Summary: This paper reports the results of a theoretical study on morphological characterization of foreground (X) and background (X\(^c)\) of a discrete binary image. Erosion asymmetry and dilation asymmetry, defined to elaborate smoothing of an image respectively by contraction and expansion, are generalized for multiscale smoothing, and their relationships with morphological skeleton and ridge (background skeleton) transformations are discussed. Then we develop algorithms identifying image topology in terms of critical scales corresponding to close-hulls and open-skulls, along with a few other salient characteristics, as respective smoothing by expansion and contraction proceeds. For empirical demonstration of these algorithms, essentially to unravel the hidden characteristics of topological and geometrical relevance, we considered deterministic and random binary Koch quadric fractals. A shape-size based zonal quantization technique for image and its background is introduced as analytical outcome of these algorithms. The ideas presented and demonstrated on binary fractals could be easily extended to the grayscale images and fractals.