Los Angeles Wavefront LASIK

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Once upon a time in 19th century (1822*, to be exact, but you do not need to know the exact time. Just trust me even your grandparents were not born), the French mathematician J. Fourier (Vive Le France), showed that any periodic function can be expressed as an infinite sum of periodic complex exponential functions. Many years(and bottles of champagne later) after he had discovered this remarkable property of (periodic) functions, his ideas were generalized to first non-periodic functions, and then periodic or non-periodic discrete time signals. It is after this generalization that it became a very suitable tool for computer calculations. In 1965, a new algorithm called fast Fourier Transform (FFT) was developed and FT became even more popular (so did Elvis). In the late 1960's, Blakemore and Campbell (1969) suggested that the neurons in the visual cortex might process spatial frequencies instead of particular features of the visual world (geekspeak).Translated to English, this means that instead of piecing the visual world together like a puzzle, the brain performs something akin to the mathematical technique of Fourier Analysis to detect the form of objects. The neurons are nerve cells. Visual Cortex is that part of the brain where the nerve cells are specialized to convert electrical signals received from the retina to vision. While this analogy between the brain and the mathematical procedure is at best a loose one (since the brain doesn't really "do" a Fourier Analysis), whatever the brain actually does when we see an object is easier to understand within this context. Thus, a review of the basic concepts of Fourier Analysis will be very helpful (Starbucks?).

Basics of Gratings (sinewave and square-wave)

A grating is a repeating sequence of light and dark bars. One adjacent pair of a light and a dark bar (Yin and Yang)makes up one cycle. These cycles repeat over and over in a grating. Typically, we think of these gratings as having sharp edges like the bars shown in Figure 1 below. Figure 1. A grating composed of black and white bars that is typically called a square-wave grating.

These gratings have become known as square-wave gratings because they have characteristics like a square such as sharp edges. This square-like nature of the light levels (or luminance), as it falls across different positions along the surface of an object can be seen in Figure 2. Clicking on Figure 2 will bring down the square-wave grating so that you can more easily compare the actual grating to the graph of the grating.

Figure 2. A graph of the luminance of a square-wave as a function of position for one cycle.

However, there is no reason that light has to vary in a square-wave fashion. What if the light varied in a sinewave fashion? Figure 3 shows one cycle of a sinewave. Figure 1 showed a square-wave grating with four cycles.

Figure 3. A graph of the luminance of a sinewave as a function of position for one cycle.

Figure 5 a sinewave grating with four cycles

Just as Fourier Analysis is a mathematical procedure used to determine the collection of sinewaves (differing in frequency and amplitude) that is neccessary to make up the square-wave pattern under consideration. Take, for example, one cycle of a square-wave which is graphed in Figure 1. This graph shows how luminance or light level changes over position as it falls across the surface of an object. Notice that even adding this one sinewave (called the fundamental because it is the lowest frequency and has the biggest amplitude) already gives the basic shape of the square-wave grating. The size of the bars and the contrast of the bars are already basically visible. What is lacking are the sharp contrasts (edges) between the white and the black bars. These edges come from sinewaves with higher frequencies and lower amplitudes. Presented below in Figure 6 are the results of a Fourier Analysis of a square-wave grating. The results of this analysis give the frequencies and amplitudes of the sinewaves necessary to make up the square-wave grating. This way of presenting the results of a Fourier Analysis is called the frequency domain. It turns out that sinewaves with frequencies that are odd multiples of the fundamental are all included in the square-wave grating. This means that if the fundamental has a cycle width of 1", the next sinewave will have a cycle width of 1/3", and the third will have a width of 1/5", etc. (The sequence is actually infinite). All frequencies higher than the fundamental are referred to as harmonics. The amplitude of each sinewave is one over the frequency. Thus, the sinewave with a frequency 11 times the fundamental will have an amplitude of 1/11th the fundamental.

Figure 6. The results of a Fourier Analysis of a square-wave grating.

Los Angeles Wavefront LASIK Disclaimer: Dr. Khanna serves eyecare patients in Los Angeles and surrounding areas. This site is for informational purposes only. It is not intended to be medical advice for those seeking LASIK surgery, but to provide information about the Los Angeles, California LASIK Surgeon, Doctor Rajesh Khanna, M.D.

Dr. Khanna proudly serves patients throughout Southern California and Orange County including: Alhambra, Altadena, Anaheim, Arcadia, Baldwin Park, Bakersfield, Beverly Hills, Burbank, Camarillo, Carson, Cerritos, Chino, Chino Hills, Compton, Corona, Costa Mesa, Diamond Bar, Downey, East Los Angeles, Fontana, Gardena, Glendale, Hacienda Heights, Hawthorne, Huntington Beach, Huntington Park, Inglewood, La Habra, Laguna Niguel, Lake Forest, Lakewood, Lancaster, Long Beach, Los Angeles, Lynwood, Mission Viejo, Montebello, Monterey Park, Newhall, Newport Beach, Norwalk, Ontario, Oxnard, Palmdale, Palos Verde, Paramount, Pasadena, Pico Rivera, Pomona, Redondo Beach, Riverside, Rosemead, Santa Barbara, Santa Clarita, Santa Monica, Santa Rosa, Simi Valley, South Gate, Thousand Oaks, Torrance, Trabuco, Upland, Ventura, and West Covina.

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