Vision_Notes_6 - VI. Image sampling For most...

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85 VI. Image sampling For most image-formation systems, the output image 5 (the optical image ), can be completely described by a four-dimensional function n(x, y, t, l) that gives the mean quantum irradiance as a function of space (x, y), time (t) and wavelength ( λ ). This is usually a complete description because, in most situations, the image noise is that of an inhomogeneous Poisson process (which is completely determined by its mean or "intensity" function). All the image information available for performing a given visual task is carried in such four-dimensional functions. In biological vision systems, and in virtually all artificial (computer) vision systems, the images formed by the optical system must (because of hardware/wetware limitations) eventually be coded into a discrete representation in space, time, and wavelength. This image sampling process is a crucial step in visual processing that can, and often does, result in significant information loss. The loss results because it is often impossible to sample all four dimensions with sufficiently high resolution. Thus, compromises must always be struck. In the biological vision systems, the first stage of sampling is carried out by the photoreceptors, in artificial systems, usually by some other two-dimensional array of elements (e.g., a CCD array or an array of photodiodes). The principles of sampling are much the same in biological and artificial systems. A fundamental principle of sampling is captured by the so-called Wittaker-Shannon sampling theorem which is useful for understanding the information loss due to discrete sampling. A. The sampling theorem Below are statements of the Whittaker-Shannon sampling theorem for one and two dimensions. Similar statements hold for higher dimensions. The sampling theorem (one-dimensional case): If a one-dimensional function, f(x), is limited to frequencies below w c cycles per unit value of x, then the function can be completely reconstructed by taking 2w c evenly spaced samples per unit value of x. The sampling theorem (two-dimensional case): If a two-dimensional function, f(x, y), is limited to frequencies below u c cycles per unit value of x (in the x direction), and to frequencies below v c cycles per unit value of y (in the y direction), then the function can be completely reconstructed by taking 4u c v c samples per unit area on the x, y plane. In other words, what the sampling theorem says is that almost any smooth continuous function can be represented exactly, with perfect precision, by the values of the function 5 For example, there would be two images in a binocular optical system.
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86 at only at discrete set of points. This may seem rather surprising. It suggests that it is, in principle, possible to discretely sample an image without any loss of information. The minimum sampling rate required for perfect reconstruction (2w c for one dimensional functions and 4u c v c for two dimensional functions) is known as the Nyquist rate .
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This note was uploaded on 08/12/2008 for the course PSY 380E taught by Professor Geisler during the Fall '07 term at University of Texas at Austin.

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Vision_Notes_6 - VI. Image sampling For most...

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