Choosing a thinning method

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Thinning reduces a binary shape to a one-pixel-wide centerline that preserves topology. Several algorithms exist; they differ in how aggressively they erode pixels, how well they preserve corners, and how fast they run. This vignette gives a short guide to choosing among the algorithms thinr provides.

What’s implemented

Method	Reference	Approach
`zhang_suen`	Zhang & Suen (1984)	2 sub-iterations, mixed-direction products
`guo_hall`	Guo & Hall (1989)	2 sub-iterations, conditions tuned for diagonals
`lee`	Lee, Kashyap & Chu (1994), 2-D	4 directional sub-iterations + Euler-invariance
`k3m`	Saeed et al. (2010)	6 phases of progressively permissive removal
`hilditch`	parallel form (Rutovitz-style)	Single pass with look-ahead crossing-number check
`opta`	Naccache & Shinghal (1984)	Safe-point thinning (SPTA)
`holt`	Holt, Stewart, Clint & Perrott (1987)	One subcycle with edge information on neighbours

zhang_suen is the default.

lee is a 2-D adaptation of Lee’s original 3-D algorithm. The 3-D case (volumetric thinning) is not implemented in this release.

The hilditch method implements the parallel form commonly cited as “Hilditch” in modern image-processing surveys; Hilditch’s 1969 paper actually describes a sequential algorithm with within-pass deletion tracking and a different crossing-number definition. See Lam, Lee & Suen (1992) for the relationship between the two.

The K3M lookup tables A_0 … A_5 are reproduced from Saeed et al. (2010), Section 3.3, page 327. OPTA’s safe-point boolean expression and Holt’s condition H are taken from the original papers; the Lam-Lee-Suen 1992 survey was used as cross-reference and one transcription error in Holt’s middle clause (north vs east) was corrected against the original.

A quick visual comparison

# A 'V' shape — exercises diagonal preservation
make_v <- function() {
  m <- matrix(0L, 11, 11)
  for (k in seq(0, 5)) {
    m[2 + k, 2 + k] <- 1L
    m[2 + k, 10 - k] <- 1L
    m[3 + k, 2 + k] <- 1L
    m[3 + k, 10 - k] <- 1L
  }
  m
}
v <- make_v()
v
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    1    0    0    0    0    0    0    0     1     0
#>  [3,]    0    1    1    0    0    0    0    0    1     1     0
#>  [4,]    0    0    1    1    0    0    0    1    1     0     0
#>  [5,]    0    0    0    1    1    0    1    1    0     0     0
#>  [6,]    0    0    0    0    1    1    1    0    0     0     0
#>  [7,]    0    0    0    0    1    1    1    0    0     0     0
#>  [8,]    0    0    0    0    1    0    1    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
#> [10,]    0    0    0    0    0    0    0    0    0     0     0
#> [11,]    0    0    0    0    0    0    0    0    0     0     0

thin(v, method = "zhang_suen")
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    0    0    0    0    0    0    0     0     0
#>  [4,]    0    0    0    0    0    0    0    0    0     0     0
#>  [5,]    0    0    0    0    0    0    0    0    0     0     0
#>  [6,]    0    0    0    0    1    1    1    0    0     0     0
#>  [7,]    0    0    0    0    0    0    0    0    0     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
#> [10,]    0    0    0    0    0    0    0    0    0     0     0
#> [11,]    0    0    0    0    0    0    0    0    0     0     0

thin(v, method = "guo_hall")
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    1    0    0    0    0    0    0    0     1     0
#>  [3,]    0    1    0    0    0    0    0    0    1     0     0
#>  [4,]    0    0    1    0    0    0    0    1    0     0     0
#>  [5,]    0    0    0    1    0    0    1    0    0     0     0
#>  [6,]    0    0    0    0    1    1    0    0    0     0     0
#>  [7,]    0    0    0    0    0    1    0    0    0     0     0
#>  [8,]    0    0    0    0    1    0    1    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
#> [10,]    0    0    0    0    0    0    0    0    0     0     0
#> [11,]    0    0    0    0    0    0    0    0    0     0     0

thin(v, method = "hilditch")
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    0    0    0    0    0    0    0     0     0
#>  [4,]    0    0    0    0    0    0    0    0    0     0     0
#>  [5,]    0    0    0    0    0    0    0    0    0     0     0
#>  [6,]    0    0    0    0    1    1    1    0    0     0     0
#>  [7,]    0    0    0    0    0    0    0    0    0     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
#> [10,]    0    0    0    0    0    0    0    0    0     0     0
#> [11,]    0    0    0    0    0    0    0    0    0     0     0

thin(v, method = "k3m")
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    0    0    0    0    0    0    0     0     0
#>  [4,]    0    0    0    0    0    0    0    0    0     0     0
#>  [5,]    0    0    0    0    1    0    1    0    0     0     0
#>  [6,]    0    0    0    0    1    0    1    0    0     0     0
#>  [7,]    0    0    0    0    1    1    1    0    0     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
#> [10,]    0    0    0    0    0    0    0    0    0     0     0
#> [11,]    0    0    0    0    0    0    0    0    0     0     0

thin(v, method = "holt")
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    0    0    0    0    0    0    0     0     0
#>  [4,]    0    0    0    0    0    0    0    0    0     0     0
#>  [5,]    0    0    0    0    0    0    0    0    0     0     0
#>  [6,]    0    0    0    0    1    1    1    0    0     0     0
#>  [7,]    0    0    0    0    0    0    0    0    0     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
#> [10,]    0    0    0    0    0    0    0    0    0     0     0
#> [11,]    0    0    0    0    0    0    0    0    0     0     0

The thinning algorithms produce broadly similar skeletons on this V — they all collapse the two diagonal strokes to single lines and meet at the bottom. Differences show up on more complex shapes:

zhang_suen is the most aggressive on simple shapes.
guo_hall and k3m preserve corners marginally better.
hilditch has the look-ahead crossing-number check, which gives different connectivity properties at junctions.
holt uses edge information about neighbouring pixels rather than a crossing-number check on the central pixel; it is specifically designed to preserve 2-pixel-wide lines.

When to use which

zhang_suen — the default. Most predictable behavior. Use for general purpose thinning. One caveat: an isolated 2×2 block is erased entirely (see “Small isolated blobs” below).
guo_hall — try this if your skeletons have lots of diagonal features and Zhang-Suen is breaking them at corners. Keeps one pixel of an isolated 2×2 block where Zhang-Suen erases it.
lee — when you want directional processing (four sub-iterations per pass, one per cardinal direction). Sometimes produces cleaner skeletons on asymmetric inputs.
k3m — strongest corner preservation in published comparative studies, at the cost of being slower (six phases per outer iteration vs. two for Zhang-Suen).
hilditch — well-cited historical algorithm; the look-ahead crossing-number check makes its connectivity slightly different from the other parallel algorithms.
opta — one-pass safe-point algorithm. Its N2 condition protects two-4-adjacent-pixel diagonal patterns, which can leave stray pixels at bar corners (a documented property of SPTA).
holt — when 2-pixel-wide lines should be preserved. The algorithm uses edge information from neighbouring pixels in a 5x5 window, allowing a single subcycle.

Small isolated blobs

The methods disagree on the smallest shapes. An isolated 2×2 block is the notable case: all four of its pixels pass the Zhang-Suen deletion gate in the same sub-iteration, so the default method erases the block completely, while Guo-Hall keeps one pixel.

block <- matrix(0L, 6, 6)
block[3:4, 3:4] <- 1L
sum(thin(block, method = "zhang_suen"))  # 0 — the block is erased
#> [1] 0
sum(thin(block, method = "guo_hall"))    # 1 — one pixel survives
#> [1] 1

This matters when a mask can contain tiny blobs — a marker remnant, a dotted-line dash, a speck left after segmentation. If such blobs must not vanish, use guo_hall (or filter blobs by size before thinning) rather than the default. Larger isolated shapes (3×3 and up) survive under every method; only the 2×2 block is fully erased.

Medial axis transform

The thinning algorithms above all produce binary 1-pixel-wide skeletons without width information. For tasks where local thickness matters, use medial_axis():

m <- matrix(0L, 9, 11)
m[3:7, 3:9] <- 1L      # 5x7 solid rectangle
medial_axis(m)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    1    1    0    0    0    1    1     0     0
#>  [4,]    0    0    1    1    1    0    1    1    1     0     0
#>  [5,]    0    0    0    1    1    1    1    1    0     0     0
#>  [6,]    0    0    1    1    1    0    1    1    1     0     0
#>  [7,]    0    0    1    1    0    0    0    1    1     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0

result <- medial_axis(m, return_distance = TRUE)
result$skeleton
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    1    1    0    0    0    1    1     0     0
#>  [4,]    0    0    1    1    1    0    1    1    1     0     0
#>  [5,]    0    0    0    1    1    1    1    1    0     0     0
#>  [6,]    0    0    1    1    1    0    1    1    1     0     0
#>  [7,]    0    0    1    1    0    0    0    1    1     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0
round(result$distance, 3)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]    0    0    0    0    0    0    0    0    0     0     0
#>  [2,]    0    0    0    0    0    0    0    0    0     0     0
#>  [3,]    0    0    1    1    1    1    1    1    1     0     0
#>  [4,]    0    0    1    2    2    2    2    2    1     0     0
#>  [5,]    0    0    1    2    3    3    3    2    1     0     0
#>  [6,]    0    0    1    2    2    2    2    2    1     0     0
#>  [7,]    0    0    1    1    1    1    1    1    1     0     0
#>  [8,]    0    0    0    0    0    0    0    0    0     0     0
#>  [9,]    0    0    0    0    0    0    0    0    0     0     0

The distance is the Euclidean distance from each foreground pixel to the nearest background pixel.

Distance transform

distance_transform() exposes the distance transform itself as a stand-alone utility, with a choice of metric:

m <- matrix(1L, 5, 5)
m[1, 1] <- 0L
distance_transform(m, metric = "manhattan")
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    1    2    3    4
#> [2,]    1    2    3    4    5
#> [3,]    2    3    4    5    6
#> [4,]    3    4    5    6    7
#> [5,]    4    5    6    7    8
distance_transform(m, metric = "chessboard")
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    1    2    3    4
#> [2,]    1    1    2    3    4
#> [3,]    2    2    2    3    4
#> [4,]    3    3    3    3    4
#> [5,]    4    4    4    4    4
round(distance_transform(m, metric = "euclidean"), 3)
#>      [,1]  [,2]  [,3]  [,4]  [,5]
#> [1,]    0 1.000 2.000 3.000 4.000
#> [2,]    1 1.414 2.236 3.162 4.123
#> [3,]    2 2.236 2.828 3.606 4.472
#> [4,]    3 3.162 3.606 4.243 5.000
#> [5,]    4 4.123 4.472 5.000 5.657

The Euclidean implementation uses the linear-time separable algorithm of Felzenszwalb and Huttenlocher (2012); the others use the classical Rosenfeld and Pfaltz (1968) two-pass sweep.

Inputs and outputs

thin() and medial_axis() accept logical, integer, and numeric matrices. Non-zero values are foreground. The return matrix matches the storage mode of the input — logical in, logical out; integer in, integer out; numeric in, numeric out.

distance_transform() always returns a numeric matrix.

Higher-dimensional arrays (e.g. 3-D volumes) are not yet supported.

Performance

A simple bench::mark() comparison on a moderate image (illustrative):

library(bench)
m <- matrix(0L, 200, 200)
m[50:150, 50:150] <- 1L  # solid square

bench::mark(
  zs       = thin(m, method = "zhang_suen"),
  gh       = thin(m, method = "guo_hall"),
  hild     = thin(m, method = "hilditch"),
  k3m      = thin(m, method = "k3m"),
  ma       = medial_axis(m),
  dt_eucl  = distance_transform(m, metric = "euclidean"),
  iterations = 5,
  check = FALSE
)

All thinning algorithms are O(iterations × pixels). Constant factors are smallest for zhang_suen and guo_hall; holt and k3m are the most expensive because of their look-aheads. The Euclidean distance transform is O(pixels) via Felzenszwalb-Huttenlocher; medial axis is O(pixels) since it just adds a single linear pass over the DT.

For 200×200 images all algorithms finish in a few milliseconds on modern hardware.

References

Thinning

Zhang, T. Y. & Suen, C. Y. (1984). A fast parallel algorithm for thinning digital patterns. Communications of the ACM, 27(3), 236–239.
Guo, Z. & Hall, R. W. (1989). Parallel thinning with two-subiteration algorithms. Communications of the ACM, 32(3), 359–373.
Lee, T.-C., Kashyap, R. L., & Chu, C.-N. (1994). Building skeleton models via 3-D medial surface axis thinning algorithms. CVGIP: Graphical Models and Image Processing, 56(6), 462–478.
Saeed, K., Tabędzki, M., Rybnik, M., & Adamski, M. (2010). K3M: A universal algorithm for image skeletonization and a review of thinning techniques. International Journal of Applied Mathematics and Computer Science, 20(2), 317–335.
Hilditch, C. J. (1969). Linear skeletons from square cupboards. In B. Meltzer & D. Michie (Eds.), Machine Intelligence 4 (pp. 403–420). Edinburgh University Press.
Naccache, N. J. & Shinghal, R. (1984). An investigation into the skeletonization approach of Hilditch. Pattern Recognition, 17(3), 279–284.
Holt, C. M., Stewart, A., Clint, M., & Perrott, R. H. (1987). An improved parallel thinning algorithm. Communications of the ACM, 30(2), 156–160.

Survey

Lam, L., Lee, S.-W., & Suen, C. Y. (1992). Thinning methodologies — a comprehensive survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(9), 869–885.

Medial axis and distance transform

Blum, H. (1967). A transformation for extracting new descriptors of shape. In Models for the Perception of Speech and Visual Form (pp. 362–380). MIT Press.
Felzenszwalb, P. F. & Huttenlocher, D. P. (2012). Distance transforms of sampled functions. Theory of Computing, 8(19), 415–428.
Rosenfeld, A. & Pfaltz, J. L. (1968). Distance functions on digital pictures. Pattern Recognition, 1(1), 33–61.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.