One-Dimensional Openings, Granulometries and Component Trees in 𝒪(1) in Per Pixel

Abstract

We introduce a new, efficient and adaptable algorithm to compute openings, granulometries and the component tree for onedimensional (1-D) signals. The algorithm requires only one scan of the signal, runs in place in O(1) per pixel, and supports any scalar data precision (integer or floating-point data). The algorithm is applied to two-dimensional images along straight lines, in arbitrary orientations. Oriented size distributions can thus be efficiently computed, and textures characterised. Extensive benchmarks are reported. They show that the proposed algorithm allows computing 1-D openings faster than existing algorithms for data precisions higher than 8 bits, and remains competitive with respect to the algorithm proposed by Van Droogenbroeck when dealing with 8-bit images. When computing granulometries, the new algorithm runs faster than any other method of the state of the art. Moreover, it allows efficient computation of 1-D component trees.

Fast Computation of Greyscale Path Openings

“…In this section we present an algorithm that su ers from none of these problems, based on the 1D algorithm presented by Morard et al [15]. The basic idea is to use the traditional binary algorithm, but instead of having the scalar λ+(x) and λ−(x), we use sets Λ+(x) and Λ−(x), represented by non-redundant ordered (ascending by grey level) lists of pairs of grey levels and path lengths.…”

Efficient and Robust Path Openings Using the Scale-Invariant Rank Operator

“…For 1D greyscale path openings (equivalent to attribute openings on length in 1D), a very efficient algorithm running in O(1) (amortized) cost per pixel was presented by Morard et al [19]. This was recently adapted to compute path openings on graphs as well [12] (albeit with a worse time complexity).…”

“…To make the above efficient, binary search trees can be used to represent + and − (instead of the stacks used by Morard et al [19]). This allows O(log(n)) insertion, lookup, and deletion, with n the length of the sequence.…”

“…However, the operator was limited to 1D signals (rows/columns of images were processed independently). In this work, we analyse the SIR operator in more detail, exposing a link to path openings [13,14,19]. In Sect.…”

Visual Quality Inspection and Fine Anomalies: Methods and Application

“…Detection methods and constraints [14] or the "maximum opening" [15][16], or the "subtraction of maximum closure of opening by original image" [17][18]. These methods are illustrated Figure 7).…”