Add comments pinpointing polarizers' orientation uncertainty computation

FOC_Capetti_test_NGC1068/FOC_reduction.py

@@ -1,256 +0,0 @@
-from pylab import *
-import numpy as np
-import matplotlib.pyplot as plt
-from astropy.io import fits
-from astropy.wcs import WCS
-from aplpy import FITSFigure 
-import scipy.ndimage
-import os as os
-
-plt.close('all')
-
-def bin_ndarray(ndarray, new_shape, operation='sum'):
-    """
-    Bins an ndarray in all axes based on the target shape, by summing or
-        averaging.
-
-    Number of output dimensions must match number of input dimensions.
-
-    Example
-    -------
-    >>> m = np.arange(0,100,1).reshape((10,10))
-    >>> n = bin_ndarray(m, new_shape=(5,5), operation='sum')
-    >>> print(n)
-
-    [[ 22  30  38  46  54]
-     [102 110 118 126 134]
-     [182 190 198 206 214]
-     [262 270 278 286 294]
-     [342 350 358 366 374]]
-
-    """
-    if not operation.lower() in ['sum', 'mean', 'average', 'avg']:
-        raise ValueError("Operation not supported.")
-    if ndarray.ndim != len(new_shape):
-        raise ValueError("Shape mismatch: {} -> {}".format(ndarray.shape,
-                                                           new_shape))
-    compression_pairs = [(d, c//d) for d,c in zip(new_shape,
-                                                  ndarray.shape)]
-    flattened = [l for p in compression_pairs for l in p]
-    ndarray = ndarray.reshape(flattened)
-    for i in range(len(new_shape)):
-        if operation.lower() == "sum":
-            ndarray = ndarray.sum(-1*(i+1))
-        elif operation.lower() in ["mean", "average", "avg"]:
-            ndarray = ndarray.mean(-1*(i+1))
-    return ndarray
-
-def plots(ax,data,vmax,vmin):
-    ax.imshow(data,vmax=vmax,vmin=vmin,origin='lower')
-
-
-
-### User input
-
-infiles = ['x274020at.c0f.fits','x274020bt.c0f.fits','x274020ct.c0f.fits','x274020dt.c0f.fits',
-             'x274020et.c0f.fits','x274020ft.c0f.fits','x274020gt.c0f.fits','x274020ht.c0f.fits',
-             'x274020it.c0f.fits']
-
-#Centroids
-#The centroids should be estimated by cross-correlating the images. 
-#Here I used the position of the central source for each file as the reference pixel position.
-ycen_0 = [304,306,303,296,295,295,294,305,304]
-xcen_0 = [273,274,273,276,274,274,274,272,271]
-#size, in pixels, of the FOV centered in x_cen_0,y_cen_0
-Dx = 200
-Dy = 200
-
-#set the new image size (Dxy x Dxy pixels binning)
-Dxy = 5  
-new_shape   = (Dx//Dxy,Dy//Dxy)
-
-#figures
-#test alignment
-vmin = 0
-vmax = 6
-font_size=40.0
-label_size=20.
-lw = 3.
-
-#pol. map
-SNRp_cut = 3 #P measumentes with SNR>3
-SNRi_cut = 30 #I measuremntes with SNR>30, which implies an uncerrtainty in P of 4.7%. 
-scalevec = 0.05 #length of vectors in pol. map
-step_vec = 1    #plot all vectors in the array. if step_vec = 2, then every other vector will be plotted
-vec_legend = 10 #% pol for legend
-figname = 'NGC1068_FOC.png'
-
-
-### SCRIPT ###
-### Step 1. Check input images before data reduction
-#this step is very simplistic.
-#Here I used the position of the central source for each file as the 
-#reference pixel position.
-#The centroids should be estimated by cross-correlating the images, 
-#and compare with the simplistic approach of using the peak pixel of the
-#object as the reference pixel.
-
-
-fig,axs = plt.subplots(3,3,figsize=(30,30),dpi=200,sharex=True,sharey=True)
-
-for jj, a in enumerate(axs.flatten()):
-    img = fits.open(infiles[jj])
-    ima = img[0].data
-    ima = ima[ycen_0[jj]-Dy:ycen_0[jj]+Dy,xcen_0[jj]-Dx:xcen_0[jj]+Dx]
-    ima = bin_ndarray(ima,new_shape=new_shape,operation='sum') #binning
-    exptime = img[0].header['EXPTIME']
-    fil = img[0].header['FILTNAM1']
-    ima = ima/exptime
-    globals()['ima_%s' % jj] = ima
-    #plots
-    plots(a,ima,vmax=vmax,vmin=vmin)
-    #position of centroid
-    a.plot([ima.shape[1]/2,ima.shape[1]/2],[0,ima.shape[0]-1],lw=1,color='black')
-    a.plot([0,ima.shape[1]-1],[ima.shape[1]/2,ima.shape[1]/2],lw=1,color='black')
-    a.text(2,2,infiles[jj][0:8],color='white',fontsize=10)
-    a.text(2,5,fil,color='white',fontsize=30)
-    a.text(ima.shape[1]-20,1,exptime,color='white',fontsize=20)
-fig.subplots_adjust(hspace=0,wspace=0)
-fig.savefig('test_alignment.png',dpi=300)
-os.system('open test_alignment.png')
-
-
-
-### Step 2. average of all images for a single polarizer to have them in the same units e/s.
-pol0   = (ima_0 + ima_1 + ima_2)/3.
-pol60  = (ima_3 + ima_4 + ima_5 + ima_6)/4.
-pol120 = (ima_7 + ima_8)/2.
-
-fig1,(ax1,ax2,ax3) = plt.subplots(1,3,figsize=(26,8),dpi=200)
-CF = ax1.imshow(pol0,vmin=vmin,vmax=vmax,origin='lower')
-cbar = plt.colorbar(CF,ax=ax1)
-cbar.ax.tick_params(labelsize=20)
-ax1.tick_params(axis='both', which='major', labelsize=20)
-ax1.text(2,2,'POL0',color='white',fontsize=10)
-
-CF = ax2.imshow(pol60,vmin=vmin,vmax=vmax,origin='lower')
-cbar = plt.colorbar(CF,ax=ax2)
-cbar.ax.tick_params(labelsize=20)
-ax2.tick_params(axis='both', which='major', labelsize=20)
-ax2.text(2,2,'POL60',color='white',fontsize=10)
-
-CF = ax3.imshow(pol120,vmin=vmin,vmax=vmax,origin='lower')
-cbar = plt.colorbar(CF,ax=ax3)
-cbar.ax.tick_params(labelsize=20)
-ax3.tick_params(axis='both', which='major', labelsize=20)
-ax3.text(2,2,'POL120',color='white',fontsize=10)
-fig1.savefig('test_combinePol.png',dpi=300)
-os.system('open test_combinePol.png')
-
-
-### Step 3. Compute Stokes IQU, P, PA, PI
-#Stokes parameters
-I_stokes = (2./3.)*(pol0 + pol60 + pol120)
-Q_stokes = (2./3.)*(2*pol0 - pol60 - pol120)
-U_stokes = (2./np.sqrt(3.))*(pol60 - pol120)
-
-#Remove nan
-I_stokes[np.isnan(I_stokes)]=0.
-Q_stokes[np.isnan(Q_stokes)]=0.
-U_stokes[np.isnan(U_stokes)]=0.
-
-#Polarimetry
-PI  = np.sqrt(Q_stokes*Q_stokes + U_stokes*U_stokes)
-P   = PI/I_stokes*100
-PA  = 0.5*arctan2(U_stokes,Q_stokes)*180./np.pi+90
-s_P  = np.sqrt(2.)*(I_stokes)**(-0.5)
-s_PA = s_P/(P/100.)*180./np.pi 
-
-fig2,((ax1,ax2,ax3),(ax4,ax5,ax6)) = plt.subplots(2,3,figsize=(40,20),dpi=200)
-CF = ax1.imshow(I_stokes,origin='lower')
-cbar = plt.colorbar(CF,ax=ax1)
-cbar.ax.tick_params(labelsize=20)
-ax1.tick_params(axis='both', which='major', labelsize=20)
-ax1.text(2,2,'I',color='white',fontsize=10)
-
-CF = ax2.imshow(Q_stokes,origin='lower')
-cbar = plt.colorbar(CF,ax=ax2)
-cbar.ax.tick_params(labelsize=20)
-ax2.tick_params(axis='both', which='major', labelsize=20)
-ax2.text(2,2,'Q',color='white',fontsize=10)
-
-CF = ax3.imshow(U_stokes,origin='lower')
-cbar = plt.colorbar(CF,ax=ax3)
-cbar.ax.tick_params(labelsize=20)
-ax3.tick_params(axis='both', which='major', labelsize=20)
-ax3.text(2,2,'U',color='white',fontsize=10)
-
-v = np.linspace(0,40,50)
-CF = ax4.imshow(P,origin='lower',vmin=0,vmax=40)
-cbar = plt.colorbar(CF,ax=ax4)
-cbar.ax.tick_params(labelsize=20)
-ax4.tick_params(axis='both', which='major', labelsize=20)
-ax4.text(2,2,'P',color='white',fontsize=10)
-
-CF = ax5.imshow(PA,origin='lower',vmin=0,vmax=180)
-cbar = plt.colorbar(CF,ax=ax5)
-cbar.ax.tick_params(labelsize=20)
-ax5.tick_params(axis='both', which='major', labelsize=20)
-ax5.text(2,2,'PA',color='white',fontsize=10)
-
-CF = ax6.imshow(PI,origin='lower')
-cbar = plt.colorbar(CF,ax=ax6)
-cbar.ax.tick_params(labelsize=20)
-ax6.tick_params(axis='both', which='major', labelsize=20)
-ax6.text(2,2,'PI',color='white',fontsize=10)
-
-fig2.savefig('test_Stokes.png',dpi=300)
-os.system('open test_Stokes.png')
-
-### Step 4. Binning and smoothing
-#Images can be binned and smoothed to improve SNR. This step can also be done 
-#using the PolX images. 
-
-
-### Step 5. Roate images to have North up
-#Images needs to be reprojected to have North up.
-#this procedure implies to rotate the Stokes QU using a rotation matrix
-
-
-### STEP 6. image to FITS with updated WCS
-new_wcs = WCS(naxis=2)
-new_wcs.wcs.crpix = [I_stokes.shape[0]/2, I_stokes.shape[1]/2]
-new_wcs.wcs.crval = [img[0].header['CRVAL1'], img[0].header['CRVAL2']]
-new_wcs.wcs.cunit = ["deg", "deg"]
-new_wcs.wcs.ctype = ["RA---TAN", "DEC--TAN"]
-new_wcs.wcs.cdelt = [img[0].header['CD1_1']*Dxy, img[0].header['CD1_2']*Dxy]
-
-#hdu_ori = WCS(img[0])
-stkI=fits.PrimaryHDU(data=I_stokes,header=new_wcs.to_header())
-pol=fits.PrimaryHDU(data=P,header=new_wcs.to_header())
-pang=fits.PrimaryHDU(data=PA,header=new_wcs.to_header())
-pol_err=fits.PrimaryHDU(data=s_P,header=new_wcs.to_header())
-pang_err=fits.PrimaryHDU(data=s_PA,header=new_wcs.to_header())
-
-
-### STEP 7. polarization map
-#quality cuts
-pxscale = stkI.header['CDELT1']
-
-#apply quality cuts
-SNRp = pol.data/pol_err.data
-pol.data[SNRp < SNRp_cut] = np.nan
-
-SNRi = stkI.data/np.std(stkI.data[0:10,0:10])
-pol.data[SNRi < SNRi_cut] = np.nan
-
-fig = plt.figure(figsize=(11,10))
-gc = FITSFigure(stkI,figure=fig)
-gc.show_contour(np.log10(SNRi),levels=np.linspace(np.log10(SNRi_cut),np.max(np.log10(SNRi)),20),\
-                filled=True,cmap='magma')
-gc.show_vectors(pol,pang,scale=scalevec,step=step_vec,color='white',linewidth=1.0)
-
-fig.savefig(figname,dpi=300)
-os.system('open '+figname)
-
-

src/FOC_reduction.py

@@ -7,7 +7,7 @@ Main script where are progressively added the steps for the FOC pipeline reducti
 #Project libraries
 import sys
 import numpy as np
-import copy
+from copy import deepcopy
 import lib.fits as proj_fits        #Functions to handle fits files
 import lib.reduction as proj_red    #Functions used in reduction pipeline
 import lib.plots as proj_plots      #Functions for plotting data
@@ -155,11 +155,10 @@ def main():
     rectangle = [vertex[2], vertex[0], shape[1], shape[0], 0., 'w']
 
     # Rotate data to have North up
-    ref_header = copy.deepcopy(headers[0])
+    ref_header = deepcopy(headers[0])
     if rotate_data:
         alpha = ref_header['orientat']
-        mrot = np.array([[np.cos(-alpha), -np.sin(-alpha)],
+        mrot = np.array([[np.cos(-alpha), -np.sin(-alpha)], [np.sin(-alpha), np.cos(-alpha)]])
-            [np.sin(-alpha), np.cos(-alpha)]])
         rectangle[0:2] = np.dot(mrot, np.asarray(rectangle[0:2]))+np.array(data_array.shape[1:])/2
         rectangle[4] = alpha
         data_array, error_array, data_mask, headers = proj_red.rotate_data(data_array, error_array, data_mask, headers, -ref_header['orientat'])
@@ -180,7 +179,7 @@ def main():
 
     ## Step 3:
     # Rotate images to have North up
-    ref_header = copy.deepcopy(headers[0])
+    ref_header = deepcopy(headers[0])
     if rotate_stokes:
         alpha = ref_header['orientat']
         mrot = np.array([[np.cos(-alpha), -np.sin(-alpha)],
@@ -193,7 +192,7 @@ def main():
 
     ## Step 4:
     # crop to desired region of interest (roi)
-#    stokescrop = proj_plots.crop_map(copy.deepcopy(stokes_test), copy.deepcopy(data_mask), snrp_cut=snrp_cut, snri_cut=snri_cut)
+#    stokescrop = proj_plots.crop_map(deepcopy(stokes_test), deepcopy(data_mask), snrp_cut=snrp_cut, snri_cut=snri_cut)
 #    stokescrop.run()
 #    stokes_crop, data_mask = stokescrop.crop()
 
@@ -202,13 +201,13 @@ def main():
     Stokes_test = proj_fits.save_Stokes(I_stokes, Q_stokes, U_stokes, Stokes_cov, P, debiased_P, s_P, s_P_P, PA, s_PA, s_PA_P, headers, figname+figtype, data_folder=data_folder, return_hdul=True)
 
     # Plot polarization map (Background is either total Flux, Polarization degree or Polarization degree error).
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype, plots_folder=plots_folder, display=None)
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype, plots_folder=plots_folder, display=None)
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_P_flux", plots_folder=plots_folder, display='Pol_Flux')
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_P_flux", plots_folder=plots_folder, display='Pol_Flux')
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_P", plots_folder=plots_folder, display='Pol_deg')
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_P", plots_folder=plots_folder, display='Pol_deg')
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_I_err", plots_folder=plots_folder, display='I_err')
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_I_err", plots_folder=plots_folder, display='I_err')
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_P_err", plots_folder=plots_folder, display='Pol_deg_err')
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_P_err", plots_folder=plots_folder, display='Pol_deg_err')
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_SNRi", plots_folder=plots_folder, display='SNRi')
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_SNRi", plots_folder=plots_folder, display='SNRi')
-    proj_plots.polarization_map(copy.deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_SNRp", plots_folder=plots_folder, display='SNRp')
+    proj_plots.polarization_map(deepcopy(Stokes_test), data_mask, rectangle=None, SNRp_cut=SNRp_cut, SNRi_cut=SNRi_cut, step_vec=step_vec, savename=figname+figtype+"_SNRp", plots_folder=plots_folder, display='SNRp')
 
     return 0
 

src/lib/convex_hull.py

@@ -3,7 +3,7 @@ Library functions for graham algorithm implementation (find the convex hull
 of a given list of points).
 """
 
-import copy
+from copy import deepcopy
 import numpy as np
 
 
@@ -141,12 +141,12 @@ def partition(s, l, r, order):
     for j in range(l, r):
         if order(s[j], s[r]):
             i = i + 1
-            temp = copy.deepcopy(s[i])
+            temp = deepcopy(s[i])
-            s[i] = copy.deepcopy(s[j])
+            s[i] = deepcopy(s[j])
-            s[j] = copy.deepcopy(temp)
+            s[j] = deepcopy(temp)
-    temp = copy.deepcopy(s[i+1])
+    temp = deepcopy(s[i+1])
-    s[i+1] = copy.deepcopy(s[r])
+    s[i+1] = deepcopy(s[r])
-    s[r] = copy.deepcopy(temp)
+    s[r] = deepcopy(temp)
     return i + 1
 
 

src/lib/reduction.py

@@ -37,7 +37,7 @@ prototypes :
         Rotate I, Q, U given an angle in degrees using scipy functions.
 """
 
-import copy
+from copy import deepcopy
 import numpy as np
 import matplotlib.pyplot as plt
 import matplotlib.dates as mdates
@@ -222,15 +222,15 @@ def crop_array(data_array, headers, error_array=None, step=5, null_val=None,
         null_val = [null_val,]*error_array.shape[0]
 
     vertex = np.zeros((data_array.shape[0],4),dtype=int)
-    for i,image in enumerate(data_array):
+    for i,image in enumerate(data_array):   # Get vertex of the rectangular convex hull of each image
         vertex[i] = image_hull(image,step=step,null_val=null_val[i],inside=inside)
     v_array = np.zeros(4,dtype=int)
-    if inside:
+    if inside:  # Get vertex of the maximum convex hull for all images
         v_array[0] = np.max(vertex[:,0]).astype(int)
         v_array[1] = np.min(vertex[:,1]).astype(int)
         v_array[2] = np.max(vertex[:,2]).astype(int)
         v_array[3] = np.min(vertex[:,3]).astype(int)
-    else:
+    else:       # Get vertex of the minimum convex hull for all images
         v_array[0] = np.min(vertex[:,0]).astype(int)
         v_array[1] = np.max(vertex[:,1]).astype(int)
         v_array[2] = np.min(vertex[:,2]).astype(int)
@@ -279,7 +279,7 @@ def crop_array(data_array, headers, error_array=None, step=5, null_val=None,
 
     crop_array = np.zeros((data_array.shape[0],new_shape[0],new_shape[1]))
     crop_error_array = np.zeros((data_array.shape[0],new_shape[0],new_shape[1]))
-    for i,image in enumerate(data_array):
+    for i,image in enumerate(data_array):   #Put the image data in the cropped array
         crop_array[i] = image[v_array[0]:v_array[1],v_array[2]:v_array[3]]
         crop_error_array[i] = error_array[i][v_array[0]:v_array[1],v_array[2]:v_array[3]]
 
@@ -732,9 +732,9 @@ def align_data(data_array, headers, error_array=None, upsample_factor=1.,
         center = np.fix(ref_center-shift).astype(int)
         res_shift = res_center-ref_center
         rescaled_image[i,res_shift[0]:res_shift[0]+shape[1],
-                res_shift[1]:res_shift[1]+shape[2]] = copy.deepcopy(image)
+                res_shift[1]:res_shift[1]+shape[2]] = deepcopy(image)
         rescaled_error[i,res_shift[0]:res_shift[0]+shape[1],
-                res_shift[1]:res_shift[1]+shape[2]] = copy.deepcopy(error_array[i])
+                res_shift[1]:res_shift[1]+shape[2]] = deepcopy(error_array[i])
         rescaled_mask[i,res_shift[0]:res_shift[0]+shape[1],
                 res_shift[1]:res_shift[1]+shape[2]] = False
         # Shift images to align
@@ -1106,14 +1106,19 @@ def compute_Stokes(data_array, error_array, data_mask, headers,
         pol_eff[1] = pol_efficiency['pol60']
         pol_eff[2] = pol_efficiency['pol120']
 
+        # Orientation and error for each polarizer  ## THIS IS WHERE WE IMPLEMENT THE ERROR THAT IS GOING WRONG
+        # POL0 = 0deg, POL60 = 60deg, POL120=120deg
         theta = np.array([180.*np.pi/180., 60.*np.pi/180., 120.*np.pi/180.])
+        # Uncertainties on the orientation of the polarizers' axes taken to be 3deg (see Nota et. al 1996, p36; Robinson & Thomson 1995)
         sigma_theta = np.array([3.*np.pi/180., 3.*np.pi/180., 3.*np.pi/180.])
         pol_flux = 2.*np.array([pol0/transmit[0], pol60/transmit[1], pol120/transmit[2]])
 
+        # Normalization parameter for Stokes parameters computation
         A = pol_eff[1]*pol_eff[2]*np.sin(-2.*theta[1]+2.*theta[2]) \
                 + pol_eff[2]*pol_eff[0]*np.sin(-2.*theta[2]+2.*theta[0]) \
                 + pol_eff[0]*pol_eff[1]*np.sin(-2.*theta[0]+2.*theta[1])
         coeff_stokes = np.zeros((3,3))
+        # Coefficients linking each polarizer flux to each Stokes parameter
         for i in range(3):
             coeff_stokes[0,i] = pol_eff[(i+1)%3]*pol_eff[(i+2)%3]*np.sin(-2.*theta[(i+1)%3]+2.*theta[(i+2)%3])/A
             coeff_stokes[1,i] = (-pol_eff[(i+1)%3]*np.sin(2.*theta[(i+1)%3]) + pol_eff[(i+2)%3]*np.sin(2.*theta[(i+2)%3]))/A
@@ -1143,6 +1148,7 @@ def compute_Stokes(data_array, error_array, data_mask, headers,
         Stokes_cov[0,2] = Stokes_cov[2,0] = coeff_stokes[0,0]*coeff_stokes[2,0]*pol_cov[0,0]+coeff_stokes[0,1]*coeff_stokes[2,1]*pol_cov[1,1]+coeff_stokes[0,2]*coeff_stokes[2,2]*pol_cov[2,2]+(coeff_stokes[0,0]*coeff_stokes[2,1]+coeff_stokes[2,0]*coeff_stokes[0,1])*pol_cov[0,1]+(coeff_stokes[0,0]*coeff_stokes[2,2]+coeff_stokes[2,0]*coeff_stokes[0,2])*pol_cov[0,2]+(coeff_stokes[0,1]*coeff_stokes[2,2]+coeff_stokes[2,1]*coeff_stokes[0,2])*pol_cov[1,2]
         Stokes_cov[1,2] = Stokes_cov[2,1] = coeff_stokes[1,0]*coeff_stokes[2,0]*pol_cov[0,0]+coeff_stokes[1,1]*coeff_stokes[2,1]*pol_cov[1,1]+coeff_stokes[1,2]*coeff_stokes[2,2]*pol_cov[2,2]+(coeff_stokes[1,0]*coeff_stokes[2,1]+coeff_stokes[2,0]*coeff_stokes[1,1])*pol_cov[0,1]+(coeff_stokes[1,0]*coeff_stokes[2,2]+coeff_stokes[2,0]*coeff_stokes[1,2])*pol_cov[0,2]+(coeff_stokes[1,1]*coeff_stokes[2,2]+coeff_stokes[2,1]*coeff_stokes[1,2])*pol_cov[1,2]
 
+        # Compute the derivative of each Stokes parameter with respect to the polarizer orientation
         dI_dtheta1 = 2.*pol_eff[0]/A*(pol_eff[2]*np.cos(-2.*theta[2]+2.*theta[0])*(pol_flux[1]-I_stokes) - pol_eff[1]*np.cos(-2.*theta[0]+2.*theta[1])*(pol_flux[2]-I_stokes))
         dI_dtheta2 = 2.*pol_eff[1]/A*(pol_eff[0]*np.cos(-2.*theta[0]+2.*theta[1])*(pol_flux[2]-I_stokes) - pol_eff[2]*np.cos(-2.*theta[1]+2.*theta[2])*(pol_flux[0]-I_stokes))
         dI_dtheta3 = 2.*pol_eff[2]/A*(pol_eff[1]*np.cos(-2.*theta[1]+2.*theta[2])*(pol_flux[0]-I_stokes) - pol_eff[0]*np.cos(-2.*theta[2]+2.*theta[0])*(pol_flux[1]-I_stokes))
@@ -1153,10 +1159,12 @@ def compute_Stokes(data_array, error_array, data_mask, headers,
         dU_dtheta2 = 2.*pol_eff[1]/A*(np.sin(2.*theta[1])*(pol_flux[2]-pol_flux[0]) - (pol_eff[0]*np.cos(-2.*theta[0]+2.*theta[1]) - pol_eff[2]*np.cos(-2.*theta[1]+2.*theta[2]))*U_stokes)
         dU_dtheta3 = 2.*pol_eff[2]/A*(np.sin(2.*theta[2])*(pol_flux[0]-pol_flux[1]) - (pol_eff[1]*np.cos(-2.*theta[1]+2.*theta[2]) - pol_eff[0]*np.cos(-2.*theta[2]+2.*theta[0]))*U_stokes)
 
+        # Compute the uncertainty associated with the polarizers' orientation (see Kishimoto 1999)
         s_I2_axis = (dI_dtheta1**2*sigma_theta[0]**2 + dI_dtheta2**2*sigma_theta[1]**2 + dI_dtheta3**2*sigma_theta[2]**2)
         s_Q2_axis = (dQ_dtheta1**2*sigma_theta[0]**2 + dQ_dtheta2**2*sigma_theta[1]**2 + dQ_dtheta3**2*sigma_theta[2]**2)
         s_U2_axis = (dU_dtheta1**2*sigma_theta[0]**2 + dU_dtheta2**2*sigma_theta[1]**2 + dU_dtheta3**2*sigma_theta[2]**2)
 
+        # Add quadratically the uncertainty to the Stokes covariance matrix ## THIS IS WHERE THE PROBLEMATIC UNCERTAINTY IS ADDED TO THE PIPELINE
         Stokes_cov[0,0] += s_I2_axis
         Stokes_cov[1,1] += s_Q2_axis
         Stokes_cov[2,2] += s_U2_axis
@@ -1361,7 +1369,7 @@ def rotate_Stokes(I_stokes, Q_stokes, U_stokes, Stokes_cov, data_mask, headers,
 
 
     #Compute new covariance matrix on rotated parameters
-    new_Stokes_cov = copy.deepcopy(Stokes_cov)
+    new_Stokes_cov = deepcopy(Stokes_cov)
     new_Stokes_cov[1,1] = np.cos(2.*alpha)**2*Stokes_cov[1,1] + np.sin(2.*alpha)**2*Stokes_cov[2,2] + 2.*np.cos(2.*alpha)*np.sin(2.*alpha)*Stokes_cov[1,2]
     new_Stokes_cov[2,2] = np.sin(2.*alpha)**2*Stokes_cov[1,1] + np.cos(2.*alpha)**2*Stokes_cov[2,2] - 2.*np.cos(2.*alpha)*np.sin(2.*alpha)*Stokes_cov[1,2]
     new_Stokes_cov[0,1] = new_Stokes_cov[1,0] = np.cos(2.*alpha)*Stokes_cov[0,1] + np.sin(2.*alpha)*Stokes_cov[0,2]
@@ -1383,7 +1391,7 @@ def rotate_Stokes(I_stokes, Q_stokes, U_stokes, Stokes_cov, data_mask, headers,
     mrot = np.array([[np.cos(-alpha), -np.sin(-alpha)],
         [np.sin(-alpha), np.cos(-alpha)]])
     for header in headers:
-        new_header = copy.deepcopy(header)
+        new_header = deepcopy(header)
         new_header['orientat'] = header['orientat'] + ang
 
         new_wcs = WCS(header).deepcopy()