Contour operations

This page contains short definitions for each contour operation available on this app. These definitions are part of a book chapter about Contour Theory in Brazil by Marcos Sampaio, to be released in 2024.

Translation

Translation consist of contour points integers renumbering from zero in the most compact form. See Marvin and Laprade in the bibliography.

For instance, the translation of a given contour segment < 4 8 2 > is < 1 2 0 >.

Equivalent class

Equivalent class is a set of a contour and its equivalent contours. Two contour segments are equivalent when all the relations of their points are equivalent. By definition, contour segments related by reflection operations are also equivalent. An equivalent class prime form represents all the class members. See Marvin and Laprade in the bibliography.

For instance, a given contour segment < 2 1 3 0 >, < 7 5 9 4 > and < 3 0 2 1 > belongs to < 0 3 1 2 > class.

Adjacent series

Adjacent series consists of a linear representation of the contour segment points relations. Adjacent Series is also known as contour adjacent series (CAS), internal diagonal (INT), and linear representation. See Friedmann (1985, 1987) and Polansky and Bassein (1992) in the bibliography for further information.

For instance, a given contour segment < 0 1 2 0 > has < + - - > adjacent series.

Inversion

Inversion consists of a reflection operation. It renumbers contour points by subtracting from the contour's highest-value point.

For instance, the given contour < 0 3 1 2 >'s inversion is < 3 0 2 1 >.

Original contour segment:

Inverted contour segment:

Retrogression

Retrogression consists of a reflection operation. It is a reversed contour.

For instance, the given contour < 0 2 1 3 >'s retrogression is < 3 1 2 0 >.

Original contour segment:

Retrograted contour segment:

Window reduction (3- and 5-sized)

Window reduction is an interactive contour points removal process where a fixed-width window slides through the contour points. See Bor (2009) in the bibliography for further information.

For instance, the 3-sized window algorithm reduces contour segment A < 0 1 3 2 4 6 5 7 8 > in B < 0 3 2 6 5 8 > and the 5-sized window algorithm reduces the same A contour segment in C < 0 8 >.

Original contour segment:

Three-sized window reduction:

Five-sized window reduction:

Direction index

The direction index is a numeric value that represents the global direction of all the contour points. See Index of ascent/descent (IAD) in Schultz (2009) in the bibliography.

For instance, the direction index of the contour segment < 0 3 2 1 > is 0.5, and of the contour segment < 3 2 1 0 > is 0.

Oscillation index

The oscillation index is a numeric value representing the proportion of a given contour segment's global direction change. See Sampaio (2012) in the bibliography.

Oscillation spectrum

The contour segment is considered a sample of a continuous wave. The oscillation spectrum describes the distribution of frequency components composing this wave. See Schmuckler (2010) in the bibliography.

For instance, let the contours A < 1 2 3 5 4 0 > and B < 1 4 2 3 5 0 > (examples by Schmuckler). The A contour is a half-cycle wave, and the B is a complete cycle. The amplitude values of A's oscillation spectrum are 1.09 0.50 0.17, and B's, 0.60 1.04 0.17. The first partial indicates the half-cycle component, the second the complete partial, and so on.

Contour < 1 2 3 5 4 0 >:

Contour < 1 4 2 3 5 0 >:

Oscillation similarity

The oscillation similarity represents the similarity between two given contours. It is the value of the statistical correlation of the given contours' oscillation spectrum. See Schmuckler (2010) and Sampaio (2018) in the bibliography.

For instance, the oscillation similarity between the contours A < 1 2 3 5 4 0 > and B < 1 4 2 3 5 0 > is 0.352.

Global pattern similarity

The global pattern similarity represents the similarity between two given contours. Ratcliff-Obershelp's algorithm returns the similarity value from given contours' linear representations. See Sampaio (2018) in the bibliography for further information.

For instance, the global pattern similarity between the contours A < 1 2 3 5 4 0 > and B < 1 4 2 3 5 0 > is 0.8.