"""
Barten (1999) Contrast Sensitivity Function
===========================================
Defines the *Barten (1999)* contrast sensitivity function:
- :func:`colour.contrast.contrast_sensitivity_function_Barten1999`
References
----------
- :cite:`Barten1999` : Barten, P. G. (1999). Contrast Sensitivity of the
Human Eye and Its Effects on Image Quality. SPIE. doi:10.1117/3.353254
- :cite:`Barten2003` : Barten, P. G. J. (2003). Formula for the contrast
sensitivity of the human eye. In Y. Miyake & D. R. Rasmussen (Eds.),
Proceedings of SPIE (Vol. 5294, pp. 231-238). doi:10.1117/12.537476
- :cite:`Cowan2004` : Cowan, M., Kennel, G., Maier, T., & Walker, B. (2004).
Contrast Sensitivity Experiment to Determine the Bit Depth for Digital
Cinema. SMPTE Motion Imaging Journal, 113(9), 281-292. doi:10.5594/j11549
- :cite:`InternationalTelecommunicationUnion2015` : International
Telecommunication Union. (2015). Report ITU-R BT.2246-4 - The present
state of ultra-high definition television BT Series Broadcasting service
(Vol. 5, pp. 1-92).
https://www.itu.int/dms_pub/itu-r/opb/rep/R-REP-BT.2246-4-2015-PDF-E.pdf
- :cite:`Watson2012` : Watson, A. B., & Yellott, J. I. (2012). A unified
formula for light-adapted pupil size. Journal of Vision, 12(10), 12.
doi:10.1167/12.10.12
"""
from __future__ import annotations
import numpy as np
from colour.hints import ArrayLike, NDArrayFloat
from colour.utilities import as_float, as_float_array
__author__ = "Colour Developers"
__copyright__ = "Copyright 2013 Colour Developers"
__license__ = "BSD-3-Clause - https://opensource.org/licenses/BSD-3-Clause"
__maintainer__ = "Colour Developers"
__email__ = "colour-developers@colour-science.org"
__status__ = "Production"
__all__ = [
"optical_MTF_Barten1999",
"pupil_diameter_Barten1999",
"sigma_Barten1999",
"retinal_illuminance_Barten1999",
"maximum_angular_size_Barten1999",
"contrast_sensitivity_function_Barten1999",
]
[docs]
def optical_MTF_Barten1999(
u: ArrayLike, sigma: ArrayLike = 0.01
) -> NDArrayFloat:
"""
Return the optical modulation transfer function (MTF) :math:`M_{opt}` of
the eye using *Barten (1999)* method.
Parameters
----------
u
Spatial frequency :math:`u`, the cycles per degree.
sigma
Standard deviation :math:`\\sigma` of the line-spread function
resulting from the convolution of the different elements of the
convolution process.
Returns
-------
:class:`numpy.ndarray`
Optical modulation transfer function (MTF) :math:`M_{opt}` of the eye.
References
----------
:cite:`Barten1999`, :cite:`Barten2003`, :cite:`Cowan2004`,
:cite:`InternationalTelecommunicationUnion2015`,
Examples
--------
>>> optical_MTF_Barten1999(4, 0.01) # doctest: +ELLIPSIS
0.9689107...
"""
u = as_float_array(u)
sigma = as_float_array(sigma)
return as_float(np.exp(-2 * np.pi**2 * sigma**2 * u**2))
[docs]
def pupil_diameter_Barten1999(
L: ArrayLike,
X_0: ArrayLike = 60,
Y_0: ArrayLike | None = None,
) -> NDArrayFloat:
"""
Return the pupil diameter for given luminance and object or stimulus
angular size using *Barten (1999)* method.
Parameters
----------
L
Average luminance :math:`L` in :math:`cd/m^2`.
X_0
Angular size of the object :math:`X_0` in degrees in the x direction.
Y_0
Angular size of the object :math:`X_0` in degrees in the y direction.
Returns
-------
:class:`numpy.ndarray`
Pupil diameter.
References
----------
:cite:`Barten1999`, :cite:`Barten2003`, :cite:`Cowan2004`,
:cite:`InternationalTelecommunicationUnion2015`, :cite:`Watson2012`
Notes
-----
- The *Log* function is using base 10 as indicated by :cite:`Watson2012`.
Examples
--------
>>> pupil_diameter_Barten1999(100, 60, 60) # doctest: +ELLIPSIS
2.7931307...
"""
L = as_float_array(L)
X_0 = as_float_array(X_0)
Y_0 = X_0 if Y_0 is None else as_float_array(Y_0)
return as_float(5 - 3 * np.tanh(0.4 * np.log10(L * X_0 * Y_0 / 40**2)))
[docs]
def sigma_Barten1999(
sigma_0: ArrayLike = 0.5 / 60,
C_ab: ArrayLike = 0.08 / 60,
d: ArrayLike = 2.1,
) -> NDArrayFloat:
"""
Return the standard deviation :math:`\\sigma` of the line-spread function
resulting from the convolution of the different elements of the convolution
process using *Barten (1999)* method.
The :math:`\\sigma` quantity depends on the pupil diameter :math:`d` of the
eye lens. For very small pupil diameters, :math:`\\sigma` increases
inversely proportionally with pupil size because of diffraction, and for
large pupil diameters, :math:`\\sigma` increases about linearly with pupil
size because of chromatic aberration and others aberrations.
Parameters
----------
sigma_0
Constant :math:`\\sigma_{0}` in degrees.
C_ab
Spherical aberration of the eye :math:`C_{ab}` in
:math:`degrees\\div mm`.
d
Pupil diameter :math:`d` in millimeters.
Returns
-------
:class:`numpy.ndarray`
Standard deviation :math:`\\sigma` of the line-spread function
resulting from the convolution of the different elements of the
convolution process.
Warnings
--------
This definition expects :math:`\\sigma_{0}` and :math:`C_{ab}` to be given
in degrees and :math:`degrees\\div mm` respectively. However, in the
literature, the values for :math:`\\sigma_{0}` and
:math:`C_{ab}` are usually given in :math:`arc min` and
:math:`arc min\\div mm` respectively, thus they need to be divided by 60.
References
----------
:cite:`Barten1999`, :cite:`Barten2003`, :cite:`Cowan2004`,
:cite:`InternationalTelecommunicationUnion2015`,
Examples
--------
>>> sigma_Barten1999(0.5 / 60, 0.08 / 60, 2.1) # doctest: +ELLIPSIS
0.0087911...
"""
sigma_0 = as_float_array(sigma_0)
C_ab = as_float_array(C_ab)
d = as_float_array(d)
return as_float(np.hypot(sigma_0, C_ab * d))
[docs]
def retinal_illuminance_Barten1999(
L: ArrayLike,
d: ArrayLike = 2.1,
apply_stiles_crawford_effect_correction: bool = True,
) -> NDArrayFloat:
"""
Return the retinal illuminance :math:`E` in Trolands for given average
luminance :math:`L` and pupil diameter :math:`d` using *Barten (1999)*
method.
Parameters
----------
L
Average luminance :math:`L` in :math:`cd/m^2`.
d
Pupil diameter :math:`d` in millimeters.
apply_stiles_crawford_effect_correction
Whether to apply the correction for *Stiles-Crawford* effect.
Returns
-------
:class:`numpy.ndarray`
Retinal illuminance :math:`E` in Trolands.
Notes
-----
- This definition is for use with photopic viewing conditions and thus
corrects for the Stiles-Crawford effect by default, i.e. directional
sensitivity of the cone cells with lower response of cone cells
receiving light from the edge of the pupil.
References
----------
:cite:`Barten1999`, :cite:`Barten2003`, :cite:`Cowan2004`,
:cite:`InternationalTelecommunicationUnion2015`,
Examples
--------
>>> retinal_illuminance_Barten1999(100, 2.1) # doctest: +ELLIPSIS
330.4115803...
>>> retinal_illuminance_Barten1999(100, 2.1, False) # doctest: +ELLIPSIS
346.3605900...
"""
d = as_float_array(d)
L = as_float_array(L)
E = (np.pi * d**2) / 4 * L
if apply_stiles_crawford_effect_correction:
E *= 1 - (d / 9.7) ** 2 + (d / 12.4) ** 4
return as_float(E)
[docs]
def maximum_angular_size_Barten1999(
u: ArrayLike,
X_0: ArrayLike = 60,
X_max: ArrayLike = 12,
N_max: ArrayLike = 15,
) -> NDArrayFloat:
"""
Return the maximum angular size :math:`X` of the object considered using
*Barten (1999)* method.
Parameters
----------
u
Spatial frequency :math:`u`, the cycles per degree.
X_0
Angular size :math:`X_0` in degrees of the object in the x direction.
X_max
Maximum angular size :math:`X_{max}` in degrees of the integration
area in the x direction.
N_max
Maximum number of cycles :math:`N_{max}` over which the eye can
integrate the information.
Returns
-------
:class:`numpy.ndarray`
Maximum angular size :math:`X` of the object considered.
References
----------
:cite:`Barten1999`, :cite:`Barten2003`, :cite:`Cowan2004`,
:cite:`InternationalTelecommunicationUnion2015`,
Examples
--------
>>> maximum_angular_size_Barten1999(4) # doctest: +ELLIPSIS
3.5729480...
"""
u = as_float_array(u)
X_0 = as_float_array(X_0)
X_max = as_float_array(X_max)
N_max = as_float_array(N_max)
return as_float(
(1 / X_0**2 + 1 / X_max**2 + u**2 / N_max**2) ** -0.5
)
[docs]
def contrast_sensitivity_function_Barten1999(
u: ArrayLike,
sigma: ArrayLike = sigma_Barten1999(0.5 / 60, 0.08 / 60, 2.1),
k: ArrayLike = 3.0,
T: ArrayLike = 0.1,
X_0: ArrayLike = 60,
Y_0: ArrayLike | None = None,
X_max: ArrayLike = 12,
Y_max: ArrayLike | None = None,
N_max: ArrayLike = 15,
n: ArrayLike = 0.03,
p: ArrayLike = 1.2274 * 10**6,
E: ArrayLike = retinal_illuminance_Barten1999(20, 2.1),
phi_0: ArrayLike = 3 * 10**-8,
u_0: ArrayLike = 7,
) -> NDArrayFloat:
"""
Return the contrast sensitivity :math:`S` of the human eye according to
the contrast sensitivity function (CSF) described by *Barten (1999)*.
Contrast sensitivity is defined as the inverse of the modulation threshold
of a sinusoidal luminance pattern. The modulation threshold of this pattern
is generally defined by 50% probability of detection. The contrast
sensitivity function or CSF gives the contrast sensitivity as a function of
spatial frequency. In the CSF, the spatial frequency is expressed in
angular units with respect to the eye. It reaches a maximum between 1 and
10 cycles per degree with a fall off at higher and lower spatial
frequencies.
Parameters
----------
u
Spatial frequency :math:`u`, the cycles per degree.
sigma
Standard deviation :math:`\\sigma` of the line-spread function
resulting from the convolution of the different elements of the
convolution process.
k
Signal-to-noise (SNR) ratio :math:`k`.
T
Integration time :math:`T` in seconds of the eye.
X_0
Angular size :math:`X_0` in degrees of the object in the x direction.
Y_0
Angular size :math:`Y_0` in degrees of the object in the y direction.
X_max
Maximum angular size :math:`X_{max}` in degrees of the integration
area in the x direction.
Y_max
Maximum angular size :math:`Y_{max}` in degrees of the integration
area in the y direction.
N_max
Maximum number of cycles :math:`N_{max}` over which the eye can
integrate the information.
n
Quantum efficiency of the eye :math:`n`.
p
Photon conversion factor :math:`p` in
:math:`photons\\div seconds\\div degrees^2\\div Trolands` that
depends on the light source.
E
Retinal illuminance :math:`E` in Trolands.
phi_0
Spectral density :math:`\\phi_0` in :math:`seconds degrees^2` of the
neural noise.
u_0
Spatial frequency :math:`u_0` in :math:`cycles\\div degrees` above
which the lateral inhibition ceases.
Returns
-------
:class:`numpy.ndarray`
Contrast sensitivity :math:`S`.
Warnings
--------
This definition expects :math:`\\sigma_{0}` and :math:`C_{ab}` used in the
computation of :math:`\\sigma` to be given in degrees and
:math:`degrees\\div mm` respectively. However, in the literature, the
values for :math:`\\sigma_{0}` and :math:`C_{ab}` are usually given in
:math:`arc min` and :math:`arc min\\div mm` respectively, thus they need to
be divided by 60.
Notes
-----
- The formula holds for bilateral viewing and for equal dimensions of
the object in x and y direction. For monocular vision, the contrast
sensitivity is a factor :math:`\\sqrt{2}` smaller.
- *Barten (1999)* CSF default values for the :math:`k`,
:math:`\\sigma_{0}`, :math:`C_{ab}`, :math:`T`, :math:`X_{max}`,
:math:`N_{max}`, :math:`n`, :math:`\\phi_{0}` and :math:`u_0` constants
are valid for a standard observer with good vision and with an age
between 20 and 30 years.
- The other constants have been filled using reference data from
*Figure 31* in :cite:`InternationalTelecommunicationUnion2015` but
must be adapted to the current use case.
- The product of :math:`u`, the cycles per degree, and :math:`X_0`,
the number of degrees, gives the number of cycles :math:`P_c` in a
pattern. Therefore, :math:`X_0` can be made a variable dependent on
:math:`u` such as :math:`X_0 = P_c / u`.
References
----------
:cite:`Barten1999`, :cite:`Barten2003`, :cite:`Cowan2004`,
:cite:`InternationalTelecommunicationUnion2015`,
Examples
--------
>>> contrast_sensitivity_function_Barten1999(4) # doctest: +ELLIPSIS
360.8691122...
Reproducing *Figure 31* in \
:cite:`InternationalTelecommunicationUnion2015` illustrating the minimum
detectable contrast according to *Barten (1999)* model with the assumed
conditions for UHDTV applications. The minimum detectable contrast
:math:`MDC` is then defined as follows::
:math:`MDC = 1 / CSF * 2 * (1 / 1.27)`
where :math:`2` is used for the conversion from modulation to contrast and
:math:`1 / 1.27` is used for the conversion from sinusoidal to rectangular
waves.
>>> from scipy.optimize import fmin
>>> settings_BT2246 = {
... "k": 3.0,
... "T": 0.1,
... "X_max": 12,
... "N_max": 15,
... "n": 0.03,
... "p": 1.2274 * 10**6,
... "phi_0": 3 * 10**-8,
... "u_0": 7,
... }
>>>
>>> def maximise_spatial_frequency(L):
... maximised_spatial_frequency = []
... for L_v in L:
... X_0 = 60
... d = pupil_diameter_Barten1999(L_v, X_0)
... sigma = sigma_Barten1999(0.5 / 60, 0.08 / 60, d)
... E = retinal_illuminance_Barten1999(L_v, d, True)
... maximised_spatial_frequency.append(
... fmin(
... lambda x: (
... -contrast_sensitivity_function_Barten1999(
... u=x,
... sigma=sigma,
... X_0=X_0,
... E=E,
... **settings_BT2246
... )
... ),
... 0,
... disp=False,
... )[0]
... )
... return as_float(np.array(maximised_spatial_frequency))
...
>>>
>>> L = np.logspace(np.log10(0.01), np.log10(100), 10)
>>> X_0 = Y_0 = 60
>>> d = pupil_diameter_Barten1999(L, X_0, Y_0)
>>> sigma = sigma_Barten1999(0.5 / 60, 0.08 / 60, d)
>>> E = retinal_illuminance_Barten1999(L, d)
>>> u = maximise_spatial_frequency(L)
>>> (
... 1
... / contrast_sensitivity_function_Barten1999(
... u=u, sigma=sigma, E=E, X_0=X_0, Y_0=Y_0, **settings_BT2246
... )
... * 2
... * (1 / 1.27)
... )
... # doctest: +ELLIPSIS
array([ 0.0218764..., 0.0141848..., 0.0095244..., 0.0066805..., \
0.0049246...,
0.0038228..., 0.0031188..., 0.0026627..., 0.0023674..., \
0.0021814...])
"""
u = as_float_array(u)
k = as_float_array(k)
T = as_float_array(T)
X_0 = as_float_array(X_0)
Y_0 = X_0 if Y_0 is None else as_float_array(Y_0)
X_max = as_float_array(X_max)
Y_max = X_max if Y_max is None else as_float_array(Y_max)
N_max = as_float_array(N_max)
n = as_float_array(n)
p = as_float_array(p)
E = as_float_array(E)
phi_0 = as_float_array(phi_0)
u_0 = as_float_array(u_0)
M_opt = optical_MTF_Barten1999(u, sigma)
M_as = 1 / (
maximum_angular_size_Barten1999(u, X_0, X_max, N_max)
* maximum_angular_size_Barten1999(u, Y_0, Y_max, N_max)
)
S = (M_opt / k) / np.sqrt(
2
/ T
* M_as
* (1 / (n * p * E) + phi_0 / (1 - np.exp(-((u / u_0) ** 2))))
)
return as_float(S)