Why lp-methods in signal and image processing: A fuzzy-based explanation

In signal and image processing, it is often beneficial to use semi-heuristic lp-methods, i.e., methods that minimize the sum of the p-th powers of the discrepancies. In this paper, we show that a fuzzy-based analysis of the corresponding intuitive idea leads exactly to the lp-methods. I. FORMULATION OF THE PROBLEM In general, signal and image reconstruction are ill-posed problems. While cameras and other image-capturing devices are getting better and better every day, none of them is perfect, there is always some blur. This blur comes from the fact that while we would like to capture the intensity I(x, y) at each spatial location (x, y), the signal captured by a reallife measuring device is influenced not only by the intensity I(x, y) at the desired location (x, y), but also by the intensities I(x′, y′) at nearby locations (x′, y′). As a result, instead of reflecting the intensity I(x, y) at the desired point (x, y), the signal s(x, y) measured by the device is a combination of intensities at the point (x, y) and at the nearby points: s(x, y) = ∫ w(x, y, x′, y′) · I(x′, y′) dx′ dy′, for appropriate weights w(x, y, x′, y′). When we take a photo of a friend with a modern sophisticated cell phone camera, this blur is barely visible – and does not constitute a serious problem. However, when a spaceship takes a photo of a distant plant, the blur is very visible – and needs to be eliminated. In such situations, we need to reconstruct the original image I(x, y) from the blurred image s(x, y). From the purely mathematical viewpoint, this reconstruction problem is ill-posed in the sense that large changes in I(x, y) can lead to very small changes in s(x, y) – and, as a result, unless we impose additional constraints on the original image I(x, y), we cannot reconstruct the image with any reasonable accuracy. This mathematical feature is easy to explain: blurring averages the image. Instead of the original intensity at the point (x, y), we have an average intensity over all the neighbors (x′, y′) of the original point (x, y). It is known that such averaging eliminates high-frequency components. Thus, if instead of the original signal I(x, y), we consider a different signal I∗(x, y) = I(x, y) + c · sin(ωx · x+ ωy · y), with a high-frequency component added, the resulting signal s∗(x, y) will be practically the same s∗(x, y) ≈ s(x, y) – but the original image, for large c, may be very different. To be able to reconstruct the image reasonably uniquely, we cannot allow all possible dependencies I(x, y), we need to impose some additional conditions on the original image. This imposition of additional conditions that helps reconstruct the original image is known as regularization; see, e.g., [9]. Similarly, the problem of reconstructing a 1-D signal x(t) from observations is ill-posed. Tikhonov’s regularization: a brief reminder. If a signal or an image is smooth (differentiable), then a natural idea is to require that the corresponding derivatives are, on average, small, i.e., e.g., that the mean square value of the derivative does not exceed a certain constant C. Let us describe this requirement in precise terms. If we have n values d1, . , dn, then the mean square value is d def = √ d1 + . + d 2 n n . The requirement that this mean square value is bounded by C, i.e., that √ d1 + . + d 2 n n ≤ C is equivalent to d1 + . + d 2 n n ≤ C and, thus, to d1 + . + d 2 n ≤ c, where c def = n · C. When we go from the discrete data to the continuous signal or image, then the sum turns into an integral. So, for 1-D signals, we have a constraint ∫ (ẋ(t)) dt ≤ c, and for 2-D images I(x, y), we have a similar constraint ∫ (( ∂I ∂x )2 + ( ∂I ∂y )2) dx dy ≤ c. Out of all signals or images that satisfy this constraint, we want to find a one which is the best fit with the observation, i.e., for which, e.g., the mean square error is the smallest: