Download (direct link):
The Wyckoff Signal/Noise Criteria
V(X0, Ó0) e (x, y), 3kiq(x0, Ó0) such that
1. kiq(x0, Ó0) » nqt, and
2. Ci(q(x0,Ó0)) » ci(±f -1(nfi)j.
The first criterion indicates that for every pixel in the output image, at least one of the input images provides sufficient exposure at that pixel location to overcome sensor noise, nqi. The second criterion states that of those at least one input image provides an exposure that falls favorably (i.e., is neither overexposed nor underexposed) on the response curve of the camera, so as not to be overcome by camera noise n f. The manner in which differently exposed images of the same subject matter are combined is illustrated, by way of an example involving three input images, in Figure 4.4.
Moreover it has been shown  that the constants ki as well as the unknown nonlinear response function of the camera can be determined, up to a single unknown scalar constant, given nothing more than two or more pictures of the same subject matter in which the pictures differ only in exposure. Thus the reciprocal exposures used to tonally register (tonally align) the multiple input images are estimates 1/k in Figure 4.4. These exposure estimates are generally made by applying an estimation algorithm to the input images, either while simultaneously estimating f or as a separate estimation process (since f only has to be estimated once for each camera, but the exposure ki is estimated for every picture i that is taken).
matter CAMERA set to exposure 1
Sensor noise Image noise
matter CAMERA set to exposure 2
matter CAMERA set to exposure 3
Assumptions (see text for corresponding equations): for every point in the imput image set,
1. There exists at least one image in the set for which the exposure is sufficient to overcome sensor noise, and
2. At least one image that satisfies the above has an exposure that is not lost in image noise by being to far into the toe or shoulder region of the response curve.
Optional anti-homomorphic Wyckoff filter to act on estimated photoquantity may be inserted
here-^ äï^—homomorphic ----------Wyckoff filter
THE WYCKOFF PRINCIPLE AND THE RANGE OF LIGHT
Owing to the large dynamic range that some Wyckoff sets can cover, small errors in f tend to have adverse effects on the overall estimate q. Thus it is preferable to estimate f as a separate process (i.e., by taking hundreds of exposures with the camera under computer program control). Once f is known (previously measured), then ki can be estimated for a particular set of images.
The final estimate for q, depicted in Figure 4.4, is given by
J2?iqi ^[Ci(q(x, y))/ki]f~1(fi(x, y)) q(x, y) = -i—— = -i-—-, (4.10)
where ci is given by
dfi(x,y) d f(kiq(x,y))
Ci(log(q(x, y))) = —---- = —----. (4.11)
d logq(x,y) d logq(x,y)
From this expression we can see that c(log(q)) are just shifted versions of c(log(q)), or dilated versions of c(q).
The intuition behind the certainty function is that it captures the slope of the response function, which indicates how quickly the output (pixel value or the like) of the camera varies for given input. In the noisy camera, especially a digital camera where quantization noise is involved, generally the camera’s
Figure 4.4 The Wyckoff principle. Multiple differently exposed images of the same subject matter are captured by a single camera. In this example there are three different exposures. The first exposure (camera set to exposure 1) gives rise to an exposure k1q, the second to k2q, and the third to k3q. Each exposure has a different realization of the same noise process associated with it, and the three noisy pictures that the camera provides are denoted f1, f2, and f3. These three differently exposed pictures comprise a noisy Wyckoff set. To combine them into a single estimate, the effect of f is undone with an estimate f that represents our best guess of what the function f is. While many video cameras use something close to the standard f = kq045 function, it is preferable to attempt to estimate f for the specific camera in use. Generally, this estimate is made together with an estimate of the exposures k,. After re-expanding the dynamic ranges with f-1, the inverse of the estimated exposures 1 /k, is applied. In this way the darker images are made lighter and the lighter images are made darker, so they all (theoretically) match. At this point the images will all appear as if they were taken with identical exposure, except for the fact that the pictures that were brighter to start with will be noisy in lighter areas of the image and those that had been darker to start with will be noisy in dark areas of the image. Thus rather than simply applying ordinary signal averaging, a weighted average is taken. The weights are the spatially varying certainty functions c(x,y). These certainty functions turn out to be the derivative of the camera response function shifted up or down by an amount kj. In practice, since f is an estimate, so is Cj, and it is denoted C in the figure. The weighted sum is q(x,y), the estimate of the photoquantity q(x,y). To view this quantity on a video display, it is first adjusted in exposure, and it may be adjusted to a different exposure level than any of the exposure levels used in taking the input images. In this figure, for illustrative purposes, it is set to the estimated exposure of the first image, k1. The result is then range-compressed with f for display on an expansive medium (display).