def get(fixed=None, visualize=False, order=2, moving=None):
"""
get point SSD similarity
"""
# data
qf = fixed[0]
N = qf.shape[0]
DIM = qf.shape[1]
## visualization
reggrid = None
if visualize:
plt.figure(0)
plt.clf()
plt.plot(qf[:, 0], qf[:, 1], "bo")
if moving: # moving points needed in order to find grid size
qm = moving[0]
reggrid = pg.getGrid(
np.vstack((qf, qm))[:, 0].min() - 1,
np.vstack((qf, qm))[:, 0].max() + 1,
np.vstack((qf, qm))[:, 1].min() - 1,
np.vstack((qf, qm))[:, 1].max() + 1,
xpts=10,
ypts=10,
)
pg.plotGrid(*reggrid)
f = partial(psim, fixed=fixed, N=N, DIM=DIM, visualize=visualize, order=order, grid=reggrid)
sim = {"f": f, "N": N, "DIM": DIM}
return sim
def psim(state, N=None, DIM=None, fixed=None, visualize=False, order=2, state0=None, grid=None):
qm, qm_1, qm_2, pm, mum_1, mum_2 = tj.state_to_weinstein_darboux(state, N, DIM)
qf = fixed[0]
if order >= 1:
qf_1 = fixed[1]
if order >= 2:
qf_2 = fixed[2]
w = [1, 0.5, 0.2]
# weighting between different order terms
# value
v0 = qm - qf
m0 = w[0] * np.einsum("ia,ia", v0, v0) # 1./N ??
if order >= 1:
v1 = qm_1 - qf_1
m1 = w[1] * np.einsum("iab,iab", v1, v1) # 1./N ??
if order >= 2:
v2 = qm_2 - qf_2
m2 = w[2] * np.einsum("iabg,iabg", v2, v2) # 1./N ??
# gradient
dq0 = w[0] * 2.0 * v0 # 1./N ??
if order >= 1:
dq1 = w[1] * 2.0 * v1 # 1./N ??
if order >= 2:
dq2 = w[2] * 2.0 * v2 # 1./N ??
# print "point sim: m0 " + str(m0) + ", m1 " + str(m1) + ", m2 " + str(m2)
## visualization
if visualize:
plt.figure(1)
plt.clf()
plt.plot(qf[:, 0], qf[:, 1], "bo")
plt.plot(qm[:, 0], qm[:, 1], "rx")
# grid
if state0 != None and grid != None:
(reggrid, Nx, Ny) = grid
(_, _, mgridts) = tj.integrate(state0, pts=reggrid)
mgridT = mgridts[-1:].reshape(-1, DIM)
pg.plotGrid(mgridT, Nx, Ny)
# generate vertices of a circle
N_vert = 20
circle_verts = np.zeros([2, N_vert + 1])
theta = np.linspace(0, 2 * np.pi, N_vert)
circle_verts[0, 0:N_vert] = 0.2 * np.cos(theta)
circle_verts[1, 0:N_vert] = 0.2 * np.sin(theta)
verts = np.zeros([2, N_vert + 1])
units = np.ones(N_vert + 1)
for i in range(0, len(qm)):
plt.arrow(
qm[i, 0], qm[i, 1], 0.2 * pm[i, 0], 0.2 * pm[i, 1], head_width=0.2, head_length=0.2, fc="b", ec="b"
)
if qm_1 != None:
verts = np.dot(qm_1[i, :, :], circle_verts) + np.outer(qm[i, :], units)
plt.plot(verts[0], verts[1], "r-")
border = 0.4
plt.xlim(min(np.vstack((qf, qm))[:, 0]) - border, max(np.vstack((qf, qm))[:, 0]) + border)
plt.ylim(min(np.vstack((qf, qm))[:, 1]) - border, max(np.vstack((qf, qm))[:, 1]) + border)
plt.axis("equal")
plt.draw()
if order == 0:
return (m0, (dq0,))
elif order == 1:
return (m0 + m1, (dq0, dq1))
else:
return (m0 + m1 + m2, (dq0, dq1, dq2))
def get(pointsPerAxis, immname, imfname, immT=None, visualize=False, border=0, normalize=False, order=2, smoothscaleFactor=0.5, SIGMAF=2., h=None, splineS=None, visRes=25j):
"""
get image similarity measure
"""
logging.info("Image sim parameters: visualize %s, border %d, normalize %s, order %d, smoothscaleFactor %g, sigmaF %g, splineS %s, visRes %s",visualize,border,normalize,order,smoothscaleFactor,SIGMAF,splineS,visRes)
# load
imm = np.double(np.load(immname))
imf = np.double(np.load(imfname))
imshape = imf.shape
Nx = np.shape(imm)[0]-2*border
Ny = np.shape(imm)[1]-2*border
assert Nx == Ny
assert Nx > 0
N = Nx*Ny
Ns = pointsPerAxis**2 # sample points
relsmoothscale = smoothscaleFactor/sqrt(Ns)
if not h:
h = sqrt(N)/sqrt(Ns) # length scale h, dist h between particles
logging.info("h: " + str(h))
smoothscale = relsmoothscale*sqrt(N)
logging.info("smoothscale: " + str(smoothscale))
SIGMA = SIGMAF*sqrt(N)/sqrt(Ns)
logging.info("SIGMA: " + str(SIGMA))
# smooth
imms = smooth(imm, smoothscale)
imfs = smooth(imf, smoothscale)
# normalize
if normalize:
imms = imms-np.min(imms)
if np.max(imms) > 1e-5:
imms = imms/np.max(imms)
imfs = imfs-np.min(imfs)
if np.max(imfs) > 1e-5:
imfs = imfs/np.max(imfs)
# grid
indent = Nx/(2*sqrt(Ns))
sgrid = np.mgrid[border+indent:border+Nx-indent:complex(0,sqrt(Ns)),border+indent:border+Ny-indent:complex(0,sqrt(Ns))]
sgridxy = (sgrid[0,:,0], sgrid[0,0,:])
# plot
if visualize:
# plotting setup
mpl.rcParams['image.interpolation'] = 'nearest'
mpl.rcParams['image.origin'] = 'lower'
# splot befine spline interpolation
plt.figure(0)
plt.imshow(imfs.T)
plt.figure(1)
plt.imshow(imms.T)
#plt.show(block=True)
# spline
imfs = splineIM(imfs, splineS=splineS)
imms = splineIM(imms, T=immT, splineS=splineS)
# original images for visualization
imf = splineIM(imf, kind='cubic', splineS=splineS)
imm = splineIM(imm, T=immT, kind='cubic', splineS=splineS)
# downsample
samplesgrid = partial(sample,sgrid, hscaling=h*sqrt(Ns)/sqrt(N))
simfs = samplesgrid(imfs)
#sDimfs = [apply_2d_slices(samplesgrid, Dimfs[i]) for i in range(np.shape(Dimfs)[0])]
sDimfs = [apply_2d_slices(partial(samplesgrid, imfs), derivs[i]) for i in range(len(derivs))]
f = partial(imsim, N=Ns, imshape=imshape, DIM=2, h=h, imms=imms, simfs=simfs, sDimfs=sDimfs, hscaling=h*sqrt(Ns)/sqrt(N), SIGMA=SIGMA, order=order)
## visualization
reggrid = None
if visualize:
plt.figure(1)
reggrid = pg.getGrid(border,border+Nx,border,border+Ny,xpts=40,ypts=40)
pg.plotGrid(*reggrid)
# attach grids
f = partial(f, sgrid=sgrid, grid=reggrid)
# image grid for warping
pointsPerAxis = complex(0,visRes)
imborder = 0 # -15
imgrid = d2zip(np.mgrid[border+imborder:border+Nx-imborder:pointsPerAxis,border+imborder:border+Ny-imborder:pointsPerAxis])
f = partial(f, imgrid=imgrid, imf=imf, imm=imm, imfs=imfs)
plt.figure(0)
plt.clf()
x = np.arange(imshape[0])
y = np.arange(imshape[1])
scimfs = samplecross((x,y),imfs).reshape(imshape)
cmin = np.min([np.min(simfs),np.min(scimfs)])
cmax = np.max([np.max(simfs),np.max(scimfs)])
plt.imshow(scimfs.T,vmin=cmin,vmax=cmax)
plt.gray()
plt.colorbar()
q = d2zip(sgrid)
plt.plot(q[:,0],q[:,1],'bo')
plt.xlim(0,imshape[0])
plt.ylim(0,imshape[1])
# Jacobians
DIM = 2
N = q.shape[0]
q_1 = np.outer(np.ones(N),np.eye(DIM)).reshape([N,DIM,DIM])
plotJacobians(q,q_1)
plt.figure(11)
plt.clf()
x = np.arange(imshape[0])
y = np.arange(imshape[1])
scimms = samplecross((x,y),imms).reshape(imshape)
plt.imshow(scimms.T)
plt.gray()
plt.colorbar()
plt.plot(d2zip(sgrid)[:,0],d2zip(sgrid)[:,1],'bo')
plt.xlim(0,imshape[0])
plt.ylim(0,imshape[1])
plt.figure(12)
plt.clf()
scDimms = samplecross((x,y),imms,(1,0),hscaling=h*sqrt(Ns)/sqrt(N)).reshape(imshape)
plt.imshow(scDimms.T)
plt.gray()
plt.colorbar()
plt.figure(13)
plt.clf()
scDimms = samplecross((x,y),imms,(2,0),hscaling=h*sqrt(Ns)/sqrt(N)).reshape(imshape)
plt.imshow(scDimms.T)
plt.gray()
plt.colorbar()
sim = {'f': f, 'N': Ns, 'SIGMA': SIGMA, 'DIM': 2, 'order': order, 'initial': (d2zip(sgrid),)}
return sim
def imsim( state, N=None, imshape=None, DIM=None, h=None, imms=None, Dimms=None, imf=None, imm=None, simfs=None, sDimfs=None, sgrid=None, visualize=False, state0=None, grid=None, order=None, imgrid=None, hscaling=None, SIGMA=None, imfs=None):
q,q_1,q_2,p,mu_1,mu_2 = tj.state_to_weinstein_darboux( state,N,DIM )
sampleq = partial(sample,d2unzip(q,N), hscaling=hscaling)
simms = sampleq(imms)
#sDimms = [apply_2d_slices(sampleq, Dimms[i]) for i in range(np.shape(Dimms)[0])]
sDimms = [apply_2d_slices(partial(sampleq, imms), derivs[i]) for i in range(len(derivs))]
d = DIM
delta = np.identity(DIM)
one = np.ones([DIM])
one_minus_delta = np.ones([DIM,DIM])-np.eye(DIM)
# value
v0 = simfs-simms
m0 = (h**d)*np.einsum('i,i',v0,v0)
if order >= 1:
v1 = sDimfs[0]-np.einsum('bi,iba->ai',sDimms[0],q_1)
m1 = (h**(d+2))/12*np.einsum('ai,ai',v1,v1)
if order >= 2:
G = sDimfs[1] \
-np.einsum('dci,idb,ica->abi',sDimms[1],q_1,q_1) \
-np.einsum('ci,icab->abi',sDimms[0],q_2)
m2 = (h**(d+2))/12*np.einsum('i,aai->',v0,G) \
+ (h**(d+4))/(5*2**6)*np.einsum('aai,aai->',G,G) \
+ (h**(d+4))/(9*2**6)*np.einsum('ab,aai,bbi->',one_minus_delta,G,G) \
+ (h**(d+4))/(9*2**6)*np.einsum('ab,abi,abi->',one_minus_delta,G,G)
# debug output
if order >= 0:
logging.info("m0: " + str(m0))
if order >= 1:
logging.info("m1: " + str(m1))
#logging.info("sDimfs[0]: " + str(sDimfs[0]))
#logging.info("moving: " + str(np.einsum('bi,iba->ai',sDimms[0],q_1)))
if order >= 2:
logging.info("m2: " + str(m2))
#logging.info("G: " + str(G))
#logging.info("sDimfs[1]: " + str(sDimfs[1]))
#logging.info("moving: " + str(np.einsum('dci,idb,ica->abi',sDimms[1],q_1,q_1)+np.einsum('ci,icab->abi',sDimms[0],q_2)))
# gradient
# dq0
g00 = -2*(h**d)*np.einsum('i,ai->ia',v0,sDimms[0])
dq0 = g00
if order >= 1:
g01 = -(h**(d+2))/6*np.einsum('bai,ibe,ec,ci->ia',sDimms[1],q_1,delta,v1)
dq0 = dq0+g01
if order >= 2:
g02 = -(h**(d+2))/12*np.einsum('ai,ddi->ia',sDimms[0],G)
G1 = -np.einsum('bcai,ibe,icd->deai',sDimms[2],q_1,q_1) \
-np.einsum('cai,icde->deai',sDimms[1],q_2)
g03 = (h**(d+2))/12*np.einsum('i,ddai->ia',v0,G1)
g04 = (h**(d+4))/(5*2**5)*np.einsum('ddi,ddai->ia',G,G1) \
+(h**(d+4))/(9*2**5)*np.einsum('de,ddi,eeai->ia',one_minus_delta,G,G1) \
+(h**(d+4))/(9*2**5)*np.einsum('de,dei,deai->ia',one_minus_delta,G,G1)
dq0 = dq0+g02+g03+g04
# rescale
dq0 = hscaling*dq0
# dq1
if order >= 1:
g10 = -(h**(d+2))/6*np.einsum('ai,bi->iab',sDimms[0],v1)
dq1 = g10
if order >= 2:
G2 = -np.einsum('aci,ice,db->deabi',sDimms[1],q_1,delta) \
-np.einsum('aci,icd,eb->deabi',sDimms[1],q_1,delta)
g11 = (h**(d+2))/12*np.einsum('i,ddabi->iab',v0,G2)
g12 = (h**(d+4))/(5*2**5)*np.einsum('ddi,ddabi->iab',G,G2) \
+(h**(d+4))/(9*2**5)*np.einsum('de,ddi,eeabi->iab',one_minus_delta,G,G2) \
+(h**(d+4))/(9*2**5)*np.einsum('de,dei,deabi->iab',one_minus_delta,G,G2)
dq1 = dq1+g11+g12
# dq2
if order >= 2:
G3 = -np.einsum('bd,ce,ai->deabci',delta,delta,sDimms[0])
g21 = (h**(d+2))/12*np.einsum('i,bc,bcabci->iabc',v0,delta,G3)
g22 = (h**(d+4))/(5*2**5)*np.einsum('ddi,ddabci->iabc',G,G3) \
+(h**(d+4))/(9*2**5)*np.einsum('de,ddi,eeabci->iabc',one_minus_delta,G,G3) \
+(h**(d+4))/(9*2**5)*np.einsum('de,dei,deabci->iabc',one_minus_delta,G,G3)
dq2 = g21+g22
# visualization
if visualize:
logging.info("iteration visualization output")
x = np.arange(imshape[0])
y = np.arange(imshape[1])
plt.figure(1)
plt.clf()
scimms = samplecross((x,y),imms).reshape(imshape)
cmin = np.min([np.min(simms),np.min(simfs),np.min(scimms)])
cmax = np.max([np.max(simms),np.max(simfs),np.max(scimms)])
plt.imshow(scimms.T,vmin=cmin,vmax=cmax)
plt.plot(d2zip(sgrid)[:,0],d2zip(sgrid)[:,1],'bo')
plt.plot(q[:,0],q[:,1],'rx')
plt.gray()
plt.colorbar()
plotJacobians(q,q_1)
#plt.quiver(q[:,0],q[:,1],dq0[:,0],dq0[:,1],color='y')
plt.xlim(0,imshape[0])
plt.ylim(0,imshape[1])
#plt.quiver(q[:,0],q[:,1],g00[:,0],g00[:,1])
plt.figure(2)
plt.clf()
plt.imshow(rse(simfs,N).T,vmin=cmin,vmax=cmax)
plt.colorbar()
plt.figure(3)
plt.clf()
plt.imshow(rse(simms,N).T,vmin=cmin,vmax=cmax)
plt.colorbar()
plt.figure(4)
plt.clf()
plt.imshow(rse(v0,N).T)
plt.colorbar()
plt.figure(5)
plt.clf()
plt.imshow(rse(sDimms[0][0,:],N).T)
plt.colorbar()
# grid plot
if sgrid != None:
qf = d2zip(sgrid)
plt.figure(6)
plt.clf()
plt.plot(qf[:,0],qf[:,1],'bo')
plt.plot(q[:,0],q[:,1],'rx')
# grid
if state0 != None and grid != None:
(reggrid,Nx,Ny) = grid
(_,_,mgridts) = tj.integrate(state0,pts=reggrid)
mgridT = mgridts[-1:].reshape(-1,DIM)
pg.plotGrid(mgridT,Nx,Ny)
## generate vertices of a circle
#N_vert = 20
#circle_verts = np.zeros( [ 2 , N_vert + 1 ] )
#theta = np.linspace(0,2*np.pi, N_vert )
#circle_verts[0,0:N_vert] = SIGMA*np.cos(theta)
#circle_verts[1,0:N_vert] = SIGMA*np.sin(theta)
#verts = np.zeros([2, N_vert + 1])
#units = np.ones( N_vert + 1)
#for i in range(0,len(q)):
# plt.arrow(q[i,0], q[i,1], 0.2*p[i,0], 0.2*p[i,1],\
# head_width=0.2, head_length=0.2,\
# fc='b', ec='b')
# if (q_1 != None):
# verts = np.dot(q_1[i,:,:], circle_verts ) \
# + np.outer(q[i,:],units)
# plt.plot(verts[0],verts[1],'r-')
border = 0.4
plt.xlim(min(np.vstack((qf,q))[:,0])-border,max(np.vstack((qf,q))[:,0])+border)
plt.ylim(min(np.vstack((qf,q))[:,1])-border,max(np.vstack((qf,q))[:,1])+border)
plt.axis('equal')
# warped images
if state0 != None and imgrid != None and imf != None and imm != None:
# fixed image, interpolated
plt.figure(20)
plt.clf()
simf = sample(d2unzip(imgrid),imf,hscaling=hscaling);
plt.imshow(simf.reshape(sqrt(simf.shape[0]),sqrt(simf.shape[0])).T)
plt.colorbar()
# fixed image, interpolated
plt.figure(21)
plt.clf()
simf = sample(d2unzip(imgrid),imfs,hscaling=hscaling);
plt.imshow(simf.reshape(sqrt(simf.shape[0]),sqrt(simf.shape[0])).T)
plt.colorbar()
# moving image, interpolated without transformation
plt.figure(22)
plt.clf()
simf = sample(d2unzip(imgrid),imm,hscaling=hscaling);
plt.imshow(simf.reshape(sqrt(simf.shape[0]),sqrt(simf.shape[0])).T)
plt.colorbar()
# moving image, interpolated without transformation
plt.figure(23)
plt.clf()
simf = sample(d2unzip(imgrid),imms,hscaling=hscaling);
plt.imshow(simf.reshape(sqrt(simf.shape[0]),sqrt(simf.shape[0])).T)
plt.colorbar()
# moving image, interpolated
plt.figure(24)
plt.clf()
(_,_,mimgridts) = tj.integrate(state0,pts=imgrid)
mimgridT = mimgridts[-1:].reshape(-1,DIM)
simm = sample(d2unzip(mimgridT),imm,hscaling=hscaling);
plt.imshow(simm.reshape(sqrt(simm.shape[0]),sqrt(simm.shape[0])).T)
plt.colorbar()
# moving image, interpolated
plt.figure(25)
plt.clf()
simm = sample(d2unzip(mimgridT),imms,hscaling=hscaling);
plt.imshow(simm.reshape(sqrt(simm.shape[0]),sqrt(simm.shape[0])).T)
plt.colorbar()
plt.draw()
#plt.show(block=False)
# save figures
for i in plt.get_fignums():
plt.figure(i)
try:
os.mkdir('output/%s' % os.getpid() )
except:
None
plt.savefig('output/%s/figure%d.eps' % (os.getpid(),i) )
if order == 0:
return (m0, (dq0, ))
elif order == 1:
return (m0+m1, (dq0,dq1))
else:
return (m0+m1+m2, (dq0,dq1,dq2))
def plotDeformedGrid(grid,Nx,Ny,state):
(reggrid,Nx,Ny) = grid
(_,_,mgridts) = tj.integrate(state0,pts=reggrid)
mgridT = mgridts[-1:].reshape(-1,DIM)
pg.plotGrid(mgridT,Nx,Ny)
# plot
save = True
xlim = (-2.5,2.5)
ylim = (-2.5,2.5)
reggrid = pg.getGrid(xlim[0],xlim[1],ylim[0],ylim[1],xpts=40,ypts=40)
(ggrid,gNx,gNy) = reggrid
(_,_,mgridts) = tj.integrate(state0,pts=ggrid)
mgridT = mgridts[-1:].reshape(-1,DIM)
#plt.figure(1)
#pg.plotGrid(*reggrid)
#pg.axis('equal')
plt.figure(2)
pg.plotGrid(mgridT,gNx,gNy,coloring=True)
plt.plot(y_span[:,0:DIM*N:2],y_span[:,1:DIM*N:2],'r-',linewidth=2)
plt.plot(q[:,0],q[:,1],'bo',markersize=10,markeredgewidth=3)
plt.plot(y_span[-1,0:DIM*N:2],y_span[-1,1:DIM*N:2],'rx',markersize=10,markeredgewidth=3)
plt.axis('equal')
plt.xlim(xlim)
plt.ylim(ylim)
plt.show(block=not save)
# save result
np.save('output/state_data',y_span)
np.save('output/time_data',t_span)
np.save('output/setup',[N,DIM,SIGMA])
# save figures
def psim( state, N=None, DIM=None, rank=None, fixed=None, moving=None, visualize=False, state0=None, grid=None, extra_points=None):
x,Xa,xi,xia = mpp.state_to_weinstein_darboux( state )
qm = x.reshape([N,DIM])
qf = fixed[0]
# value
v0 = qm-qf
m0 = np.einsum('ia,ia',v0,v0) # 1./N ??
# gradient
dq0 = 2.*v0 # 1./N ??
#print "point sim: m0 " + str(m0) + ", m1 " + str(m1) + ", m2 " + str(m2)
## visualization
if visualize:
plt.figure(1)
plt.clf()
if state0 != None:
# grid
if grid != None:
(reggrid,Nx,Ny) = grid
(t_span,y_span,mgridts) = mpp.integrate(state0,pts=reggrid)
mgridT = mgridts[-1:].reshape(-1,DIM)
pg.plotGrid(mgridT,Nx,Ny,coloring=True)
else:
(t_span,y_span) = mpp.integrate(state0)
# plot curves and frames
Q = y_span[:,0:N*DIM].reshape([-1,N,DIM])
#for i in range(N): # comment out for mean
# plt.plot(Q[:,i,0],Q[:,i,1],'k-')
for i in range(len(t_span)):
x_i,Xa_i,xi_i,xia_i = mpp.state_to_weinstein_darboux( y_span[i,:] )
x_i = x_i.reshape([N,DIM])
Xa_i = Xa_i.reshape([N*DIM,mpp.rank])
# plot frame
colors = plt.get_cmap()(np.linspace(0,1,mpp.rank))
#if i % 4 == 0 or i == len(t_span)-1: # plot every 4th frame
if i == len(t_span)-1: # plot last frame
for j in range(mpp.rank):
Fr = Xa_i[:,j].reshape(N,DIM)
for k in range(N):
plt.quiver(x_i[k,0],x_i[k,1],Fr[k,0],Fr[k,1],color=colors[j],angles='xy', scale_units='xy', scale=3, width=.003) # scale=3 for paper plots
if state0 != None and extra_points != None:
(t_span,y_span,eps) = mpp.integrate(state0,pts=extra_points)
eps = eps[-1:].reshape(-1,DIM)
# for mean:
#plt.plot(extra_points[:,0],extra_points[:,1],'--',color='gray')
plt.plot(eps[:,0],eps[:,1],'gray')
## for ccs:
#plt.plot(extra_points[:,0],extra_points[:,1],color='gray')
#plt.plot(eps[:,0],eps[:,1],'--',color='gray')
plt.plot(qf[:,0],qf[:,1],'bo')
plt.plot(qm[:,0],qm[:,1],'ro')
if moving != None:
qq = moving[0]
plt.plot(qq[:,0],qq[:,1],'go')
border = 0.6 # .6 for paper plots
plt.axis('equal')
plt.xlim(min(np.vstack((qf,qm))[:,0])-border,max(np.vstack((qf,qm))[:,0])+border)
plt.ylim(min(np.vstack((qf,qm))[:,1])-border,max(np.vstack((qf,qm))[:,1])+border)
plt.axis('off')
plt.draw()
#plt.show(block=False)
# save figures
for i in plt.get_fignums():
plt.figure(i)
try:
os.mkdir('output/%s' % os.getpid() )
except:
None
plt.savefig('output/%s/figure%d.eps' % (os.getpid(),i) )
return (m0, (dq0, ))
