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We derive theoretically and demonstrate experimentally an approach to range-from-focus with an important improvement
over all previous methods. Previous methods rely on subjective measures of sharpness to focus a selected locale of the
image. Our method uses measured physical features of the optical signal to generate an objective focus-error distance map.
To compute range-from-focus-error distance it is not necessary to focus any part of the image: range is calculated directly
from the lens formula by substituting the difference between the lens-to-sensor distance and the focus-error distance for the
usual lens-to-image distance. Our method senses focus-error distance in parallel for all locales of the image, thus providing
a complete range image. The method is based on our recognition that when an image sensor is driven in longitudinal
oscillation ("dithered") the Fourier amplitude of the first harmonic component of the signal is proportional to the first power
of the ratio of dither amplitude to focus-error distance, whereas the Fourier amplitude of the second harmonic component is
proportional to the square of this ratio. The ratio of the first harmonic sin ot amplitude A1, to the second harmonic cos 2cot
amplitude B2 is thus a constant (-4) multiple of the ratio of the focus-error distance to the dither amplitude. The
focus-error distance measurement via the ratio of the first-to-second harmonic amplitudes is extremely robust in the sense
that the scene's gray level structure, the spatial and temporal structure of the illumination, and technical noise sources (most
of which affect the Fourier amplitudes multiplicatively) all appear identically in both amplitudes, thus cancelling in the
ratio. Extracting the two Fourier amplitudes and taking their ratio could be accomplished, pixel-by-pixel, by some
ambitious but not outrageous analog computing circuitry that we describe. We derive the method for a point scene model,
and we demonstrate the method with apparatus that instantiates this modeL
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