def symmM_to_posdefM(A = None, atol = 1.0e-9, rtol = 1.0e-9, method = 'make', forcesymm = True):
"""
Convert a symmetric matrix to a positive definite one.
Args:
:A:
| ndarray
:atol:
| float, optional
| The absolute tolerance parameter (see Notes of numpy.allclose())
:rtol:
| float, optional
| The relative tolerance parameter (see Notes of numpy.allclose())
:method:
| 'make' or 'nearest', optional (see notes for more info)
:forcesymm:
| True or False, optional
| If A is not symmetric, force symmetry using:
| A = numpy.triu(A) + numpy.triu(A).T - numpy.diag(numpy.diag(A))
Returns:
:returns:
| ndarray with positive-definite matrix.
Notes on supported methods:
1. `'make': A Python/Numpy port of Muhammad Asim Mubeen's matlab function
Spd_Mat.m
<https://nl.mathworks.com/matlabcentral/fileexchange/45873-positive-definite-matrix>`_
2. `'nearest': A Python/Numpy port of John D'Errico's `nearestSPD`
MATLAB code.
<https://stackoverflow.com/questions/43238173/python-convert-matrix-to-positive-semi-definite>`_
"""
if A is not None:
A = np2d(A)
# Make sure matrix A is symmetric up to a certain tolerance:
sn = check_symmetric(A, atol = atol, rtol = rtol)
if ((A.shape[0] != A.shape[1]) | (sn != True)):
if (forcesymm == True) & (A.shape[0] == A.shape[1]):
A = np.triu(A) + np.triu(A).T - np.diag(np.diag(A))
else:
raise Exception('symmM_to_posdefM(): matrix A not symmetric.')
if check_posdef(A, atol = atol, rtol = rtol) == True:
return A
else:
if method == 'make':
# A Python/Numpy port of Muhammad Asim Mubeen's matlab function Spd_Mat.m
#
# See: https://nl.mathworks.com/matlabcentral/fileexchange/45873-positive-definite-matrix
Val, Vec = np.linalg.eig(A)
Val = np.real(Val)
Vec = np.real(Vec)
Val[np.where(Val==0)] = _EPS #making zero eigenvalues non-zero
p = np.where(Val<0)
Val[p] = -Val[p] #making negative eigenvalues positive
return np.dot(Vec,np.dot(np.diag(Val) , Vec.T))
elif method == 'nearest':
# A Python/Numpy port of John D'Errico's `nearestSPD` MATLAB code [1], which
# credits [2].
#
# [1] https://www.mathworks.com/matlabcentral/fileexchange/42885-nearestspd
#
# [2] N.J. Higham, "Computing a nearest symmetric positive semidefinite
# matrix" (1988): https://doi.org/10.1016/0024-3795(88)90223-6
#
# See: https://stackoverflow.com/questions/43238173/python-convert-matrix-to-positive-semi-definite
B = (A + A.T) / 2.0
_, s, V = np.linalg.svd(B)
H = np.dot(V.T, np.dot(np.diag(s), V))
A2 = (B + H) / 2.0
A3 = (A2 + A2.T) / 2.0
if check_posdef(A3, atol = atol, rtol = rtol) == True:
return A3
spacing = np.spacing(np.linalg.norm(A))
I = np.eye(A.shape[0])
k = 1
while not check_posdef(A3, atol = atol, rtol = rtol):
mineig = np.min(np.real(np.linalg.eigvals(A3)))
A3 += I * (-mineig * k**2.0+ spacing)
k += 1
return A3
def cam18sl(data, datab = None, Lb = [100], fov = 10.0, inputtype = 'xyz', direction = 'forward', outin = 'Q,aW,bW', parameters = None):
"""
Convert between CIE 2006 10° XYZ tristimulus values (or spectral data)
and CAM18sl color appearance correlates.
Args:
:data:
| ndarray of CIE 2006 10° absolute XYZ tristimulus values or spectral data
or color appearance attributes of stimulus
:datab:
| ndarray of CIE 2006 10° absolute XYZ tristimulus values or spectral data
of stimulus background
:Lb:
| [100], optional
| Luminance (cd/m²) value(s) of background(s) calculated using the CIE 2006 10° CMFs
| (only used in case datab == None and the background is assumed to be an Equal-Energy-White)
:fov:
| 10.0, optional
| Field-of-view of stimulus (for size effect on brightness)
:inputtpe:
| 'xyz' or 'spd', optional
| Specifies the type of input:
| tristimulus values or spectral data for the forward mode.
:direction:
| 'forward' or 'inverse', optional
| -'forward': xyz -> cam18sl
| -'inverse': cam18sl -> xyz
:outin:
| 'Q,aW,bW' or str, optional
| 'Q,aW,bW' (brightness and opponent signals for amount-of-neutral)
| other options: 'Q,aM,bM' (colorfulness) and 'Q,aS,bS' (saturation)
| Str specifying the type of
| input (:direction: == 'inverse') and
| output (:direction: == 'forward')
:parameters:
| None or dict, optional
| Set of model parameters.
| - None: defaults to luxpy.cam._CAM18SL_PARAMETERS
| (see references below)
Returns:
:returns:
| ndarray with color appearance correlates (:direction: == 'forward')
| or
| XYZ tristimulus values (:direction: == 'inverse')
Notes:
| * Instead of using the CIE 1964 10° CMFs in some places of the model,
| the CIE 2006 10° CMFs are used througout, making it more self_consistent.
| This has an effect on the k scaling factors (now different those in CAM15u)
| and the illuminant E normalization for use in the chromatic adaptation transform.
| (see future erratum to Hermans et al., 2018)
| * The paper also used an equation for the amount of white W, which is
| based on a Q value not expressed in 'bright' ('cA' = 0.937 instead of 123).
| This has been corrected for in the luxpy version of the model, i.e.
| _CAM18SL_PARAMETERS['cW'][0] has been changed from 2.29 to 1/11672.
| (see future erratum to Hermans et al., 2018)
References:
1. `Hermans, S., Smet, K. A. G., & Hanselaer, P. (2018).
"Color appearance model for self-luminous stimuli."
Journal of the Optical Society of America A, 35(12), 2000–2009.
<https://doi.org/10.1364/JOSAA.35.002000>`_
"""
if parameters is None:
parameters = _CAM18SL_PARAMETERS
outin = outin.split(',')
#unpack model parameters:
cA, cAlms, cHK, cM, cW, ca, calms, cb, cblms, cfov, k, naka, unique_hue_data = [parameters[x] for x in sorted(parameters.keys())]
# precomputations:
Mlms2xyz = np.linalg.inv(_CMF['2006_10']['M'])
MAab = np.array([cAlms,calms,cblms])
invMAab = np.linalg.inv(MAab)
#-------------------------------------------------
# setup EEW reference field and default background field (Lr should be equal to Lb):
# Get Lb values:
if datab is not None:
if inputtype != 'xyz':
Lb = spd_to_xyz(datab, cieobs = '2006_10', relative = False)[...,1:2]
else:
Lb = datab[...,1:2]
else:
if isinstance(Lb,list):
Lb = np2dT(Lb)
# Setup EEW ref of same luminance as datab:
if inputtype == 'xyz':
wlr = getwlr(_CAM18SL_WL3)
else:
if datab is None:
wlr = data[0] # use wlr of stimulus data
else:
wlr = datab[0] # use wlr of background data
datar = np.vstack((wlr,np.ones((Lb.shape[0], wlr.shape[0])))) # create eew
xyzr = spd_to_xyz(datar, cieobs = '2006_10', relative = False) # get abs. tristimulus values
datar[1:] = datar[1:]/xyzr[...,1:2]*Lb
# Create datab if None:
if (datab is None):
if inputtype != 'xyz':
datab = datar.copy()
else:
datab = spd_to_xyz(datar, cieobs = '2006_10', relative = False)
datar = datab.copy()
# prepare data and datab for loop over backgrounds:
# make axis 1 of datab have 'same' dimensions as data:
if (data.ndim == 2):
data = np.expand_dims(data, axis = 1) # add light source axis 1
if inputtype == 'xyz':
if datab.shape[0] == 1: #make datab and datar have same lights source dimension (used to store different backgrounds) size as data
datab = np.repeat(datab,data.shape[1],axis=0)
datar = np.repeat(datar,data.shape[1],axis=0)
else:
if datab.shape[0] == 2:
datab = np.vstack((datab[0],np.repeat(datab[1:], data.shape[1], axis = 0)))
if datar.shape[0] == 2:
datar = np.vstack((datar[0],np.repeat(datar[1:], data.shape[1], axis = 0)))
# Flip light source/ background dim to axis 0:
data = np.transpose(data, axes = (1,0,2))
#-------------------------------------------------
#initialize camout:
dshape = list(data.shape)
dshape[-1] = len(outin) # requested number of correlates
if (inputtype != 'xyz') & (direction == 'forward'):
dshape[-2] = dshape[-2] - 1 # wavelength row doesn't count & only with forward can the input data be spectral
camout = np.nan*np.ones(dshape)
for i in range(data.shape[0]):
# get rho, gamma, beta of background and reference white:
if (inputtype != 'xyz'):
xyzb = spd_to_xyz(np.vstack((datab[0], datab[i+1:i+2,:])), cieobs = '2006_10', relative = False)
xyzr = spd_to_xyz(np.vstack((datar[0], datar[i+1:i+2,:])), cieobs = '2006_10', relative = False)
else:
xyzb = datab[i:i+1,:]
xyzr = datar[i:i+1,:]
lmsb = np.dot(_CMF['2006_10']['M'],xyzb.T).T # convert to l,m,s
rgbb = (lmsb / _CMF['2006_10']['K']) * k # convert to rho, gamma, beta
#lmsr = np.dot(_CMF['2006_10']['M'],xyzr.T).T # convert to l,m,s
#rgbr = (lmsr / _CMF['2006_10']['K']) * k # convert to rho, gamma, beta
#rgbr = rgbr/rgbr[...,1:2]*Lb[i] # calculated EEW cone excitations at same luminance values as background
rgbr = np.ones(xyzr.shape)*Lb[i] # explicitely equal EEW cone excitations at same luminance values as background
if direction == 'forward':
# get rho, gamma, beta of stimulus:
if (inputtype != 'xyz'):
xyz = spd_to_xyz(data[i], cieobs = '2006_10', relative = False)
elif (inputtype == 'xyz'):
xyz = data[i]
lms = np.dot(_CMF['2006_10']['M'],xyz.T).T # convert to l,m,s
rgb = (lms / _CMF['2006_10']['K']) * k # convert to rho, gamma, beta
# apply von-kries cat with D = 1:
if (rgbb == 0).any():
Mcat = np.eye(3)
else:
Mcat = np.diag((rgbr/rgbb)[0])
rgba = np.dot(Mcat,rgb.T).T
# apply naka-rushton compression:
rgbc = naka_rushton(rgba, n = naka['n'], sig = naka['sig'](rgbr.mean()), noise = naka['noise'], scaling = naka['scaling'])
#rgbc = np.ones(rgbc.shape)*rgbc.mean() # test if eew ends up at origin
# calculate achromatic and color difference signals, A, a, b:
Aab = np.dot(MAab, rgbc.T).T
A,a,b = asplit(Aab)
a = ca*a
b = cb*b
# calculate colorfullness like signal M:
M = cM*((a**2.0 + b**2.0)**0.5)
# calculate brightness Q:
Q = cA*(A + cHK[0]*M**cHK[1]) # last term is contribution of Helmholtz-Kohlrausch effect on brightness
# calculate saturation, s:
s = M / Q
# calculate amount of white, W:
W = 1 / (1.0 + cW[0]*(s**cW[1]))
# adjust Q for size (fov) of stimulus (matter of debate whether to do this before or after calculation of s or W, there was no data on s, M or W for different sized stimuli: after)
Q = Q*(fov/10.0)**cfov
# calculate hue, h and Hue quadrature, H:
h = hue_angle(a,b, htype = 'deg')
if 'H' in outin:
H = hue_quadrature(h, unique_hue_data = unique_hue_data)
else:
H = None
# calculate cart. co.:
if 'aM' in outin:
aM = M*np.cos(h*np.pi/180.0)
bM = M*np.sin(h*np.pi/180.0)
if 'aS' in outin:
aS = s*np.cos(h*np.pi/180.0)
bS = s*np.sin(h*np.pi/180.0)
if 'aW' in outin:
aW = W*np.cos(h*np.pi/180.0)
bW = W*np.sin(h*np.pi/180.0)
if (outin != ['Q','aW','bW']):
camout[i] = eval('ajoin(('+','.join(outin)+'))')
else:
camout[i] = ajoin((Q,aW,bW))
elif direction == 'inverse':
# get Q, M and a, b depending on input type:
if 'aW' in outin:
Q,a,b = asplit(data[i])
Q = Q / ((fov/10.0)**cfov) #adjust Q for size (fov) of stimulus back to that 10° ref
W = (a**2.0 + b**2.0)**0.5
s = (((1.0 / W) - 1.0)/cW[0])**(1.0/cW[1])
M = s*Q
if 'aM' in outin:
Q,a,b = asplit(data[i])
Q = Q / ((fov/10.0)**cfov) #adjust Q for size (fov) of stimulus back to that 10° ref
M = (a**2.0 + b**2.0)**0.5
if 'aS' in outin:
Q,a,b = asplit(data[i])
Q = Q / ((fov/10.0)**cfov) #adjust Q for size (fov) of stimulus back to that 10° ref
s = (a**2.0 + b**2.0)**0.5
M = s*Q
if 'h' in outin:
Q, WsM, h = asplit(data[i])
Q = Q / ((fov/10.0)**cfov) #adjust Q for size (fov) of stimulus back to that 10° ref
if 'W' in outin:
s = (((1.0 / WsM) - 1.0)/cW[0])**(1.0/cW[1])
M = s*Q
elif 's' in outin:
M = WsM*Q
elif 'M' in outin:
M = WsM
# calculate achromatic signal, A from Q and M:
A = Q/cA - cHK[0]*M**cHK[1]
# calculate hue angle:
h = hue_angle(a,b, htype = 'rad')
# calculate a,b from M and h:
a = (M/cM)*np.cos(h)
b = (M/cM)*np.sin(h)
a = a/ca
b = b/cb
# create Aab:
Aab = ajoin((A,a,b))
# calculate rgbc:
rgbc = np.dot(invMAab, Aab.T).T
# decompress rgbc to (adapted) rgba :
rgba = naka_rushton(rgbc, n = naka['n'], sig = naka['sig'](rgbr.mean()), noise = naka['noise'], scaling = naka['scaling'], direction = 'inverse')
# apply inverse von-kries cat with D = 1:
rgb = np.dot(np.diag((rgbb/rgbr)[0]),rgba.T).T
# convert rgb to lms to xyz:
lms = rgb/k*_CMF['2006_10']['K']
xyz = np.dot(Mlms2xyz,lms.T).T
camout[i] = xyz
if camout.shape[0] == 1:
camout = np.squeeze(camout,axis = 0)
return camout
def cam_sww16(data, dataw = None, Yb = 20.0, Lw = 400.0, Ccwb = None, relative = True, \
parameters = None, inputtype = 'xyz', direction = 'forward', \
cieobs = '2006_10'):
"""
A simple principled color appearance model based on a mapping
of the Munsell color system.
| This function implements the JOSA A (parameters = 'JOSA') published model.
Args:
:data:
| ndarray with input tristimulus values
| or spectral data
| or input color appearance correlates
| Can be of shape: (N [, xM], x 3), whereby:
| N refers to samples and M refers to light sources.
| Note that for spectral input shape is (N x (M+1) x wl)
:dataw:
| None or ndarray, optional
| Input tristimulus values or spectral data of white point.
| None defaults to the use of CIE illuminant C.
:Yb:
| 20.0, optional
| Luminance factor of background (perfect white diffuser, Yw = 100)
:Lw:
| 400.0, optional
| Luminance (cd/m²) of white point.
:Ccwb:
| None, optional
| Degree of cognitive adaptation (white point balancing)
| If None: use [..,..] from parameters dict.
:relative:
| True or False, optional
| True: xyz tristimulus values are relative (Yw = 100)
:parameters:
| None or str or dict, optional
| Dict with model parameters.
| - None: defaults to luxpy.cam._CAM_SWW_2016_PARAMETERS['JOSA']
| - str: 'best-fit-JOSA' or 'best-fit-all-Munsell'
| - dict: user defined model parameters
| (dict should have same structure)
:inputtype:
| 'xyz' or 'spd', optional
| Specifies the type of input:
| tristimulus values or spectral data for the forward mode.
:direction:
| 'forward' or 'inverse', optional
| -'forward': xyz -> cam_sww_2016
| -'inverse': cam_sww_2016 -> xyz
:cieobs:
| '2006_10', optional
| CMF set to use to perform calculations where spectral data
is involved (inputtype == 'spd'; dataw = None)
| Other options: see luxpy._CMF['types']
Returns:
:returns:
| ndarray with color appearance correlates (:direction: == 'forward')
| or
| XYZ tristimulus values (:direction: == 'inverse')
Notes:
| This function implements the JOSA A (parameters = 'JOSA')
published model.
| With:
| 1. A correction for the parameter
| in Eq.4 of Fig. 11: 0.952 --> -0.952
|
| 2. The delta_ac and delta_bc white-balance shifts in Eq. 5e & 5f
| should be: -0.028 & 0.821
|
| (cfr. Ccwb = 0.66 in:
| ab_test_out = ab_test_int - Ccwb*ab_gray_adaptation_field_int))
References:
1. `Smet, K. A. G., Webster, M. A., & Whitehead, L. A. (2016).
A simple principled approach for modeling and understanding uniform color metrics.
Journal of the Optical Society of America A, 33(3), A319–A331.
<https://doi.org/10.1364/JOSAA.33.00A319>`_
"""
# get model parameters
args = locals().copy()
if parameters is None:
parameters = _CAM_SWW16_PARAMETERS['JOSA']
if isinstance(parameters,str):
parameters = _CAM_SWW16_PARAMETERS[parameters]
parameters = put_args_in_db(parameters,args) #overwrite parameters with other (not-None) args input
#unpack model parameters:
Cc, Ccwb, Cf, Mxyz2lms, cLMS, cab_int, cab_out, calpha, cbeta,cga1, cga2, cgb1, cgb2, cl_int, clambda, lms0 = [parameters[x] for x in sorted(parameters.keys())]
# setup default adaptation field:
if (dataw is None):
dataw = _CIE_ILLUMINANTS['C'].copy() # get illuminant C
xyzw = spd_to_xyz(dataw, cieobs = cieobs,relative=False) # get abs. tristimulus values
if relative == False: #input is expected to be absolute
dataw[1:] = Lw*dataw[1:]/xyzw[:,1:2] #dataw = Lw*dataw # make absolute
else:
dataw = dataw # make relative (Y=100)
if inputtype == 'xyz':
dataw = spd_to_xyz(dataw, cieobs = cieobs, relative = relative)
# precomputations:
Mxyz2lms = np.dot(np.diag(cLMS),math.normalize_3x3_matrix(Mxyz2lms, np.array([[1, 1, 1]]))) # normalize matrix for xyz-> lms conversion to ill. E weighted with cLMS
invMxyz2lms = np.linalg.inv(Mxyz2lms)
MAab = np.array([clambda,calpha,cbeta])
invMAab = np.linalg.inv(MAab)
#initialize data and camout:
data = np2d(data).copy() # stimulus data (can be upto NxMx3 for xyz, or [N x (M+1) x wl] for spd))
dataw = np2d(dataw).copy() # white point (can be upto Nx3 for xyz, or [(N+1) x wl] for spd)
# make axis 1 of dataw have 'same' dimensions as data:
if (data.ndim == 2):
data = np.expand_dims(data, axis = 1) # add light source axis 1
if inputtype == 'xyz':
if dataw.shape[0] == 1: #make dataw have same lights source dimension size as data
dataw = np.repeat(dataw,data.shape[1],axis=0)
else:
if dataw.shape[0] == 2:
dataw = np.vstack((dataw[0],np.repeat(dataw[1:], data.shape[1], axis = 0)))
# Flip light source dim to axis 0:
data = np.transpose(data, axes = (1,0,2))
# Initialize output array:
dshape = list(data.shape)
dshape[-1] = 3 # requested number of correlates: l_int, a_int, b_int
if (inputtype != 'xyz') & (direction == 'forward'):
dshape[-2] = dshape[-2] - 1 # wavelength row doesn't count & only with forward can the input data be spectral
camout = np.nan*np.ones(dshape)
# apply forward/inverse model for each row in data:
for i in range(data.shape[0]):
# stage 1: calculate photon rates of stimulus and adapting field, lmst & lmsf:
if (inputtype != 'xyz'):
if relative == True:
xyzw_abs = spd_to_xyz(np.vstack((dataw[0],dataw[i+1])), cieobs = cieobs, relative = False)
dataw[i+1] = Lw*dataw[i+1]/xyzw_abs[0,1] # make absolute
xyzw = spd_to_xyz(np.vstack((dataw[0],dataw[i+1])), cieobs = cieobs, relative = False)
lmsw = 683.0*np.dot(Mxyz2lms,xyzw.T).T/_CMF[cieobs]['K']
lmsf = (Yb/100.0)*lmsw # calculate adaptation field and convert to l,m,s
if (direction == 'forward'):
if relative == True:
data[i,1:,:] = Lw*data[i,1:,:]/xyzw_abs[0,1] # make absolute
xyzt = spd_to_xyz(data[i], cieobs = cieobs, relative = False)/_CMF[cieobs]['K']
lmst = 683.0*np.dot(Mxyz2lms,xyzt.T).T # convert to l,m,s
else:
lmst = lmsf # put lmsf in lmst for inverse-mode
elif (inputtype == 'xyz'):
if relative == True:
dataw[i] = Lw*dataw[i]/100.0 # make absolute
lmsw = 683.0* np.dot(Mxyz2lms, dataw[i].T).T /_CMF[cieobs]['K'] # convert to lms
lmsf = (Yb/100.0)*lmsw
if (direction == 'forward'):
if relative == True:
data[i] = Lw*data[i]/100.0 # make absolute
lmst = 683.0* np.dot(Mxyz2lms, data[i].T).T /_CMF[cieobs]['K'] # convert to lms
else:
lmst = lmsf # put lmsf in lmst for inverse-mode
# stage 2: calculate cone outputs of stimulus lmstp
lmstp = math.erf(Cc*(np.log(lmst/lms0) + Cf*np.log(lmsf/lms0)))
lmsfp = math.erf(Cc*(np.log(lmsf/lms0) + Cf*np.log(lmsf/lms0)))
lmstp = np.vstack((lmsfp,lmstp)) # add adaptation field lms temporarily to lmsp for quick calculation
# stage 3: calculate optic nerve signals, lam*, alphp, betp:
lstar,alph, bet = asplit(np.dot(MAab, lmstp.T).T)
alphp = cga1[0]*alph
alphp[alph<0] = cga1[1]*alph[alph<0]
betp = cgb1[0]*bet
betp[bet<0] = cgb1[1]*bet[bet<0]
# stage 4: calculate recoded nerve signals, alphapp, betapp:
alphpp = cga2[0]*(alphp + betp)
betpp = cgb2[0]*(alphp - betp)
# stage 5: calculate conscious color perception:
lstar_int = cl_int[0]*(lstar + cl_int[1])
alph_int = cab_int[0]*(np.cos(cab_int[1]*np.pi/180.0)*alphpp - np.sin(cab_int[1]*np.pi/180.0)*betpp)
bet_int = cab_int[0]*(np.sin(cab_int[1]*np.pi/180.0)*alphpp + np.cos(cab_int[1]*np.pi/180.0)*betpp)
lstar_out = lstar_int
if direction == 'forward':
if Ccwb is None:
alph_out = alph_int - cab_out[0]
bet_out = bet_int - cab_out[1]
else:
Ccwb = Ccwb*np.ones((2))
Ccwb[Ccwb<0.0] = 0.0
Ccwb[Ccwb>1.0] = 1.0
alph_out = alph_int - Ccwb[0]*alph_int[0] # white balance shift using adaptation gray background (Yb=20%), with Ccw: degree of adaptation
bet_out = bet_int - Ccwb[1]*bet_int[0]
camout[i] = np.vstack((lstar_out[1:],alph_out[1:],bet_out[1:])).T # stack together and remove adaptation field from vertical stack
elif direction == 'inverse':
labf_int = np.hstack((lstar_int[0],alph_int[0],bet_int[0]))
# get lstar_out, alph_out & bet_out for data:
lstar_out, alph_out, bet_out = asplit(data[i])
# stage 5 inverse:
# undo cortical white-balance:
if Ccwb is None:
alph_int = alph_out + cab_out[0]
bet_int = bet_out + cab_out[1]
else:
Ccwb = Ccwb*np.ones((2))
Ccwb[Ccwb<0.0] = 0.0
Ccwb[Ccwb>1.0] = 1.0
alph_int = alph_out + Ccwb[0]*alph_int[0] # inverse white balance shift using adaptation gray background (Yb=20%), with Ccw: degree of adaptation
bet_int = bet_out + Ccwb[1]*bet_int[0]
lstar_int = lstar_out
alphpp = (1.0 / cab_int[0]) * (np.cos(-cab_int[1]*np.pi/180.0)*alph_int - np.sin(-cab_int[1]*np.pi/180.0)*bet_int)
betpp = (1.0 / cab_int[0]) * (np.sin(-cab_int[1]*np.pi/180.0)*alph_int + np.cos(-cab_int[1]*np.pi/180.0)*bet_int)
lstar_int = lstar_out
lstar = (lstar_int /cl_int[0]) - cl_int[1]
# stage 4 inverse:
alphp = 0.5*(alphpp/cga2[0] + betpp/cgb2[0]) # <-- alphpp = (Cga2.*(alphp+betp));
betp = 0.5*(alphpp/cga2[0] - betpp/cgb2[0]) # <-- betpp = (Cgb2.*(alphp-betp));
# stage 3 invers:
alph = alphp/cga1[0]
bet = betp/cgb1[0]
sa = np.sign(cga1[1])
sb = np.sign(cgb1[1])
alph[(sa*alphp)<0.0] = alphp[(sa*alphp)<0] / cga1[1]
bet[(sb*betp)<0.0] = betp[(sb*betp)<0] / cgb1[1]
lab = ajoin((lstar, alph, bet))
# stage 2 inverse:
lmstp = np.dot(invMAab,lab.T).T
lmstp[lmstp<-1.0] = -1.0
lmstp[lmstp>1.0] = 1.0
lmstp = math.erfinv(lmstp) / Cc - Cf*np.log(lmsf/lms0)
lmst = np.exp(lmstp) * lms0
# stage 1 inverse:
xyzt = np.dot(invMxyz2lms,lmst.T).T
if relative == True:
xyzt = (100.0/Lw) * xyzt
camout[i] = xyzt
# if flipaxis0and1 == True: # loop over shortest dim.
# camout = np.transpose(camout, axes = (1,0,2))
# Flip light source dim back to axis 1:
camout = np.transpose(camout, axes = (1,0,2))
if camout.shape[0] == 1:
camout = np.squeeze(camout,axis = 0)
return camout