
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- Phim_base[ip,ip+1] = -(D2) # smooth in log space
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-
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- #Phim_base /= np.max(Phim_base)
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- #Phim_base += a1*np.eye(N)
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+ Phim_base[ip,ip+1] = -(D2)
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+ Phim_base /= np.max(Phim_base) # normalize
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+ Phim_base += a1*np.eye(N)
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elif smooth == "Smooth":
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- print ("Smooth model")
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+ #print ("Smooth model")
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for ip in range(N):
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if ip > 0:
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Phim_base[ip,ip-1] = -1 # smooth in log space
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if ip == 0:
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- Phim_base[ip,ip ] = 1.0 # Encourage a little low model
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+ Phim_base[ip,ip ] = 2.05 # Encourage a little low model
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elif ip == N-1:
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- Phim_base[ip,ip ] = 8.0 # Penalize long decays
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+ Phim_base[ip,ip ] = 2.5 # Penalize long decays
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else:
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- Phim_base[ip,ip ] = 2.0 # Smooth and small
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+ Phim_base[ip,ip ] = 2.1 # Smooth and small
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if ip < N-1:
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Phim_base[ip,ip+1] = -1 # smooth in log space
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- #print(Phim_base)
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- else:
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- print ("SMALLEST")
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- # Smallest model
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+ elif smooth == "Smallest":
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for ip in range(N):
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Phim_base[ip,ip ] = 1.
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+ else:
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+ print("non valid model constraint:", smooth)
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+ exit()
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Phi_m = alpha*Phim_base
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WmTWm = Phim_base # np.dot(Phim_base, Phim_base.T)
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b_pre = np.dot(A, x)
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-
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phid = np.linalg.norm( np.dot(Wd, (b-b_pre)) )**2
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phim = np.linalg.norm( np.dot(Phim_base, (x-xr)) )**2
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@@ -121,24 +118,28 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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phib = PhiB(mu1, mu2, 0, 1e8, x)
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mu1 = ((phid + alpha*phim) / phib)
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- #print ("iteration", -1, mu1, mu2, phib, phid, phim, len(b))
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+ PHIM = []
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+ PHID = []
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+ MOD = []
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+
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+ ALPHA = []
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+ ALPHA.append(alpha)
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for i in range(MAXITER):
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-
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- b_pre = np.dot(A, x)
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- phid = np.linalg.norm(np.dot(Wd, (b-b_pre)))**2
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- # technically phim should not be on WmTWm matrix. So no need to square result.
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- phim = np.linalg.norm(np.dot(Phim_base, (x-xr)))#**2
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- phib = PhiB(mu1, mu2, 0, 1e8, x)
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+
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Phi_m = alpha*Phim_base
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+ # reset mu1 at each iteration
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+ # Calvetti -> No ; Li -> Yes
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+ # without this, non monotonic convergence occurs...which is OK if you really trust your noise
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mu1 = ((phid + alpha*phim) / phib)
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WmTWm = Phim_base # np.dot(Phim_base, Phim_base.T)
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- #ztilde = x
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- #print("ztilde", ztilde)
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phid_old = phid
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inner = 0
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- First = True
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+
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+ First = True # guarantee entry
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+
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+ xp = np.copy(x) # prior step x
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while ( (phib / (phid+alpha*phim)) > EPSILON or First==True ):
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@@ -154,14 +155,14 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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AA = ATWdTWdA + mu1*X2 + mu2*Y2 + Phi_m
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M = np.eye( N ) * (1./np.diag(ATWdTWdA + mu1*X2 + mu2*Y2 + Phi_m))
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- # Solve system (newton step)
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- b2 = np.dot(A.transpose(), np.dot(WdTWd, b-b_pre) ) + 2.*mu1*np.diag(X1) + 2.*mu2*np.diag(Y1) - alpha*np.dot(WmTWm,(x-xr))
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- ztilde = iter.cg(AA, b2, M=M) # tol=1e-3*phid, maxiter=200, callback=callback) #, tol=1e-2, maxiter) #, x0=x) #, tol=ttol) #, M=M, x0=x)
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- h = (ztilde[0])
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+ # Solve system (newton step) (Li)
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+ b2 = np.dot(A.conj().transpose(), np.dot(WdTWd, b-b_pre) ) + 2.*mu1*np.diag(X1) + 2.*mu2*np.diag(Y1) - alpha*np.dot(WmTWm,(x-xr))
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+ ztilde = iter.cg(AA, b2, M = M)
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+ h = (ztilde[0].real)
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- # Solve system (direct solution)
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+ # Solve system (direct solution) (Calvetti)
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#b2 = np.dot(A.conj().transpose(), np.dot(WdTWd, b)) + 2.*mu1*np.diag(X1) + 2.*mu2*np.diag(Y1) - alpha*np.dot(WmTWm,(x-xr))
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- #ztilde = iter.cg(AA, b2, x0=x) #, tol=1e-2) #, x0=x) #, tol=ttol) #, M=M, x0=x)
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+ #ztilde = iter.cg(AA, b2, M=M, x0=x)
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#h = (ztilde[0].real - x)
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# step size
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@@ -169,8 +170,7 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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##########################################################
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# Update and fix any over/under stepping
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- x = x+d*h
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- #x = np.max( ( (minVal+1e-120)*np.ones(N), x+d*h), 0)
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+ x += d*h
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# Determine mu steps to take
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s1 = mu1 * (np.dot(X2, ztilde[0].real) - 2.*np.diag(X1))
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@@ -181,32 +181,87 @@ def logBarrier(A, b, T2Bins, x_0=0, xr=0, alpha=10, mu1=10, mu2=10, smooth=False
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mu2 = SIGMA/N * np.abs(np.dot(s2, x))
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b_pre = np.dot(A, x)
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- phid = np.linalg.norm(np.dot(Wd, (b-b_pre)))**2
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- phim = np.linalg.norm(np.dot(Phim_base, (x-xr)))#**2
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+ phid = np.linalg.norm( np.dot(Wd, (b-b_pre)))**2
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+ phim = np.linalg.norm( np.dot(Phim_base, (x-xr)) )**2
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phib = PhiB(mu1, mu2, minVal, maxVal, x)
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inner += 1
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+ PHIM.append(phim)
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+ PHID.append(phid)
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+ MOD.append(np.copy(x))
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+
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# determine alpha
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scale = (len(b)/phid)
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- alpha *= scale #**(1/6)
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-
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- score = np.sqrt(phid/(len(b)+1.)) # unbiased
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-
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- # check stopping criteria
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- if score < 1:
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- print ("*overshot* optimal solution found") #, alpha, score)
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- break
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- if score < 1.1: # or np.linalg.norm(x_old-x) < 1e-5 or phid > phid_old:
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- #print ("overshoot")
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- #alpha *= 10
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|
202
|
|
- print ("optimal solution found") #, alpha, score)
|
|
203
|
|
- break
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|
204
|
|
- if i > 10 and (np.sqrt(phid_old/(len(b)+1.)) - score) < 1e-2: # 1e-2
|
|
205
|
|
- print ("slow convergence") #, alpha, score, i, scale, score-np.sqrt(phid_old/len(b)))
|
|
206
|
|
- break
|
|
207
|
|
-
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|
208
|
|
- print ( "alpha","phid", "iter", "search", "prior" )
|
|
209
|
|
- print ( alpha, score, i, scale, np.sqrt(phid_old/(len(b)+1)))
|
|
|
195
|
+ #alpha *= np.sqrt(scale)
|
|
|
196
|
+ alpha *= min(scale, .95) # was .85...
|
|
|
197
|
+ ALPHA.append(alpha)
|
|
|
198
|
+ #alpha = ALPHA[i+1]
|
|
|
199
|
+
|
|
|
200
|
+
|
|
|
201
|
+# if np.sqrt(phid/len(b)) < 0.97:
|
|
|
202
|
+# ibreak = -1
|
|
|
203
|
+# print ("------------overshot--------------------", alpha, np.sqrt(phid/len(b)), ibreak)
|
|
|
204
|
+# alpha *= 2. #0
|
|
|
205
|
+# x -= d*h
|
|
|
206
|
+# b_pre = np.dot(A, x)
|
|
|
207
|
+# phid = np.linalg.norm( np.dot(Wd, (b-b_pre)))**2
|
|
|
208
|
+# phim = np.linalg.norm( np.dot(Phim_base, (x-xr)) )#**2
|
|
|
209
|
+# mu1 = ((phid + alpha*phim) / phib)
|
|
|
210
|
+ if lambdastar == "discrepency":
|
|
|
211
|
+ if np.sqrt(phid/len(b)) < 1.00 or alpha < 1e-5:
|
|
|
212
|
+ ibreak = 1
|
|
|
213
|
+ print ("optimal solution found", alpha, np.sqrt(phid/len(b)), ibreak)
|
|
|
214
|
+ break
|
|
|
215
|
+ # slow convergence, bail and use L-curve
|
|
|
216
|
+ # TI- only use L-curve. Otherwise results for perlin noise are too spurious for paper.
|
|
|
217
|
+ if lambdastar == "lcurve":
|
|
|
218
|
+ if i>20 and ((np.sqrt(phid_old/len(b))-np.sqrt(phid/len(b))) < 1e-4):
|
|
|
219
|
+ #if np.sqrt(phid/len(b)) < 3.0 and ((np.sqrt(phid_old/len(b))-np.sqrt(phid/len(b))) < 1e-3):
|
|
|
220
|
+ ibreak = 1
|
|
|
221
|
+ MOD = np.array(MOD)
|
|
|
222
|
+ print ("###########################") #slow convergence", alpha, "phid_old", np.sqrt(phid_old/len(b)), "phid", np.sqrt(phid/len(b)), ibreak)
|
|
|
223
|
+ print ("Using L-curve criteria")
|
|
|
224
|
+ kappa = curvaturefd(np.log(np.array(PHIM)), np.log(np.array(PHID)), ALPHA[0:-1])
|
|
|
225
|
+ #kappa2 = curvatureg(np.log(np.array(PHIM)), np.log(np.array(PHID)))
|
|
|
226
|
+ #kappa = curvature( np.array(PHIM), np.array(PHID))
|
|
|
227
|
+ x = MOD[ np.argmax(kappa) ]
|
|
|
228
|
+ b_pre = np.dot(A, x)
|
|
|
229
|
+ phid = np.linalg.norm( np.dot(Wd, (b-b_pre)))**2
|
|
|
230
|
+ phim = np.linalg.norm( np.dot(Phim_base, (x-xr)) )**2
|
|
|
231
|
+ mu1 = ((phid + alpha*phim) / phib)
|
|
|
232
|
+ print ("L-curve selected", alpha, "phid_old", np.sqrt(phid_old/len(b)), "phid", np.sqrt(phid/len(b)), ibreak)
|
|
|
233
|
+ print ("###########################")
|
|
|
234
|
+ if np.sqrt(phid/len(b)) <= 1:
|
|
|
235
|
+ ibreak=0
|
|
|
236
|
+
|
|
|
237
|
+ #plt.figure()
|
|
|
238
|
+ #plt.plot( (np.array(PHIM)), np.log(np.array(PHID)/len(b)), '.-')
|
|
|
239
|
+ #plt.plot( ((np.array(PHIM))[np.argmax(kappa)]) , np.log( (np.array(PHID)/len(b))[np.argmax(kappa)] ), '.', markersize=12)
|
|
|
240
|
+ #plt.axhline()
|
|
|
241
|
+ #plt.plot( np.array(PHIM), np.array(PHID), '.-')
|
|
|
242
|
+ #plt.plot( np.array(PHIM)[np.argmax(kappa)], np.array(PHID)[np.argmax(kappa)], '.', markersize=12)
|
|
|
243
|
+ #plt.savefig('lcurve.pdf')
|
|
|
244
|
+ break
|
|
|
245
|
+
|
|
|
246
|
+ PHIM = np.array(PHIM)
|
|
|
247
|
+ PHID = np.array(PHID)
|
|
|
248
|
+
|
|
|
249
|
+ if (i == MAXITER-1 ):
|
|
|
250
|
+ ibreak = 2
|
|
|
251
|
+ print("Reached max iterations!!", alpha, np.sqrt(phid/len(b)), ibreak)
|
|
|
252
|
+ kappa = curvaturefd(np.log(np.array(PHIM)), np.log(np.array(PHID)), ALPHA[0:-1])
|
|
|
253
|
+ x = MOD[ np.argmax(kappa) ]
|
|
|
254
|
+ b_pre = np.dot(A, x)
|
|
|
255
|
+ phid = np.linalg.norm( np.dot(Wd, (b-b_pre)))**2
|
|
|
256
|
+ phim = np.linalg.norm( np.dot(Phim_base, (x-xr)) )**2
|
|
|
257
|
+ mu1 = ((phid + alpha*phim) / phib)
|
|
|
258
|
+
|
|
|
259
|
+ if lambdastar == "lcurve":
|
|
|
260
|
+ return x, ibreak, np.sqrt(phid/len(b)), PHIM, PHID/len(b), np.argmax(kappa)
|
|
|
261
|
+ else:
|
|
|
262
|
+ return x, ibreak, np.sqrt(phid/len(b))
|
|
|
263
|
+
|
|
210
|
264
|
|
|
211
|
|
- return x
|
|
212
|
265
|
|
|
|
266
|
+if __name__ == "__main__":
|
|
|
267
|
+ print("Test")
|
Trevor Irons
6 年前
