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Differences between Zernike, Fourier have limited significance for clinicians
Given infinitely fast computers, the two expansion series are equally good at describing wavefront aberrations. In the real world, Fourier is faster, an expert says.
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What is the difference between the Zernike and Fourier expansion series polynomials, and how much should the ophthalmologist be concerned about them?
The answer to the second question, according to one optics expert, is “not much.” Both Zernike and Fourier do what they are intended to do: describe complex three-dimensional surfaces in mathematical terms, making it possible to design customized corneal ablations.
The answer to the first question is more complex, involving long-established concepts from mathematics and physics. But the practical difference for the ophthalmologist may be less important than some may have been led to believe.
“This subject may be the least important clinically to the ophthalmologist, out of anything we have talked about in the Quality of Vision series,” said Jack T. Holladay, MD, MSEE, FACS, “but it is one that we hear a lot of talk about, so it is helpful to understand what the controversy is about.”
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In optics and in ophthalmology, the Zernike and Fourier formulas are ways of describing three-dimensional surfaces, Dr. Holladay said. The general term for these formulas is “expansion series.”
Expansion series can be relatively simple, like the formulas used by computers and calculators to derive sines, cosines and other trigonometric functions, Dr. Holladay said.
“In the old days, we used to use trig tables to find the sine of, say, 30°,” Dr. Holladay said. “We would look up 30°, and the table would tell us that the sine of that angle is 0.5. But calculators and computers don’t contain huge trig tables in memory to be used for looking up those functions; they calculate them as needed using expansion series.”
The expansion series to calculate the sine of an angle takes this form:
Sin x = x – x3/3! + x5/5! + … + xn/n!
Dr. Holladay noted that in working with these types of equations, angles must be expressed in radians, not in degrees. “This is similar to working in strabismus, where we have to use prism diopters,” he said.
The circumference of a circle, which is 360°, is C = 2r, where C is circumference and r is the radius of the circle.
360° = 2r
r =360°/2 = 57.3° (1 radian = 57.3°)
To find the sine of 30° using the expansion series above:
30° =/6 radians = 0.5236 radian
Sin (30°) = Sin (/6 ) = 0.5236 – (0.5236)3/6 + (0.5236)5/120 = 0.5236 – 0.0239 + 0.0066 = 0.5063
The actual sine of 30° is exactly 0.5000. With three terms in the expansion series, the calculation is accurate to almost three decimal places, Dr. Holladay noted. More terms are added to the formula depending on the precision required.
The general equation for an expansion series using a polynomial takes the following form:
a + b(x) + c(x2) + d(x3) + ….
The two-dimensional formula (x and y) used to generate a sine wave can be made to describe a three-dimensional surface by adding a third term (z) for height, Dr. Holladay said.
“Any three-dimensional surface, whether the surface of the cornea or the wavefront from an entire eye, can be represented using a polynomial in which the height is z and the coordinates are x and y,” he said.
A general quadratic equation to represent any 3-D surface would look like the following:
z = ax2 + bxy + cy2 + dx + ey + f
The graph of this equation can be a circle, ellipse, parabola, hyperbola, etc., Dr. Holladay said. These are often referred to as conic sections because they can be obtained as the intersection of a plane with a right circular cone.
Well-known concepts
The Zernike and Fourier polynomials are also expansion series, but the individual terms in the series are more complex than the terms in the equations outlined above. While these concepts may be relatively new to most ophthalmologists, the formulas themselves have been around for decades, even centuries.
Jean Baptiste Joseph Fourier, born in 1768, first elucidated the concept of the Fourier series in the early decades of the 1800s. He said that any function can be described in terms of an infinite series of sines and cosines.
Comparison shows Zernike and Fourier representations of a complex shape. Images: Visx Inc. | |
In a Fourier single series, calculating a function of x,
f(x) = (a * sin x) + (b * sin x2) + …..
Using this formula, any two-dimensional waveform can be described exactly, given an infinite number of terms, Dr. Holladay said. In a similar way, describing z as a function of x and y, any three-dimensional surface can be represented using an infinite number of terms, he said.
The precision of the description can never be more than half the highest frequency of the sample, Dr. Holladay said.
“So if we want the height of the cornea or the wavefront to be accurate to within 1 µm, we would need 64 terms for the Fourier equation to give that level of accuracy,” he said.
Because sine waves are simple forms, and because the Fourier equation is so useful in fields such as engineering, over the years a number of shortcuts, called fast Fourier transforms, have been developed to make the calculations simpler, Dr. Holladay said.
“The whole basis for electrical engineering designs is based on fast Fourier transforms,” Dr. Holladay said. “These have been available for over a hundred years.”
The Zernike polynomial, named after Dutch physicist Frits Zernike, is more recent in origin, having been described in the first half of the 20th century. It specifically relates to optics, Dr. Holladay said.
“Zernike is also an expansion series that lets you represent any 3-D surface,” he said. “The advantage of Zernike over Fourier is that the first terms in the series – tilt, prism, sphere, cylinder, spherical aberration and coma – are all aberrations that we have been using for years to describe optical systems.”
In contrast, in the Fourier series, no single term exists to represent astigmatism, spherical aberration and so on, he said, although they can be described by the sums of certain terms. But this involves a second set of calculations in addition to the calculations that went into generating the Fourier series in the first place, Dr. Holladay noted.
“For ophthalmology, this is a disadvantage of the Fourier equation,” he said. “First you have to calculate the series itself, the you need a second set of terms to pull out the cylinder, sphere and other optical aberrations.”
Complex surfaces
While the Zernike series has the advantage of describing basic low-order optical errors such as sphere and cylinder easily, that advantage is lost when dealing with a complex, irregular surface, Dr. Holladay said.
“If you have an irregular surface that doesn’t conform to one of the terms of the Zernike equation, such as coma or quadrafoil, then you have to use many terms to describe it,” he said. “An infinite number of terms would still give you a perfect description of the wavefront, but the complexity of the mathematical calculation goes up exponentially, much more so than with the 64 terms needed for the Fourier series. It’s not fast, and it takes a lot of computer power.”
As a result of the complexity of the description, the Zernike equation has difficulty mapping irregularities in detail, especially those in the periphery of the area being analyzed, Dr. Holladay said.
“So when you compare the 64 terms in Fourier to the 12th-order Zernike, you get more precision with the Fourier for an extremely irregular surface,” he said.
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Dr. Holladay described an example of the difficulty that the Zernike equation has in mapping a complex corneal surface:
“You’re familiar with the illustrations that show the Zernike shapes – the aberrations are all regular. So if we have a patient with keratoconus, where there is a protrusion or bump at 6 o’clock on the cornea, the first term in the Zernike series that can describe that is coma. But in addition to the inferior bump, coma also has a valley above. So we have to add a second Zernike term, trefoil, with three bumps, to cancel out that valley created by coma in the superior cornea. But now, because of the trefoil, we have extra bumps at 4 and 8 o’clock. So we have to add quadrafoil, with two valleys, to counter those bumps. But quadrafoil adds two additional valleys, so we have to add pentafoil. … As you can see, we need to add more and more terms to describe the original error, which was keratoconus — one inferior bump.”
The complexity is also deceptive in a way, Dr. Holladay said, because the patient never had trefoil, quadrafoil, pentafoil and so on to start with. These are just terms used to cancel out parts of the lower terms that were not wanted, he said.
“These terms were just added to compensate for the fact that the coma didn’t describe the corneal surface properly,” he said. “So the calculation, which as I said gets exponentially more complex and time- and energy-consuming, must go far enough up into the higher-order aberrations to describe the surface accurately. This is why Zernike is not optimal, not set up to work on complex surfaces.”
The true value
The bottom line, Dr. Holladay said, is how well a laser’s software program describes the cornea surface or wavefront in question, regardless of what formula it is using.
“The real question is, What is the root-mean-square error, the difference between the actual surface and the surface as described by Zernike or Fourier? If the RMS error is the same, it’s the same. That’s the true value of the fit,” he said.
“Using terms like Zernike and Fourier to make clinical decisions is usually not helpful,” he said. “They are used by the engineer, the mathematician; they don’t really have applications for the ophthalmologist. They can lead you to make conclusions that result in a mistake.”
From a theoretical standpoint, Dr. Holladay said, if computers were as fast and precise as necessary, there should be no difference between Fourier and Zernike in describing three-dimensional surfaces.
“But with real computers in the real world, Fourier transforms can do it more rapidly,” he said.
“On the other hand,” he said “it is helpful to remember that, although Fourier uses 64 terms and Zernike uses 12 orders, it’s not fair to say Zernike is a lower-order calculation than Fourier. They are both describing surfaces with the same amount of complexity. But because the Fourier calculations are simpler, Fourier is faster.”
For Your Information:
- Jack T. Holladay, MD, MSEE, FACS, can be reached at Holladay LASIK Institute, Bellaire Triangle Building, 6802 Mapleridge, Suite 200, Bellaire, TX 77401; 713-668-7337; 713-668-7336; e-mail:docholladay@docholladay.com.
- Tim Donald, OSN Copy Chief, is writing the QOV series.