def zero_background(image,sigma=3):
""" Using sigma clipping, estimate, then remove the background.
"""
im_mask = sf.sigma_clip(image,sigma=sigma,max_iters=np.size(image))
im_mask = np.reshape(im_mask,image.shape)
bg = np.mean(image[im_mask])
bg_std = np.std(image[im_mask])
bg_mask = image < (bg+2*sigma/3*bg_std)
image[bg_mask] = 0
return image
def remove_bg(image,mask=None,sigma=3):
""" Another attempt to remove a constant background using sigma clipping
If needed, expand to 2D linear, maybe quadratic...
"""
if mask is not None:
image[mask] = np.nan
im_mask = sf.sigma_clip(image,sigma=sigma,max_iters=np.size(image))
im_mask = np.reshape(im_mask,image.shape)
bg = np.mean(image[im_mask])
bg_std = np.std(image[im_mask])
# print bg, bg_std
bg_mask = image < (bg+2*sigma/3*bg_std)
# plt.imshow(bg_mask,interpolation='none')
# plt.show()
image[bg_mask] = 0
return image
def fit_spline_psf(raw_img,hcenters,vcenters,sigmas,powers,readnoise,
gain,plot_results=False,verbose=False):
""" function to fit parameters for radial bspline psf.
"""
### 1. Estimate spline amplitudes, centers, w/ circular model
actypix = raw_img.shape[1]
#r_breakpoints = [0, 1.2, 2.5, 3.7, 5, 8, 10]
## 2.3, 3
#r_breakpoints = np.hstack(([0, 1.5, 2.4, 3],np.arange(3.5,10,0.5))) #For cpad=8
##########################################################################
############## All this hardcoded stuff should be flexible (TODO) ########
##########################################################################
# r_breakpoints = np.hstack(([0, 1.5, 2.4, 3],np.arange(3.5,6.6,1))) #For cpad=5
r_breakpoints = np.hstack(([0, 1.5, 2.4, 3],np.arange(3.5,8.6,1))) #For cpad=6
#r_breakpoints = np.hstack(([0, 1.2, 2.3, 3],np.arange(3.5,10,0.5))) #For cpad=8
theta_orders = [0]
cpad = 6
bp_space = 2 #beakpoint spacing in pixels
invar = 1/(raw_img+readnoise**2)
### Initial spline coeff guess
spl_coeffs, s_scale, fit_params, new_hcenters, new_vcenters = spline_coeff_fit(raw_img,hcenters,vcenters,invar,r_breakpoints,sigmas,powers,theta_orders=theta_orders,cpad=cpad,bp_space=bp_space,return_new_centers=True)
#'''
### 2. Set up and initialize while loop (other steps embedded within loop)
num_bases = spl_coeffs.shape[1]
new_hscale = (new_hcenters-actypix/2)/actypix
peak_mask = np.ones((len(new_hscale)),dtype=bool) #Can be used to mask "bad" peaks
params1 = lmfit.Parameters()
### Loop to add horizontal/vertical centers
for j in range(len(new_hscale)):
harr = np.arange(-cpad,cpad+1)+int(np.floor(new_hcenters[j]))
varr = np.arange(-cpad,cpad+1)+int(np.floor(new_vcenters[j])) ### Shouldn't need +1...
params1.add('vc{}'.format(j), value = new_vcenters[j]-varr[0])
params1.add('hc{}'.format(j), value = new_hcenters[j]-harr[0])
### and add initial ellitical parameter guesses (for quadratic variation)
params1.add('q0', value=0.9, min=0, max=1)
params1.add('PA0', value=0, min=-np.pi, max=np.pi)
params1.add('q1', value=0, min=-1, max=1)
params1.add('PA1', value=0, min=-np.pi, max=np.pi)
params1.add('q2', value=0, min=-1, max=1)
params1.add('PA2', value=0, min=-np.pi, max=np.pi)
params = lmfit.Parameters()
params.add('hc', value = params1['hc0'].value)
params.add('vc', value = params1['vc0'].value)
params.add('q', value = 1, min=0, max=1)
params.add('PA', value=0, min=-np.pi, max=np.pi)
### Start while loop - iterate until convergence
chi_new = np.ones((sum(peak_mask))) #Can build this from first fit if desired
chi_old = np.zeros((sum(peak_mask)))
chi_min = 100
coeff_matrix_min = np.zeros((3,np.shape(spl_coeffs)[1])).T
params_min = lmfit.Parameters()
dlt_chi = 1e-3 #difference between successive chi_squared values to cut off
mx_loops = 50 #eventually must cutoff
loop_cnt = 0
fit_bg = False ## True fits a constant background at each subimage
while abs(np.sum(chi_new)-np.sum(chi_old)) > dlt_chi and loop_cnt < mx_loops:
if verbose:
print("starting loop {}".format(loop_cnt))
print(" chi_old mean = {}".format(np.mean(chi_old)))
print(" chi_new mean = {}".format(np.mean(chi_new)))
print(" delta_chi = {}".format((np.sum(chi_new)-np.sum(chi_old))))
chi_old = np.copy(chi_new)
### 3. Build profile, data, and noise matrices at each pixel point and sum
dim_s = (2*cpad+1)**2
dim_h = sum(peak_mask)*dim_s
profile_matrix = np.zeros((dim_h,3*num_bases+fit_bg*len(new_hscale))) #hardcoded for quadratic
# last_profile = np.zeros((dim_s,3*num_bases+fit_bg))
data_array = np.zeros((dim_h))
noise_array = np.zeros((dim_h))
data_for_fitting = np.zeros((2*cpad+1,2*cpad+1,len(new_hscale)))
invar_for_fitting = np.zeros((2*cpad+1,2*cpad+1,len(new_hscale)))
d_scale = np.zeros(len(new_hscale)) # Will build from data
# bg_data = np.zeros(len(new_hscale))
for k in range(len(new_hscale)):
### Slice subset of image data around each peak
harr = np.arange(-cpad,cpad+1)+int(np.floor(new_hcenters[k]))
varr = np.arange(-cpad,cpad+1)+int(np.floor(new_vcenters[k]))
harr = harr[harr>=0]
harr = harr[harr<raw_img.shape[1]]
varr = varr[varr>=0]
varr = varr[varr<raw_img.shape[0]]
data_for_fitting[:,:,k] = raw_img[varr[0]:varr[-1]+1,harr[0]:harr[-1]+1]#/s_scale[k]
# invar_for_fitting[:,:,k] = invar[varr[0]:varr[-1]+1,harr[0]:harr[-1]+1]#/s_scale[k]
d_scale[k] = np.sum(data_for_fitting[:,:,k])
invar_for_fitting[:,:,k] = s_scale[k]/(abs(data_for_fitting[:,:,k])+readnoise**2/s_scale[k])
# rarr = sf.make_rarr(np.arange(2*cpad+1),np.arange(2*cpad+1),cpad,cpad)
# bg_mask = rarr > 3
# bg_data[k] = poisson_bg(data_for_fitting[:,:,k],mask=bg_mask)
### bound s_scale to (hopefully) prevent runaway growth
# for k in range(len(new_hscale)):
# sig_factor = 1 #Constrain s_scale to be within this man stddevs
# d_min = d_scale[k]-np.sqrt(d_scale[k])*sig_factor
# d_max = d_scale[k]+np.sqrt(d_scale[k])*sig_factor
# if s_scale[k] < d_min:
# s_scale[k] = d_min
# elif s_scale[k] > d_max:
# s_scale[k] = d_max
# s_scale *= np.sum(d_scale)/np.sum(s_scale)
for k in range(len(new_hscale)):
### Pull in best center estimates
params['hc'].value = params1['hc{}'.format(k)].value
params['vc'].value = params1['vc{}'.format(k)].value
### Pull in best elliptical parameter estimates
if loop_cnt == 0:
params['q'].value = 1
else:
params['q'].value = params1['q0'].value + params1['q1'].value*new_hscale[k] + params1['q2'].value*new_hscale[k]**2
params['PA'].value = params1['PA0'].value + params1['PA1'].value*new_hscale[k] + params1['PA2'].value*new_hscale[k]**2
### Scale data
# data_for_fitting[:,:,k] -= bg_data[k] ### remove bg first
data_for_fitting[:,:,k] /= s_scale[k]
# invar_for_fitting[:,:,k] *= s_scale[k]
### Setup arrays for spline analysis
r_arr, theta_arr, dim1, r_inds = spline.build_rarr_thetaarr(data_for_fitting[:,:,k],params)
### Build data, noise, and profile array
data_array[k*dim_s:(k+1)*dim_s] = np.ravel(data_for_fitting[:,:,k])[r_inds] #scaled, sorted data array
noise_array[k*dim_s:(k+1)*dim_s] = np.ravel(invar_for_fitting[:,:,k])[r_inds]
profile_base = spline.build_radial_profile(r_arr,theta_arr,r_breakpoints,theta_orders,(2*cpad+1)**2,order=4)
profile_matrix[k*dim_s:(k+1)*dim_s,0:num_bases] = profile_base
profile_matrix[k*dim_s:(k+1)*dim_s,num_bases:2*num_bases] = profile_base*new_hscale[k]
profile_matrix[k*dim_s:(k+1)*dim_s,2*num_bases:3*num_bases] = profile_base*(new_hscale[k]**2)
if fit_bg:
profile_matrix[k*dim_s:(k+1)*dim_s,3*num_bases+k*fit_bg] = 1
# plt.imshow(profile_matrix,interpolation='none')
# plt.show()
### 4. Using matrices from step 3. perform chi^2 fitting for coefficients
next_coeffs, next_chi = sf.chi_fit(data_array,profile_matrix,np.diag(noise_array))
if fit_bg:
bg_array = next_coeffs[3*num_bases:]
# print bg_array*s_scale
trunc_coeffs = next_coeffs[0:3*num_bases]
else:
trunc_coeffs = np.copy(next_coeffs)
dd2 = int(np.size(trunc_coeffs)/3)
coeff_matrix = trunc_coeffs.reshape(3,dd2).T
# if fit_bg: ### Don't save background fit term
# bg_array = coeff_matrix[:,-1]
# print bg_array*s_scale
# coeff_matrix = coeff_matrix[:,:-1]
# last_coeffs = np.dot(coeff_matrix,(np.vstack((ones(len(new_hscale)),new_hscale,new_hscale**2))))
### Check each of the profiles with next_coeffs + adjust scale factor
profile_matrix = np.zeros((dim_s,3*num_bases+fit_bg*len(new_hscale))) #hardcoded for quadratic
data_array = np.zeros((dim_h))
noise_array = np.zeros((dim_h))
chi2_first = np.zeros(len(new_hscale))
# fit_sums = 0
# print("Temp fit sums:")
for k in range(len(new_hscale)):
### Pull in best center estimates
params['hc'].value = params1['hc{}'.format(k)].value
params['vc'].value = params1['vc{}'.format(k)].value
### Pull in best elliptical parameter estimates
if loop_cnt == 0:
params['q'].value = 1
else:
params['q'].value = params1['q0'].value + params1['q1'].value*new_hscale[k] + params1['q2'].value*new_hscale[k]**2
params['PA'].value = params1['PA0'].value + params1['PA1'].value*new_hscale[k] + params1['PA2'].value*new_hscale[k]**2
### Setup arrays for spline analysis
r_arr, theta_arr, dim1, r_inds = spline.build_rarr_thetaarr(data_for_fitting[:,:,k],params)
### Build data, noise, and profile array
data_array[k*dim_s:(k+1)*dim_s] = np.ravel(data_for_fitting[:,:,k])[r_inds] #scaled, sorted data array
noise_array[k*dim_s:(k+1)*dim_s] = np.ravel(invar_for_fitting[:,:,k])[r_inds]
profile_base = spline.build_radial_profile(r_arr,theta_arr,r_breakpoints,theta_orders,(2*cpad+1)**2,order=4)
profile_matrix[:,0:num_bases] = profile_base
profile_matrix[:,num_bases:2*num_bases] = profile_base*new_hscale[k]
profile_matrix[:,2*num_bases:3*num_bases] = profile_base*(new_hscale[k]**2)
if fit_bg:
profile_matrix[:,3*num_bases:] = 0
profile_matrix[:,3*num_bases+k*fit_bg] = 1
tmp_fit = np.dot(profile_matrix,next_coeffs)
# print np.sum(tmp_fit)
# fit_sums += np.sum(tmp_fit)
resort_inds = np.argsort(r_inds)
tmp_fit = np.reshape(tmp_fit[resort_inds],data_for_fitting[:,:,k].shape)
# plt.figure("Arc, iteration {}".format(k))
## plt.imshow(np.hstack((tmp_fit,small_img/s_scale[k])),interpolation='none')
chi2_first[k] = np.sum(((tmp_fit-data_for_fitting[:,:,k])**2)*invar_for_fitting[:,:,k])#*s_scale[k]**2
# plt.imshow((tmp_fit-small_img/s_scale[k])*small_inv,interpolation='none')
# plt.show()
# plt.close()
# print "chi2 first:", chi2_first
# next_coeffs *= fit_sums/(k+1)
# s_scale /= fit_sums/(k+1)
### Optional place to check coefficients variation over order
#for i in range(8):
# plt.plot(new_hscale,last_coeffs[i])
#
#plt.show()
#plt.close()
#first_fit = np.dot(last_profile,next_coeffs)
#print next_coeffs
#print params['vc'].value
#print params['hc'].value
#print r_arr[0:10]
#print profile_base[0]
#print profile_matrix[0,:]/(k+1)
#print last_profile[0]
#print first_fit[0]
#resort_inds = np.argsort(r_inds)
#scale1 = np.max(small_img)/np.max(first_fit)
##print scale1, scale, scale1/scale
#first_fit = np.reshape(first_fit[resort_inds],small_img.shape)
#print np.sum(first_fit), k, scale1, s_scale[k]
#first_fit /= np.sum(first_fit)
##plt.imshow(first_fit,interpolation='none')
#plt.imshow(np.hstack((small_img/s_scale[k],first_fit,(small_img/s_scale[k]-first_fit)*small_inv)),interpolation='none')
#plt.show()
#plt.imshow((small_img/s_scale[k]-first_fit)*small_inv,interpolation='none')
#plt.show()
#test_xs = (np.arange(xpix)-xpix/2)/xpix
#for i in range(num_bases):
# test_ys = next_coeffs[i]+next_coeffs[num_bases+i]*test_xs+next_coeffs[2*num_bases+i]*test_xs**2
# plt.plot(test_xs,test_ys)
#plt.show()
### 5. Now do a nonlinear fit for hc, vc, q, and PA
#data_for_lmfit = np.zeros((np.size(small_img),len(new_hscale)))
#invar_for_lmfit = np.zeros((np.size(small_img),len(new_hscale)))
# for k in range(len(new_hscale)):
# harr = np.arange(-cpad,cpad+1)+int(np.floor(new_hcenters[k]))
# varr = np.arange(-cpad,cpad+1)+int(np.floor(new_vcenters[k]))
# data_for_lmfit[:,:,k] = raw_img[varr[0]:varr[-1]+1,harr[0]:harr[-1]+1]/s_scale[k]
# invar_for_lmfit[:,:,k] = invar[varr[0]:varr[-1]+1,harr[0]:harr[-1]+1]*(s_scale[k])
# r_arr, theta_arr, dim1, r_inds = spline.build_rarr_thetaarr(small_img,params)
# data_for_lmfit[:,k] = np.ravel(small_img)[r_inds]/s_scale[k]
# invar_for_lmfit[:,k] = np.ravel(small_inv)[r_inds]/np.sqrt(s_scale[k])
# resort_inds = np.argsort(r_inds)
# plt.imshow(np.resize(data_for_lmfit[:,k][resort_inds],np.shape(small_img)))
# plt.show()
# plt.close()
### Make proper inputs for minimizer function
#centers = np.vstack((new_hcenters,new_vcenters)).T
args = (data_for_fitting,invar_for_fitting,r_breakpoints,new_hscale,next_coeffs)
kws = dict()
kws['theta_orders'] = theta_orders
kws['fit_bg'] = fit_bg
minimizer_results = lmfit.minimize(spline.spline_poly_residuals,params1,args=args,kws=kws)
### Re-initialize params1, put in elliptical values. Will add hc/vc at end
### (using mask, so #of values for centers will differ)
params1['q0'].value = minimizer_results.params['q0'].value
params1['q1'].value = minimizer_results.params['q1'].value
params1['q2'].value = minimizer_results.params['q2'].value
params1['PA0'].value = minimizer_results.params['PA0'].value
params1['PA1'].value = minimizer_results.params['PA1'].value
params1['PA2'].value = minimizer_results.params['PA2'].value
#hc_ck = minimizer_results.params['hc0'].value + minimizer_results.params['hc1'].value*new_hscale + minimizer_results.params['hc2'].value*new_hscale**2
#vc_ck = minimizer_results.params['vc0'].value + minimizer_results.params['vc1'].value*new_hscale + minimizer_results.params['vc2'].value*new_hscale**2
q_ck = minimizer_results.params['q0'].value + minimizer_results.params['q1'].value*new_hscale + minimizer_results.params['q2'].value*new_hscale**2
PA_ck = minimizer_results.params['PA0'].value + minimizer_results.params['PA1'].value*new_hscale + minimizer_results.params['PA2'].value*new_hscale**2
# print q_ck
# print PA_ck
### Convert so q is less than 1
if np.max(q_ck) > 1:
q_ck_tmp = 1/q_ck #change axis definition
if np.max(q_ck_tmp) > 1:
print "q array always over 1!"
else:
q_ck = q_ck_tmp
PA_ck = PA_ck + np.pi/2 #change axis definition
q_coeffs = np.polyfit(new_hscale,q_ck,2)
PA_coeffs = np.polyfit(new_hscale,PA_ck,2)
params1['q0'].value = q_coeffs[2]
params1['q1'].value = q_coeffs[1]
params1['q2'].value = q_coeffs[0]
params1['PA0'].value = PA_coeffs[2]
params1['PA1'].value = PA_coeffs[1]
params1['PA2'].value = PA_coeffs[0]
# print q_ck
# print PA_ck
#plt.plot(np.arange(5),np.arange(5))
#plt.show()
#plt.plot(hc_ck,vc_ck,new_hcenters,new_vcenters)
#plt.show()
#ecc = minimizer_results.params['q'].value
#pos_ang = minimizer_results.params['PA'].value
### Check to see if elliptical values worked out well
chi_new = np.zeros(len(new_hscale))
for i in range(len(new_hscale)):
params['vc'].value = minimizer_results.params['vc{}'.format(i)].value
params['hc'].value = minimizer_results.params['hc{}'.format(i)].value
# harr = np.arange(-cpad,cpad+1)+int(np.floor(new_hcenters[i]))
# varr = np.arange(-cpad,cpad+1)+int(np.floor(new_vcenters[i]))
# params['vc'].value = new_vcenters[i]-varr[0]+1
# params['hc'].value = new_hcenters[i]-harr[0]
x_coord = new_hscale[i]
img_matrix = data_for_fitting[:,:,i]
invar_matrix = invar_for_fitting[:,:,i]
q = params1['q0'].value + params1['q1'].value*x_coord + params1['q2'].value*x_coord**2
PA = params1['PA0'].value + params1['PA1'].value*x_coord + params1['PA2'].value*x_coord**2
params['q'].value = q
params['PA'].value = PA
sp_coeffs = np.dot(coeff_matrix,np.array(([1,new_hscale[i],new_hscale[i]**2])))
if fit_bg:
sp_coeffs = np.hstack((sp_coeffs,bg_array[i]))
# r_arr, theta_arr, dim1, r_inds = spline.build_rarr_thetaarr(small_img,params)
# profile_base = spline.build_radial_profile(r_arr,theta_arr,r_breakpoints,theta_orders,(2*cpad+1)**2,order=4)
fitted_image = spline.spline_2D_radial(img_matrix,invar_matrix,r_breakpoints,params,theta_orders,order=4,return_coeffs=False,spline_coeffs=sp_coeffs,sscale=None,fit_bg=fit_bg)
### Update s_scale
chi_new[i] = np.sum(((img_matrix-fitted_image)**2)*invar_matrix)*s_scale[i]/(np.size(img_matrix)-len(sp_coeffs)-2)#*s_scale[i]**2
# print chi_new[i]
# print s_scale[i]
# print np.max(invar_matrix)*3.63**2/s_scale[i]
# print chi_new[i]*s_scale[i]
# print chi_new[i]*s_scale[i]**2
### Set new scale - drive sum of image toward unity
s_scale[i] = s_scale[i]*np.sum(fitted_image)
# plt.imshow(np.hstack((img_matrix,fitted_image)),interpolation='none')#,(img_matrix-fitted_image)*invar_matrix)),interpolation='none')
# plt.imshow(invar_matrix,interpolation='none')
# plt.plot(img_matrix[:,5])
# plt.plot(fitted_image[:,5])
# plt.show()
# plt.close()
#print chi2_first
#print chi2_second
#print s_scale
#print s_scale2
### Mask/eliminate points with high chi2
peak_mask = sf.sigma_clip(chi_new,sigma=3,max_iters=1)
if sum(peak_mask) < 4:
print("Too few peaks for fitting")
exit(0)
# break
### Update new_hscale, s_scale, new_h/vcenters
s_scale = s_scale[peak_mask]
cnts = len(new_hscale)
new_hscale = np.zeros((sum(peak_mask)))
lp_idx = 0
for j in range(cnts):
if not peak_mask[j]:
if verbose:
print "skipping point {}".format(j)
continue
else:
harr = np.arange(-cpad,cpad+1)+int(np.floor(new_hcenters[j]))
params1.add('hc{}'.format(lp_idx), value = minimizer_results.params['hc{}'.format(j)].value)
params1.add('vc{}'.format(lp_idx), value = minimizer_results.params['vc{}'.format(j)].value)
new_hscale[lp_idx] = (params1['hc{}'.format(lp_idx)].value+harr[0]-1-actypix/2)/actypix
lp_idx += 1
new_hcenters = new_hcenters[peak_mask]
new_vcenters = new_vcenters[peak_mask]
### Record minimum values (some subsequent iterations give higher chi2)
if loop_cnt == 0:
coeff_matrix_min = np.copy(coeff_matrix)
params_min = lmfit.Parameters(params1)
if np.sum(chi_new) < chi_min:
if verbose:
print "Better fit on loop ", loop_cnt
chi_min = np.sum(chi_new)
coeff_matrix_min = np.copy(coeff_matrix)
params_min = lmfit.Parameters(params1)
loop_cnt += 1
### End of loop
if verbose:
print("End of Loop")
### Check that q, PA, aren't driving toward unphysical answers
# test_hscale = np.arange(-1,1,0.01)
#q = params_min['q0'].value + params_min['q1'].value*test_hscale + params_min['q2'].value*test_hscale**2
#PA = params_min['PA0'].value + params_min['PA1'].value*test_hscale + params_min['PA2'].value*test_hscale**2
#bg = coeff_matrix_min[0,-1] + coeff_matrix_min[1,-1]*test_hscale + coeff_matrix_min[2,-1]*test_hscale**2
#plt.plot(test_hscale,q)
#plt.show()
#plt.plot(test_hscale,PA)
#plt.show()
#plt.plot(test_hscale,bg)
#plt.show()
#plt.close()
if plot_results:
### Plot final answers for evaluation
for i in range(len(new_hscale)):
params['vc'].value = minimizer_results.params['vc{}'.format(i)].value
params['hc'].value = minimizer_results.params['hc{}'.format(i)].value
# harr = np.arange(-cpad,cpad+1)+int(np.floor(new_hcenters[i]))
# varr = np.arange(-cpad,cpad+1)+int(np.floor(new_vcenters[i]))
# params['vc'].value = new_vcenters[i]-varr[0]+1
# params['hc'].value = new_hcenters[i]-harr[0]
x_coord = new_hscale[i]
img_matrix = data_for_fitting[:,:,i]
invar_matrix = invar_for_fitting[:,:,i]
q = params_min['q0'].value + params_min['q1'].value*x_coord + params_min['q2'].value*x_coord**2
PA = params_min['PA0'].value + params_min['PA1'].value*x_coord + params_min['PA2'].value*x_coord**2
params['q'].value = q
params['PA'].value = PA
sp_coeffs = np.dot(coeff_matrix_min,np.array(([1,new_hscale[i],new_hscale[i]**2])))
if fit_bg:
sp_coeffs = np.hstack((sp_coeffs,bg_array[i]))
# r_arr, theta_arr, dim1, r_inds = spline.build_rarr_thetaarr(small_img,params)
# profile_base = spline.build_radial_profile(r_arr,theta_arr,r_breakpoints,theta_orders,(2*cpad+1)**2,order=4)
fitted_image = spline.spline_2D_radial(img_matrix,invar_matrix,r_breakpoints,params,theta_orders,order=4,return_coeffs=False,spline_coeffs=sp_coeffs,sscale=None,fit_bg=fit_bg)
### Update s_scale
# print chi_new[i]
# print chi_new[i]*s_scale[i]
# print chi_new[i]*s_scale[i]**2
chi_sq_red = np.sum(((img_matrix-fitted_image))**2*invar_matrix)/(np.size(img_matrix)-len(sp_coeffs)-2)*(s_scale[i])
print "Reduced Chi^2 on iteration ", i, " is: ", chi_sq_red
# plt.plot(fitted_image[:,cpad]/np.max(fitted_image[:,cpad]))
# plt.plot(np.sum(fitted_image,axis=1)/np.max(np.sum(fitted_image,axis=1)))
# plt.show()
# plt.imshow(np.hstack((img_matrix,fitted_image,(img_matrix-fitted_image))),interpolation='none')
plt.imshow((img_matrix-fitted_image)*invar_matrix,interpolation='none')
# plt.imshow((img_matrix-fitted_image)*invar_matrix,interpolation='none')
plt.show()
plt.close()
centers, ellipse = params_to_array(params_min)
results = np.hstack((np.ravel(coeff_matrix_min),np.ravel(ellipse)))
return results