Calculating Persistent Homology with a Cubical Complex

Sample dataset

For this vignette, we will generate a lattice representing a 2-dimensional image; this will be stored in a variable named sample_image.

# create dataset
sample_image <- matrix(0, nrow = 10, ncol = 10)
i <- 2:9
j <- c(2, 9)
sample_image[i, j] <- 1
sample_image[j, i] <- 1

# view as matrix
sample_image
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    0    0    0    0    0    0    0    0    0     0
#>  [2,]    0    1    1    1    1    1    1    1    1     0
#>  [3,]    0    1    0    0    0    0    0    0    1     0
#>  [4,]    0    1    0    0    0    0    0    0    1     0
#>  [5,]    0    1    0    0    0    0    0    0    1     0
#>  [6,]    0    1    0    0    0    0    0    0    1     0
#>  [7,]    0    1    0    0    0    0    0    0    1     0
#>  [8,]    0    1    0    0    0    0    0    0    1     0
#>  [9,]    0    1    1    1    1    1    1    1    1     0
#> [10,]    0    0    0    0    0    0    0    0    0     0

# view as image
graphics::image(sample_image, useRaster = TRUE, axes = FALSE)

Above, each of the 100 matrix values is analogous to a single pixel in an image.

Calculating persistent homology

Based on the image, we expect a 1-cycle to be present in the persistent homology of sample_image. The CubicalRipser C++ library is wrapped by R using Rcpp, and performs calculations via a cubical complex created with sample_image. These calculations result in a data frame that characterizes the persistent homology of sample_image and can be performed with a single line of R code using ripserr.

# calculate persistent homology
image_phom <- cubical(sample_image)

# print `cubical` output
image_phom
#> PHom object containing persistence data for 2 features.
#> 
#> Contains:
#> * 1 0-dim feature
#> * 1 1-dim feature
#> 
#> Radius/diameter: min = 0; max = 1.

# print features
head(image_phom)
#>   dimension birth death
#> 1         0     0     1
#> 2         1     0     1

Each row in image_phom represents a single feature. The homology matrix has 3 columns:

1. dimension: if 0, represents a 0-cycle; if 1, represents a 1-cycle; and so on.
2. birth: radius of the cubical complex at which this feature begins
3. death: radius of the cubical complex at which this feature ends

Persistence of a feature is generally defined as the length of the interval of the radius within which the feature exists. This is calculated as the numerical difference between the second (birth) and third (death) columns of the homology data frame Confirmed in the output above, the homology data frame is ordered by dimension, with the birth column used to sort features of the same dimension. As expected for sample_image, the homology data frame contains a single 1-cycle. The ggtda and TDAstats R packages can be used to visualize image_phom for further insight.