Virtual Super Resolution of Scale Invariant Textured Images Using Multifractal Stochastic Processes

Abstract

We present a new method of magnification for textured images featuring scale invariance properties. This work is originally motivated by an application to astronomical images. One goal is to propose a method to quantitatively predict statistical and visual properties of images taken by a forthcoming higher resolution telescope from older images at lower resolution. This is done by performing a virtual super resolution using a family of scale invariant stochastic processes, namely compound Poisson cascades, and fractional integration. The procedure preserves the visual aspect as well as the statistical properties of the initial image. An augmentation of information is performed by locally adding random small scale details below the initial pixel size. This extrapolation procedure yields a potentially infinite number of magnified versions of an image. It allows for large magnification factors (virtually infinite) and is physically conservative: zooming out to the initial resolution yields the initial image back. The (virtually) super resolved images can be used to predict the quality of future observations as well as to develop and test compression or denoising techniques. Keywords natural images • scale invariance • multifractal analysis • extrapolation • enhancement • infinitely divisible cascades

Scale invariant images in astronomy through the lens of multifractal modeling

“…The main thrust is to replace Q r0 r1 by Q r0 r2 in (2) to get I 2 at resolution r 2 < r 1 , see [4] for details. Eventually, a renormalization step ensures that the procedure is conservative in the sense that zooming out to the initial resolution gives the original image back.…”

“…Then the fractional integration yields the field I characterized by ζ(q) = qH + τ (q). See [2,4] for a detailed presentation.…”

“…The plausibility stems from the statistical and physical consistency with the original observation. The 'virtually super-resolved' (VSR) images [4] will help the physicists to calibrate and test algorithms on forthcoming higher resolution instruments. It also allows to predict histograms of future high resolution imagers.…”

Super-resolution: a comprehensive survey

“…-Natural Image Prior [139], [342], [464], [513], [612] -Stationary Simultaneous Auto Regression (SAR) [170] which applies uniform smoothness to all the locations in the image. -Non-stationary SAR [215] in which the variance of the SAR prediction can be different from one location in the image to another.…”

Joint Inversion of Fracture Model Properties for CO2Storage Monitoring or Oil Recovery History Matching

“…In addition, using interpolations the final density map seems to be more natural from a geological point of view. Future work would require to take the smoothing factor into account while modeling the faults or to use a method that directly builds continuous correlated density maps [31]. In this method, the continuity of fault displacements is enhanced using smoother density maps.…”