Topological smoothing: overview
Shape smoothing plays an important role in image processing and pattern recognition. For example, the analysis or recognition of a shape is often perturbed by noise, thus the smoothing of object boundaries is a necessary pre-processing step. Also, when zooming or warping binary digital images, one obtains a crenelated result that must be smoothed for better visualization. The smoothing procedure can also be used to extract some shape characteristics: by making the difference between the original and the smoothed object, salient or carved parts can be detected and measured.
In all previous works, it was always assumed that the shape to be smoothed is a single object, in other words, its boundary is a simple closed curve (in 2D) or surface (in 3D). What happens if we want to apply the smoothing to a whole scene composed of several objects? If we apply any of the proposed schemes to each object independently and then merge the results, there is no guarantee that the images of two disjoint objects will be disjoint. More generally, little attention has been paid to global topological properties of smoothing procedures.
We introduce a new method for smoothing 2D and 3D objects in binary images while preserving topology. Here, objects are defined as sets of grid points, and topology preservation is ensured by the exclusive use of homotopic transformations defined in the framework of digital topology. Smoothness is obtained by the use of morphological openings and closings by metric discs or balls of increasing radius, in the manner of alternating sequential filters from the field of mathematical morphology. All these morphological filters do not preserve topology, this is why we introduce new operators: homotopic cutting and homotopic filling, which combine a filtering effect with the guarantee of topology preservation. The homotopic alternating sequential filter is a composition of homotopic cuttings and fillings by balls of increasing radius. It takes an original image X and a control image C as input, and smoothes X ``as much as possible'' while respecting the topology of X and geometrical constraints implicitly represented by C. Based on this filter, we introduce a general smoothing procedure with a single parameter which allows to control the degree of smoothing. Furthermore, the result of this procedure presents small variations in response to small variations of the parameter value. We also propose a method with no parameter for smoothing zoomed binary images in 2D or 3D while preserving topology.
For more information, please refer to [CB04].
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