A persistence lens is a hierarchy of disjoint upper and lower level sets of a continuous luminance image's Reeb graph. The boundary components of a persistence lens's interior components are Jordan curves that serve as a hierarchical segmentation of the image, and may be rendered as vector graphics. A persistence lens determines a varilet basis for the luminance image, in which image simplification is a realized by subspace projection. Image scale space, and image fractal analysis, result from applying a scale measure to each basis function.
Varilet image analysis provides vector representation of hierarchical image segmentation, by vectorizing the Jordan lens facets' boundary components. The luminance value is identical for all Jordan boundaries of the same lens region. Image vectorization methods typically use image segmentation and or edge detectors, e.g. Selinger @cite_57 . @cite_1 merge regions of similar color. @cite_15 use heuristics to group pixels into cells.
@cite_57 @cite_15 @cite_1
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We describe a novel algorithm for extracting a resolution-independent vector representation from pixel art images, which enables magnifying the results by an arbitrary amount without image degradation. Our algorithm resolves pixel-scale features in the input and converts them into regions with smoothly varying shading that are crisply separated by piecewise-smooth contour curves. In the original image, pixels are represented on a square pixel lattice, where diagonal neighbors are only connected through a single point. This causes thin features to become visually disconnected under magnification by conventional means, and creates ambiguities in the connectedness and separation of diagonal neighbors. The key to our algorithm is in resolving these ambiguities. This enables us to reshape the pixel cells so that neighboring pixels belonging to the same feature are connected through edges, thereby preserving the feature connectivity under magnification. We reduce pixel aliasing artifacts and improve smoothness by fitting spline curves to contours in the image and optimizing their control points. Most segmentation algorithms are composed of several procedures: split and merge, small region elimination, boundary smoothing,..., each depending on several parameters. The introduction of an energy to minimize leads to a drastic reduction of these parameters. The authors prove that the most simple segmentation tool, the “region merging” algorithm, made according to the simplest energy, is enough to compute a local energy minimum belonging to a compact class and to achieve the job of most of the tools mentioned above. The authors explain why “merging” in a variational framework leads to a fast multiscale, multichannel algorithm, with a pyramidal structure. The obtained algorithm is @math , where n is the number of pixels of the picture. This fast algorithm is applied to make grey level and texture segmentation and experimental results are shown.