We have established a relationship between the asymptotic (k→∞) spectral index of idealized optically thin localized radiating sources, n(x, t), and their scans f. For piecewise-constant (pc) clouds, if the envelope of the one-dimensional asymptotic spectrum of the power of n is proportional to k-2, then the asymptotic spectrum of the scan depends upon the character of the contour at point yo = (xo,yo) where it s tangent at an extremal ray. If the contour at (xo) behaves like y - yo = y(x - xo)t, then the asymptotic spectral envelope of the power spectral density (PSD) of the scan varies as k-2(&tgr;+1). For a convex curve with finite curvature at x0, have &tgr;g=1/2 and obtain the well-known result, k-3. In addition, if the radiating density varies spatially in a cloud flank with a power law &mgr;, then the asymptotic spectral envelope of the scan's PSD varies as k-2(2+&tgr;+&mgr;). Thus by examining projections of radiating clouds, one cannot distinguish between local gradients of density and the local shape of the boundary curve of the cloud. We also apply the result to the interpretation of optical scans from two PLACES events. These show the onset of steepening and striationso. However, the accurately resolved wave number range insufficient for determining the spectrum ssociated with the nonlinear dynamics of striation evolution.