Image Pre-compensation for Ocular Aberrations
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On-screen image pre-compensation has good prospect with the increasing usage of various display screen devices in our daily life. Comparing to glasses, contact glasses and ocular surgery, on-screen image pre-compensation can be easily carried out by computer calculation without any irreversible change in the eyes, as long as the ocular aberration is known. Further, since neither contact lenses nor glasses are advised to be worn all of the time, on screen pre-compensation could even supplement glasses and contact lens use. It is known that compensation for higher aberrations can lead to 'super-sight', which is the neural limit of human eye. On-screen compensation also has the prospect of achieving this with customized screens in the foreseeable future.
- Image Processing Theories
- Human Visual System
The human visual system is the combination of the optical system of the eye, and the neural processing of the light information received [Roorda (2011)], in which the latter is out of the concern of this research. The optical system of the eye is an intricate construction including the pupil, cornea, retina and lens (see Fig.1). The light come through the pupil is refracted by the lens and make an inverse image on the retina. During this process, any deficit would cause aberrations. For instance, myopia may result from the lens that the refraction is too high or that the distance from the lens and retina is too long.
Fig.1 Cross-section of eye structure
There is a limit resolution dominated by the neural receptor on the retina, which is below the diffraction limit. Although even for normal emmetropic eyes the sight is below neural limit and diffraction limit due to the minor deficit of eye structure. [Austin (2011)] For eyes with refractive issues, caused by cornea or lens from an ideal spherical shape, the aberrations would significantly dominate over this limit. Thus, in the following research, we shall omit the neural limitation. To increase the efficiency in the following, we can simply model the eye structure as such: a lens (regarding the cornea and the lens as a whole) with an adjustable size (pupil size) and an image plane (retina).
- Point Spread Function and image quality
As is stated in the previous section the aberrations would come from any deficit of eye structure. In order to quantify the distortion in mathematical means, we introduce the Point Spread Function (PSF). Fundamentally, the PSF is defined as a function describes the response of an imaging system to a point source or point object. Note that the loss of light would not be considered in the PSF. Then, if we consider the PSF does not change across the field of view, which applies to the central 1-2Â° of visual angle [Reference!!!], the image can be expressed by the convolution of the PSF and the object in this area.
Where denotes the convolution algorithm. Note that the deconvolution method is based on the inverse operation of Eq.1, which will be introduce in Section 1.2.4.
Fig.2 A contrast of PSF and MTF of an ideal emmetropic eyes (up) and a typical myopic eyes of -1.00 dioptre (down)
Now we introduce two functions that can show the quality of the image: Optical Transfer Function (OTF) and the Modulation Transfer Function (MTF). Either OTF or MTF specifies the response to a periodic sine-wave pattern passing through the lens system, as a function of its spatial frequency or period, and its orientation [WIKI]. The OTF is the Fourier transform of the PSF, and the MTF is the real magnitude of the OTF. In a 2d system, these two functions are defined as:
Where denotes the Fourier transform, and denote the phase space and Euclidian space, respectively.
Where means taking the absolute value.
- Zernike Polynomials
The Zernike polynomials are a sequence of polynomials that are orthogonal over circular pupils. Some of the polynomials are related to classical aberrations. In optometry and ophthalmology, Zernike polynomials are the standard way to describe aberrations of the cornea or lens from an ideal spherical shape, which result in refraction errors [WIKI].
The definition of orthogonal Zernike Polynomials recommended in an ANSI standard is represented as:
Where m and n denote the radial degree and the azimuthal frequency, respectively. The radial polynomials are defined as:
And the triangular functions:
Note that nm and nm must be even.
The relationship between double index (m, n) and single index (i):
Table.1 Eye aberrations presented by Zernike Polynomials
Aberrations are expressed as the distortion of the wavefront as it passes through the eye. As is stated, Zernike polynomials are the standard way [Campbell (2003)] of quantifying this distortion. The aperture function (or pupil function) can link Zernike polynomials with the PSF:
Where denotes complex aperture function (or pupil function). denotes the phase of the wavefront, and the 'i' is the imaginary unit and denotes the amplitude function, which is usually one inside the circular pupil and zero on the outside. The PSF can be expressed as the square of Fourier transform of the complex aperture function:
We now know that the PSF can be calculated with a known wavefront and the distortion of the wavefront caused by refractive error can be actually represented by several orders of Zernike Polynomials with different amplitudes, which can be precisely measured with a Shack-Hartmann wavefront analyser device.
- Deconvolution Method
We introduce a way to pre-process the image to neutralize the aberration caused by eyes, which is also called image pre-compensation. Simplistically, to 'compensate them in advance to proactively counteract degradations resulting from the ocular aberrations of different users'.
Point Spread Function (PSF) is defined as a function describes the response of an imaging system to a point source or point object. The sinusoidal function is an eigenstate of the PSF (i.e. if the input image is a sinusoidal function, no matter what the PSF is, the output image would also be a sinusoidal function)
The Image on the retina (or) can be linked with PSF by convolution as shown in Eq.1. Then we do Fourier transform on both side of the equation
Note the convolution has changed to multiplication in the phase space. If we define a new OBJ' as:
The new image is
This means If we can process the OBJ' as defined, we will have the intended image in the observer's eyes. To form the OBJ' we introduce Minimum Mean Square Error filtering (or Wiener Filter)
Where K is a constant.
- Computing Theories
- Fast Fourier Transform
As is shown in previous sections, we use two algorithms that require an amount of calculation, which is Fourier transform (inverse Fourier transform) and convolution. Since computer images can be seen as 2-demension lattices, we will use 2d Discrete Fourier Transform:
It is known that this process requires a significant amount of calculation. The conventional way of doing this would take a long time for regular PC. However, for research need, we will need to do this calculation in real-time. Thus, we introduce the Fast Fourier Transform (FFT). A definition of FFT could be: "An FFT is an algorithm computes the discrete Fourier transform (DFT) of a sequence or its inverse. Fourier analysis converts a signal from its original domain (often time or space) to representation in the frequency domain and vice versa. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors." [Van Loan (1992)]
Also, all convolution within our program will be calculated by means of the FFT through the following equation:
Fig.3 A contrast of the speed of two means of calculation with respect of data length.
The purpose of doing so is to accelerate the speed of calculation, since the conventional way of calculating convolution is much slower than the FFT. This difference of speed is shown in Fig.3.
- Nyquist Limit
As is stated, we need the image and the PSF to before doing the pre-compensation. The PSF is calculated by aperture function Eq.9. To simulate the pupil, we can use a circular apertureâ€¦. However, this circular pupil has some restrictions in computer simulation, which is the Nyquist limit.
In signal processing if we
If we want to reconstruct all Fourier components of a periodic waveform, there is a restriction that the sampling rate needs to be at least twice the highest waveform frequency. The Nyquist limit, also known as Nyquist frequency, is the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal, which is half of the sampling rate of a discrete signal processing system. [Cramér & Grenander (1959)]
For our simulation the sampling rate 'n' is represented as:
Aliasing will occur when .
- Psychometric Theories
In order to quantify the enhancement of the Deconvolution Method to the subjects, we need to measure the change of the thresholds of the eyes before and after the compensation. Specifically, in our research we need to find out the threshold of minimum contrast and size of an image that the subjects can correctly recognize. This requires the use of some psychometric theories.
- Adaptive Staircase Method
The staircase method is a widely used method in psychophysics test. The point of staircase method is to adjust the intensity of stimuli according to the response of the participant. To illustrate this method we shall use an example introduced by Cronsweet (1962):
"Suppose the problem is to determine S's absolute, intensive threshold for the sound of a click. The first stimulus that E delivers is a click of some arbitrary intensity. S responds either that he did or did not hear it. If S says 'yes' (he did hear it), the next stimulus is made less intense, and if S says 'no,' the second stimulus is made more intense. If S responds 'yes' to the second stimulus, the third is made less intense, and if he says 'no,' it is made more intense. This procedure is simply continued until some predetermined criterion or 'number of trials' is reached. The results of a series of 30 trials are shown in Fig.4. The results may be recorded directly on graph-paper; doing so helps E keep the procedure straight."
Fig. 4 An example trail by Cornsweet (1962)
There are four important characteristic of adaptive staircase method (1) Starting value; (2) Step-size; (3) Stopping condition; and (4) Modification of step-sizes. [Cornsweet 1962]
- The starting value should be near the threshold value. As is shown in Fig.4, the starting point determines how many step until it reach a level that near the threshold. The test will be most efficient if the starting value is near to that threshold.
- The step-size is 1 'db' for the example test. Step-size should meet the requirement that it is neither too big that not able to measure the threshold accurately nor too small to slow down the test process. It is advised that the step-size would be the most effective when it is the size of the differential threshold.
- The result with the staircase method would be like Fig.4 in general when it hover around a certain level of intensity of stimuli. When reached this asymptotic level, the trails should be taken into account. An efficient way is to set a number of trails that need to be record and start to count after it reach the asymptotic level.
- Under some conditions, the step-size need to be changed during the test. "For careful experimental design, the first stimulus in each of the staircases are at same intensity-level." [Cornsweet 1962] However, then the staring level would be too far from the final level. This can be avoided by using large steps at the start, and smaller steps when it approach the final level. For instance, this can be done by decrease the step from 3db to 1db at the third reversal.
It should be stated that the adaptive staircase method is a very efficient way of measurement. For a given reliability of a computed threshold-value, the staircase-method requires the presentation of many fewer stimuli than any other psychophysical method.
- Related Work
General image compensation has long been used since the invention of lens. The invention of the computer and portable display devices make it easier to perform on-screen image pre-compensation. On-screen compensation has the advantage of convenience in that it can easily be carried out with any display-screen device that can compute. In addition, acuity limits in the human vision on the fovea are found to be between 0.6 and 0.25 arc minutes [Schwiegerling 2000], which is better than the typical acuity of emmetropic eyes [Pamplona 2012]. This means that effective compensation may increase the performance of emmetropic eyes.
- Deconvolution Method
On screen image pre-compensation is based on the idea that the aberrations can be neutralized by pre-compensating the object. Specifically, it requires dividing the Fourier transform of uncorrected image by the Fourier transform of the PSF (i.e. the OTF). A detailed derivation can be found at section1.2.4. Early research by Alonso and Barreto (2003) tested subjects with defocus aberration using this method. Their results showed an improvement in observers' visual acuity compared to non-corrected images.
However, in practical use, for example, defocus, the defocus magnitude (in dioptres) as well as the pupil size, wavelength and viewing distance (visual angle) is required to calculate and scale the PSF, which means measurement and substitution of these parameters are also required to deliver the intended compensation.
- Enhancement of Deconvolution Method
Recent research has further improved the deconvolution method. Huang et al (2012) carried out work with dynamic image compensation. They fixed the viewing distance from the screen and measured the real-time pupil size with the help of a Tobii T60 eye tracker device. Then they compensated the image with this real-time pupil size data. The reliability and acuity were improved by this dynamic compensation. Unlike perfect eyes, for which bigger pupil size would lead to smaller diffraction limited PSF, for most eyes, a bigger pupil size would lead to an increase in aberrations. That is also why dynamic compensation is important.
As is mentioned in previous section, the principle of pre-compensation is to divide the Fourier transform of the image by the Fourier transform of the OTF. In order to avoid near-zero values in the OTF, most of the research used Minimum Mean Square Error filtering (Wiener filter). However, the outcome usually suffers from an apparent loss of contrast.
Recent research has revealed other ways to optimize the compensation to have higher contrast and sharper boundaries. The multi-domain approach was introduced by Alonso Jr et al. (2006). They claimed that there are unnecessary parts in pre-compensated image. Simplistically, there is compensation that is irrelevant with respect to the important information in the image. This work showed an improvement of acuity using this method with respect to recognising text. More recently, Montalto et al. (2015) applied the total variation method to process the pre-compensated image. The result is slightly better but still suffers from a trade-off between contrast and acuity. Fundamentally, the impaired human eye can be seen as a low-pass filter, and either an increase of image aliasing or a decrease of contrast is inevitable.
- Other Approaches
The research described above can be seen as an enhancement and a supplement of the original method carried out by Alonso (2003). However, as is stated, there is a limit of image pre-compensation by the PSF deconvolution method. Others has studied other on-screen methods to achieve a better outcome. Huang et al. (2012) introduced a multilayer approach based on the drawback of normal on screen pre-compensation that was shown by Yellot and Yellot (2007). This method is based on the deconvolution method, but uses a double-layer display rather than normal display. According to Fig.2, if we have two separated displays, then we have two different MTF curve. Then, the near-zero gap in MTF can be filled. This approach has showed a demonstrable improvement of contrast and brightness in their simulation. However, it required a transparent front display that does not block the light from the rear display at all, which is not plausible in practical use.
Later, Pamplona et al. (2012) investigated a light field theory approach and built a monochrome dual-stack-LCD display (also known as parallax barriers) prototype and a lenticular-based display prototype to form directional light. Huang et al. (2014) restated the potential of using light field theory on image compensation and built another prototype with a parallax barrier mask and higher resolution. The outcome of both methods were similar. They could produce colour images with only a little decrease in contrast and acuity. However, it should be stated that both methods were carried out with a fixed directional light field, which used a fixed camera to photograph the intended corrected image. It is obvious that is not feasible in practical use with moving observer. Adjustable directional light has not been implemented due to the limits imposed by diffraction and resolution. In addition, there are minor issues on the loss of brightness as well in these research.
Overall, the most applicable way of on-screen image compensation is still deconvolution method. The light field method requires very precise eye tracking to inject the light into pupil, while deconvolution only requires the observer to keep a certain distance and to be in front of the pre-compensated image.
We built a program for the test that can proceed the pre-compensation in real-time using deconvolution method. This program can pre-compensate any aberration that can be represented by Zernike polynomials
The experiment is based on adaptive staircase method. During the experiment, the program shows optotype Landolt-C in four directions (i.e. up, down, left and right) which is randomly generated at each trail. The subjects choose the direction of the Landolt-C.
Staircase: This research intend to find two thresholds: contrast and size. Though the We shall describe the staircase method for the contrast threshold. The experiment for size threshold is taken likewise.
The four characteristic for our adaptive staircase method are:
- The start value is relatively large since the subject
- The step-size
- The experiment ends in N trials after it reached the final level
- For our research, we cannot determine an ideal starting value because the subjects have different type and intensity of aberration. Thus, we have to change the size-step to make our experiment efficient.
The threshold is calculated using the record the last N trails of the experiment, which is determined by the following equation:
The program was design as such that ???
- Assumptions, Approximations and Limitations
Assumption: About Subjects
Limitation: Polychromatic issues, No. of Pixels, Staircase
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