A Sampling Strategy using Multi-Scale Annular Operators

Annular operators may be used to group image data by detecting symmetric relationships between edges. The spatial extent of operators is bounded by two concentric circular regions having diierent radii, r 1 and r 2. Edges which fall within the annular region between these circles are recorded, thereby enabling the detection of circular correlation between edge segments. In order to detect such conngurations occurring at any position within an image and within some range of circular sizes, a multi-scale sampling structure is developed based upon the detection of circles of radius r. Scale in this setting refers to the size of detected circles. A set of similarly sized operators is used to detect circles having a range of sizes r 1 < r < r 2. A covering for the image plane is then achieved by positioning individual operators on points of a triangular lattice. The resulting structure represents one layer of operators. To detect circles falling within an arbitrarily large range of sizes, multiple layers are employed. A set of parameters is used to specify a particular sampling structure in terms of the annular sizes, the spacing between annuli in a given layer, and the number of layers required. Depending upon a particular choice of parameters, the sampling structure exhibits two diierent redundant tendencies. The rst is a spatial sampling redundancy where multiple operators in the same layer respond to a single symmetric edge connguration. The second is a sampling redundancy in scale where operators within multiple layers respond to a single symmetric edge connguration. These redundancies are examined in terms of practical design requirements for minimizing the total number of operators needed to sample a given image.