This research is
sponsored by
Grant
IIS-0308155
within
the Universal Access Program
of
the National Science Foundation
Pre-Compensation for High-Order Aberrations of the Human Eye
Using On-Screen Image Deconvolution
M. Alonso Jr.1,
A. B. Barreto1,2
1Department
of Electrical and Computer Engineering, 2 Department of Biomedical
Engineering
University Park, Miami, Florida, 33199
Abstract -
Some
individuals with severe visual impairments may have difficulty
in interacting with computers, even when using traditional means
of visual correction (e.g., spectacles, contact lenses). This
is, in part, because these correction mechanisms can only
compensate for the most regular distortions or aberrations of
the image in the eye. This paper proposes an image processing
approach that will pre-compensate the images displayed on the
computer screen, so as to counter the effect of the eye’s
aberrations on the image. The characterization of the eye
required to perform this customized pre-compensation is the
eye’s Point Spread Function (PSF). The PSF can now be measured by a new generation of
ophthalmic instruments generically called “Wavefront
Analyzers.” The characterization provided by these instruments
also includes the “higher-order aberration components” and
could, therefore, lead to a more comprehensive vision
correction approach than traditional mechanisms. The methods
presented here will be explained in terms of their theoretical
foundation and illustrated with results from the correction of
aberrations introduced by a lens with known and constant PSF.
I. Introduction
Visual perception is the
primary source of information about the surrounding environment
that humans have. Through the evolutionary process, humans have
developed a very refined sense of vision, seeing in three
dimensions with high resolution and color [1]. The sense of
sight is so important that, when limited by age or disease, it
limits a person’s ability to perform otherwise ordinary tasks,
such as interacting with a computer system.
Visual perception of
objects in the physical world is determined by the formation of
their images on the retina, located inside the human eye [2].
However, the natural visual system of some individuals does not
accurately map images from the outside world onto their retinas.
The most common of these visual impairments include myopia,
hyperopia, and astigmatism. All of them result in a retinal
representation of a point of light that is not confined to a
single point on the retina. The distortion in the mapping of
external images onto the retina is represented by the eye’s
“wavefront aberration function.” Myopia, hyperopia, and
astigmatism are referred to as low-order aberrations because the
distortion they introduce can be modeled by first or second
order Zernike polynomials. Currently, spectacles or contact
lenses correct these low-order impairments by modifying external
images before they reach the eye.
There are cases,
however, in which the low-order Zernike model is not sufficient
to describe the aberration of the eye. With the recent advances
in wavefront sensing technology, it is now feasible to
accurately model the high-order aberrations present in each
person’s eye and thus obtain good models of various aberrations
currently not correctable through conventional means (e.g.,
spectacles, contact lenses). Using these models, it would be
possible to provide persons with currently uncorrectable
aberrations, a new alternative to enhance their interaction with
computers.
In contrast with the optical correction of visual limitations,
the approach described here is based on modifying the image at
its source, i.e., in applying image processing modifications on
the image to be displayed on-screen before it is shown to the
user, based on the knowledge of his/her wavefront aberration
function. The aim of the pre-compensation proposed is to modify
the intended display image in a way that is opposite to the
effect of the wavefront aberration of the eye. Once this is
achieved, the result is displayed to the viewer so that the
effect of the wavefront aberration in the viewer’s eye will
“cancel” our pre-compensation resulting in the projection of an
undistorted version of the intended image on the retina
II.
Wavefront aberration and Zernike model
It should be noted that, even if
the focusing elements of the eye were perfect, i.e., if there
were no aberrations present to distort the image, the retinal
image would still contain degradation due to the diffraction of
light as it passes through the pupil (diffraction limited) [2].
For a perfectly round pupil, the diffraction pattern of a point
source of light appears as a bright spot in the center of the
retina, surrounded by faint concentric rings. This is known as
the Airy disk [3].
If diffraction is neglected, the
light from a point source that enters an ideal eye, free of
aberrations, will converge to a focal point on the retina [3].
The Zernike polynomials are
two-dimensional functions that form a complete orthogonal basis
set defined on the unit disc. They have been used in optical
engineering for over 60 years, [6], [7]. Recently, the Zernike
polynomials have been applied to the characterization of
aberrations of the human eye [8], [9], [10]. Modern ophthalmic
wavefront analyzers provide an approximation to the wavefront
aberration function measured from the subject as a combination
of Zernike polynomials. As indicated by equations (5), (6), and
(7), this characterization of the optics of the eye can be used
to determine its PSF and OTF, thus enabling the implementation
of the deconvolution concept expressed in equations (3) and (4),
as detailed below.
III.
Imaging in the
Human Eye as a Convolution Process
A
recent shift in the interpretation of optics has been in
the characterization of optical systems, including the human
eye, as linear, shift invariant systems described by their point
spread function (PSF) [2]. Therefore, the eye can be
characterized by its PSF, T(x, y), and the retinal image, R(x,
y), can be found by convolving (denoted by ) the input to the system, i.e., the image of
the object viewed, I(x, y), with the PSF of the eye:
(1)
When
viewing a computer screen, the real world image, I(x, y), is
stored as a digital image, DI(x, y), which will be displayed to
the user. The retinal image, in this case, will be formed by
convolution of the on-screen image and the PSF of the viewer’s
eye:
(2)
In this
case, however, the on-screen image to be displayed to the user
can be manipulated in advance. If the inverse PSF of the
viewer’s eye, T-1(x, y), could be defined, then an
image, RD(x,y), which is the result of convolving the intended
digital image, DI(x, y), with this inverse function could be
shown to the viewer:
(3)
Under these circumstances,
substituting the ordinary DI(x,y) with the pre-compensated
RD(x,y), in (2):
(4)
This means that the user would perceive in R(x,y) an undistorted
version of the intended digital image, DI(x,y). The image
represented by the braces in (4) can also be considered to be
the result of deconvolving the PSF, T(x,y), from the intended
image, DI(x,y). This is the pre-compensated image to be
displayed to the viewer. In practice, the deconvolution process
may be performed more efficiently in the frequency domain, as
discussed below. In any case, it is clear that, in order to
obtain RD(x, y), the PSF of the eye must be known or estimated.
IV.
Evaluation of the PSF
of the Human Eye
The PSF of the
human eye can be found indirectly through what is known as the
wavefront aberration function, W(x, y). This function
represents the deviation of the light wavefront from a purely
spherical pattern as it passes the pupil on its way to the
retina [3]. In an unaberrated eye, the refracted light is
organized in the form of a uniform spherical wavefront,
converging to the paraxial focal point on the retina.
Recently developed “Wavefront Analyzers” based on
the Hartmann-Shack Principle have made it possible to measure
the wavefront aberration function for the human eye. The
wavefront aberration function, W(x,y), is the primary component
of the pupil function, P(x,y). The pupil function incorporates
the complete information about the imaging properties of the
optical system [4], and is defined as:
(5)
where A(x,y) is the amplitude function describing
the relative efficiency of light passing through the pupil
(usually given a value of one), and n is the index of
refraction [3].
According to the Fourier optics relationships in
the eye, knowledge of the pupil function can be used to
determine the optical transfer function, OTF, which is “one of
the most powerful descriptors of imaging performance for an
optical system” [3]. The OTF is a complex function whose
magnitude is the modulation transfer function (MTF) and whose
phase is the phase transfer function (PTF) [3].
The OTF can be
found by convolving the pupil function, P(x,y), with its complex
conjugate, P*(-x,-y) [2],[3]:
(6)
which is equivalent of saying that the OTF is the
autocorrelation of the pupil function. The PSF and the OTF are a
Fourier transform pair [5]:
(7)
V.
Inverse Filtering
Inverse filtering
is traditionally used in image processing to restore an image
U(x,y) from a degraded image G(x,y), assuming a known
degradation function, H(x,y). As stated above, the input and
output of any linear system are related through the convolution
operator. That is,
(8)
If the Fourier principles of convolution are
applied, U(x,y) can be obtained as follows:
(9)
where
 and
 denote the Fourier transform and the inverse
Fourier transform, respectively.
In the context of
pre-compensation of a digital image to be shown to the viewer,
the objective is to deconvolve the PSF of the viewer’s eye, T(x,
y), from the intended digital image, DI(x, y), in order to
derive the pre-compensated display image to show on-screen, RD(x,
y). Making the corresponding substitutions in equation (9), the
calculation of RD(x, y) is practically accomplished as:
(10)
This process is
commonly known as inverse filtering in frequency, and is
equivalent to two-dimensional deconvolution in the spatial
domain. It should be noted that, according to equation (7), the
denominator within the braces of equation (10) is the OTF of the
viewer’s eye.
But this implementation of inverse filtering has
several limitations, especially for values of the PSF or its
frequency counterpart, the OTF, which are close to zero. A
common approach to circumvent this problem is the use of Minimum
Mean Square Error filtering [11]. In the context of the problem
at hand, this approach obtains the pre-compensated image as:
(11)
where RD(fx,fy), T(fx,fy), and DI(fx,fy) are the Fourier
transforms of RD(x,y), T(x,y), and DI(x,y), respectively. In
addition to the limitation of values in the pre-compensated
image, the approach indicated in equation (11) helps reduce the
impact of modeling or measurement noise inherent to the
definition of the eye’s PSF. The K factor in equation (11)
should be proportional to the magnitude of the estimation noise
in the PSF approximation.
VI. Results and Discussion
Quantification of the results was done with the participation of
human subjects. In order to proceed with the testing of this
method, human subjects having a satisfactory standard visual
acuity of 20/20 [13], either corrected or natural, were
recruited. Then, a refractive error, specifically defocus, was
artificially introduced in their field of view by means of a
pre-selected lens. A similar procedure was used by Sonksen to
test visual function [14].
In order to provide a well-known and
standardized method of reporting visual performance, a standard
Eye Test Chart was is used. The characters of a Bailey-Lovie
visual acuity chart with ‘Sloan’ letters (Figure 1) were
reproduced as properly sized digital images for the tests.
The procedure outlined above for obtaining the
PSF of a -6.0 diopter (D) lens, as well as for generating the
pre-compensated image, was accomplished using MATLAB. Figure 1
shows the analytical PSF obtained for a -6.0 D lens, scaled and
resized. Additionally, this image has been cropped and padded to
match the image size of a line from a standard eye test chart,
shown in Figure 3.
This
is necessary due to the fact that both the image and the PSF
need to be of the same size in order to implement the minimum
mean square error deconvolution described. Another key factor to
this process is that the eye test chart lines, when displayed
on-screen, are equivalent in size to an actual eye test chart.
This is important because it ensures that the test can be
administered in a standard fashion.
Figure 4 represents the degraded image that would be seen
through the blur defined by the PSF in Figure 2. Figure 5 shows
the result of applying the pre-compensation process to the image
in Figure 3. This is the image that compensates for the
aberration introduced by the -6.0 D lens. This image was
produced using equation (11) with a K value of 0.0005. Figure 6,
on the other hand, shows the result of viewing the
pre-compensated image (Figure 5) through the same blur.
Figure 1-
Bailey-Lovie Eye Test Chart
Figure
2-
Analytical PSF of the –6.0 D lens used in the tests.
Figure
3-
Digital Representation of one line from a standard eye test
chart
Figure
4-
Simulated blur caused by the PSF in Figure 1
Figure
5-
Resulting pre-compensated image
Figure
6-
Simulated image of what the person would see when viewing Figure
3 through the -6.0 diopter lens.
The results of testing with human subjects are
shown in Figure 7. This figure shows the visual acuity in logMAR
units for fourteen subjects. Each of their eyes was studied
independently resulting in twentyeight visual acuity scores.
With respect to Figure 7, the bottom trace indicates the visual
acuities obtained for the eye test without any blur or
pre-compensation. The top trace indicates the values through the
blur, and the center trace represents the values obtain from the
pre-compensated eye test. A smaller value of visual acuity
indicates the ability of the subject to read further down the
chart. For reference, in a standard eye test, a logMAR value of
1.4 at a viewing distance of two meters, represents a misreading
of all of the largest letters on the eye test chart.
As can be seen from Figure 7, the
pre-compensation partially restored the vision of the subjects,
allowing them to read further down the eye test chart.
VII. Conclusion
This paper has proposed a
framework for pre-compensation of digital images before they are
displayed to low-vision users on a computer screen in order to
compensate for their visual limitations. The results shown in
Figure 7 demonstrates the promise of this approach, in that the
methods presented in this paper, when applied to digital
on-screen images, partially restore the vision of subjects with
an artificial, second-order aberration introduced into their
field of view. Although this example pre-compensates for a
low-order aberration, it should be noted that this framework
also contemplates the correction of higher-order optical
aberrations that are not addressed by current visual correction
methods.
Figure
7-
Results with human subjects
VIII. References.
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