Zhang-Suen thinning algorithm
This is an algorithm used to thin a black and white i.e. one bit per pixel images.
For example, with an input image of:
################# #############
################## ################
################### ##################
######## ####### ###################
###### ####### ####### ######
###### ####### #######
################# #######
################ #######
################# #######
###### ####### #######
###### ####### #######
###### ####### ####### ######
######## ####### ###################
######## ####### ###### ################## ######
######## ####### ###### ################ ######
######## ####### ###### ############# ######
It produces the thinned output:
# ########## #######
## # #### #
# # ##
# # #
# # #
# # #
############ #
# # #
# # #
# # #
# # #
# ##
# ############
### ###
- Algorithm
Assume black pixels are one and white pixels zero, and that the input image is a rectangular N by M array of ones and zeroes.
The algorithm operates on all black pixels P1 that can have eight neighbours. The neighbours are, in order, arranged as:
| P9 | P2 | P3 |
| P8 | P1 | P4 |
| P7 | P6 | P5 |
Obviously the boundary pixels of the image cannot have the full eight neighbours.
- Define = the number of transitions from white to black, (0 -> 1) in the sequence P2,P3,P4,P5,P6,P7,P8,P9,P2. (Note the extra P2 at the end - it is circular).
- Define = The number of black pixel neighbours of P1. ( = sum(P2 .. P9) )
- Step 1
All pixels are tested and pixels satisfying all the following conditions are just noted at this stage.
- (0) The pixel is black and has eight neighbours
- (1)
- (2) A(P1) = 1
- (3) At least one of P2 and P4 and P6 is white
- (4) At least one of P4 and P6 and P8 is white
After iterating over the image and collecting all the pixels satisfying all step 1 conditions, all these condition satisfying pixels are set to white.
- Step 2
All pixels are again tested and pixels satisfying all the following conditions are just noted at this stage.
- (0) The pixel is black and has eight neighbours
- (1)
- (2) A(P1) = 1
- (3) At least one of P2 and P4 and P8 is white
- (4) At least one of P2 and P6 and P8 is white
After iterating over the image and collecting all the pixels satisfying all step 2 conditions, all these condition satisfying pixels are again set to white.
- Iteration
If any pixels were set in this round of either step 1 or step 2 then all steps are repeated until no image pixels are so changed.
- Task
- Write a routine to perform Zhang-Suen thinning on an image matrix of ones and zeroes.
- Use the routine to thin the following image and show the output here on this page as either a matrix of ones and zeroes, an image, or an ASCII-art image of space/non-space characters.
00000000000000000000000000000000 01111111110000000111111110000000 01110001111000001111001111000000 01110000111000001110000111000000 01110001111000001110000000000000 01111111110000001110000000000000 01110111100000001110000111000000 01110011110011101111001111011100 01110001111011100111111110011100 00000000000000000000000000000000
- Reference
- Zhang-Suen Thinning Algorithm, Java Implementation by Nayef Reza.
- "Character Recognition Systems: A Guide for Students and Practitioners" By Mohamed Cheriet, Nawwaf Kharma, Cheng-Lin Liu, Ching Suen
D
This uses the module from the Bitmap Task.
<lang d>import std.stdio, std.algorithm, std.string, std.functional, bitmap;
alias sum = curry!(reduce!q{a + b}, 0); //*
struct BlackWhite {
ubyte c; static immutable black = typeof(this)(0); static immutable white = typeof(this)(1); alias c this;
}
alias Neighbours = BlackWhite[9]; alias Img = Image!BlackWhite;
/// Zhang-Suen thinning algorithm. Img zhangSuen(Img image1) pure /*nothrow*/ in {
assert(image1.image.all!(x => x == Img.black || x == Img.white));
} out(result) {
assert(result.nx == image1.nx && result.ny == image1.ny); assert(result.image.all!(x => x == Img.black || x == Img.white));
} body {
/// True if inf <= x <= sup.
static inInterval(T)(in T x, in T inf, in T sup) pure nothrow {
return x >= inf && x <= sup;
}
/// Return 8-neighbours+1 of point (x,y) of given image, in order.
static void neighbours(in Img I, in size_t x, in size_t y,
ref Neighbours n) pure nothrow {
n = [I[x,y-1], I[x+1,y-1], I[x+1,y], I[x+1,y+1], // P2,P3,P4,P5
I[x,y+1], I[x-1,y+1], I[x-1,y], I[x-1,y-1], // P6,P7,P8,P9
I[x,y-1]];
}
if (image1.nx <= 3 || image1.ny <= 3)
return image1;
auto image2 = Img.fromData(image1.image.dup, image1.nx, image1.ny);
Neighbours n;
bool hasChanged;
do {
hasChanged = false;
// Step 1:
foreach (immutable y; 1 .. image1.ny - 1) {
foreach (immutable x; 1 .. image1.nx - 1) {
neighbours(image1, x, y, n);
if (image1[x, y] && // Condition 0
(!n[2] || !n[4] || !n[6]) && // Condition 4
(!n[0] || !n[2] || !n[4]) && // Condition 3
n[].count([0, 1]) == 1 && // Condition 2
// n[0 .. 8].sum in iota(2, 7)) {
inInterval(n[0 .. 8].sum, 2, 6)) { // Condition 1
hasChanged = true;
image2[x, y] = Img.black;
} else
image2[x, y] = image1[x, y];
}
}
// Step 2:
foreach (immutable y; 1 .. image1.ny - 1) {
foreach (immutable x; 1 .. image1.nx - 1) {
neighbours(image2, x, y, n);
if (image2[x, y] && // Condition 0
(!n[0] || !n[4] || !n[6]) && // Condition 4
(!n[0] || !n[2] || !n[6]) && // Condition 3
n[].count([0, 1]) == 1 && // Condition 2
inInterval(n[0 .. 8].sum, 2, 6)) { // Condition 1
hasChanged = true;
image1[x, y] = Img.black;
} else
image1[x, y] = image2[x, y];
}
}
} while (hasChanged);
return image1;
}
void main() {
immutable before_txt = " ##..### ##..### ##..### ##..### ##..##. ##..##. ##..##. ##..##. ##..##. ##..##. ##..##. ##..##. ######. .......";
immutable small_rc = " ................................ .#########.......########....... .###...####.....####..####...... .###....###.....###....###...... .###...####.....###............. .#########......###............. .###.####.......###....###...... .###..####..###.####..####.###.. .###...####.###..########..###.. ................................";
immutable rc = " ........................................................... .#################...................#############......... .##################...............################......... .###################............##################......... .########.....#######..........###################......... ...######.....#######.........#######.......######......... ...######.....#######........#######....................... ...#################.........#######....................... ...################..........#######....................... ...#################.........#######....................... ...######.....#######........#######....................... ...######.....#######........#######....................... ...######.....#######.........#######.......######......... .########.....#######..........###################......... .########.....#######.######....##################.######.. .########.....#######.######......################.######.. .########.....#######.######.........#############.######.. ...........................................................";
foreach (immutable txt; [before_txt, small_rc, rc]) {
auto img = Img.fromText(txt);
"From:".writeln;
img.textualShow(/*bl=*/ '.', /*wh=*/ '#');
"\nTo thinned:".writeln;
img.zhangSuen.textualShow(/*bl=*/ '.', /*wh=*/ '#');
writeln;
}
}</lang>
- Output:
From: ##..### ##..### ##..### ##..### ##..##. ##..##. ##..##. ##..##. ##..##. ##..##. ##..##. ##..##. ######. ....... To thinned: ##..### #.....# #.....# #...### #...#.. #...#.. #...#.. #...#.. #...#.. #...#.. #...#.. #...#.. #####.. ....... From: ................................ .#########.......########....... .###...####.....####..####...... .###....###.....###....###...... .###...####.....###............. .#########......###............. .###.####.......###....###...... .###..####..###.####..####.###.. .###...####.###..########..###.. ................................ To thinned: ................................ ..#######.........######........ ..#.....#........##............. ..#......#.......#.............. ..#.....#........#.............. ..#####.#........#.............. .......##........#.............. ........#....#...##....##...#... .........#.........####......... ................................ From: ........................................................... .#################...................#############......... .##################...............################......... .###################............##################......... .########.....#######..........###################......... ...######.....#######.........#######.......######......... ...######.....#######........#######....................... ...#################.........#######....................... ...################..........#######....................... ...#################.........#######....................... ...######.....#######........#######....................... ...######.....#######........#######....................... ...######.....#######.........#######.......######......... .########.....#######..........###################......... .########.....#######.######....##################.######.. .########.....#######.######......################.######.. .########.....#######.######.........#############.######.. ........................................................... To thinned: ........................................................... ........................................................... ....#.##########.......................#######............. .....##........#...................####.......#............ .....#..........#.................##....................... .....#..........#................#......................... .....#..........#................#......................... .....#..........#................#......................... .....############...............#.......................... .....#..........#...............#.......................... .....#..........#................#......................... .....#..........#................#......................... .....#..........#................#......................... .....#............................##....................... .....#.............................############............ .......................###..........................###.... ........................................................... ...........................................................
Python
Several input images are converted. <lang python># -*- coding: utf-8 -*-
- Example from this blog post.
beforeTxt = \ 1100111 1100111 1100111 1100111 1100110 1100110 1100110 1100110 1100110 1100110 1100110 1100110 1111110 0000000\
- Thanks to this site and vim for these next two examples
smallrc01 = \ 00000000000000000000000000000000 01111111110000000111111110000000 01110001111000001111001111000000 01110000111000001110000111000000 01110001111000001110000000000000 01111111110000001110000000000000 01110111100000001110000111000000 01110011110011101111001111011100 01110001111011100111111110011100 00000000000000000000000000000000\
rc01 = \ 00000000000000000000000000000000000000000000000000000000000 01111111111111111100000000000000000001111111111111000000000 01111111111111111110000000000000001111111111111111000000000 01111111111111111111000000000000111111111111111111000000000 01111111100000111111100000000001111111111111111111000000000 00011111100000111111100000000011111110000000111111000000000 00011111100000111111100000000111111100000000000000000000000 00011111111111111111000000000111111100000000000000000000000 00011111111111111110000000000111111100000000000000000000000 00011111111111111111000000000111111100000000000000000000000 00011111100000111111100000000111111100000000000000000000000 00011111100000111111100000000111111100000000000000000000000 00011111100000111111100000000011111110000000111111000000000 01111111100000111111100000000001111111111111111111000000000 01111111100000111111101111110000111111111111111111011111100 01111111100000111111101111110000001111111111111111011111100 01111111100000111111101111110000000001111111111111011111100 00000000000000000000000000000000000000000000000000000000000\
def intarray(binstring):
Change a 2D matrix of 01 chars into a list of lists of ints
return [[1 if ch == '1' else 0 for ch in line]
for line in binstring.strip().split()]
def chararray(intmatrix):
Change a 2d list of lists of 1/0 ints into lines of 1/0 chars return '\n'.join(.join(str(p) for p in row) for row in intmatrix)
def toTxt(intmatrix):
Change a 2d list of lists of 1/0 ints into lines of '#' and '.' chars
return '\n'.join(.join(('#' if p else '.') for p in row) for row in intmatrix)
def neighbours(x, y, image):
Return 8-neighbours of point p1 of picture, in order
i = image
x1, y1, x_1, y_1 = x+1, y-1, x-1, y+1
#print ((x,y))
return [i[y1][x], i[y1][x1], i[y][x1], i[y_1][x1], # P2,P3,P4,P5
i[y_1][x], i[y_1][x_1], i[y][x_1], i[y1][x_1]] # P6,P7,P8,P9
def transitions(neighbours):
n = neighbours + neighbours[0:1] # P2, ... P9, P2 return sum((n1, n2) == (0, 1) for n1, n2 in zip(n, n[1:]))
def zhangSuen(image):
changing1 = changing2 = [(-1, -1)]
while changing1 or changing2:
# Step 1
changing1 = []
for y in range(1, len(image) - 1):
for x in range(1, len(image[0]) - 1):
P2,P3,P4,P5,P6,P7,P8,P9 = n = neighbours(x, y, image)
if (image[y][x] == 1 and # (Condition 0)
P4 * P6 * P8 == 0 and # Condition 4
P2 * P4 * P6 == 0 and # Condition 3
transitions(n) == 1 and # Condition 2
2 <= sum(n) <= 6): # Condition 1
changing1.append((x,y))
for x, y in changing1: image[y][x] = 0
# Step 2
changing2 = []
for y in range(1, len(image) - 1):
for x in range(1, len(image[0]) - 1):
P2,P3,P4,P5,P6,P7,P8,P9 = n = neighbours(x, y, image)
if (image[y][x] == 1 and # (Condition 0)
P2 * P6 * P8 == 0 and # Condition 4
P2 * P4 * P8 == 0 and # Condition 3
transitions(n) == 1 and # Condition 2
2 <= sum(n) <= 6): # Condition 1
changing2.append((x,y))
for x, y in changing2: image[y][x] = 0
#print changing1
#print changing2
return image
if __name__ == '__main__':
for picture in (beforeTxt, smallrc01, rc01):
image = intarray(picture)
print('\nFrom:\n%s' % toTxt(image))
after = zhangSuen(image)
print('\nTo thinned:\n%s' % toTxt(after))</lang>
- Output:
Just the example asked for in the task:
From: ................................ .#########.......########....... .###...####.....####..####...... .###....###.....###....###...... .###...####.....###............. .#########......###............. .###.####.......###....###...... .###..####..###.####..####.###.. .###...####.###..########..###.. ................................ To thinned: ................................ ..#######.........######........ ..#.....#........##............. ..#......#.......#.............. ..#.....#........#.............. ..#####.#........#.............. .......##........#.............. ........#....#...##....##...#... .........#.........####......... ................................
Racket
<lang racket>#lang racket (define (img-01string->vector str)
(define lines (regexp-split "\n" str))
(define h (length lines))
(define w (if (zero? h) 0 (string-length (car lines))))
(define v (for*/vector #:length (* w h)
((l (in-list lines)) (p (in-string l)))
(match p (#\0 0) (#\1 1) (#\# 1) (#\. 0))))
(values v h w))
- Task (2) asks for "or an ASCII-art image of space/non-space characters."
- Spaces don't really impress where the borders are, so we'll use a dot.
(define cell->display-char (match-lambda (0 ".") (1 "#") (else "?")))
(define (display-img v w)
(for ((p (in-vector v)) (col (in-naturals))) (printf "~a" (cell->display-char p)) (when (= (modulo col w) (sub1 w)) (newline))))
- returns vector of ([P1's idx] P1 P2 ... P9)
(define (Pns v w r c)
(define i (+ c (* r w)))
(define-syntax-rule (vi+ x) (vector-ref v (+ i x)))
(define-syntax-rule (vi- x) (vector-ref v (- i x)))
(vector i (vi+ 0) (vi- w) (vi+ (- 1 w))
(vi+ 1) (vi+ (+ w 1)) (vi+ w)
(vi+ (- w 1)) (vi- 1) (vi- (+ w 1))))
- Second argument to in-vector is the start offset;
- We skip offset 0 (idx) and 1 (P1)
(define (B Ps) (for/sum ((Pn (in-vector Ps 2))) Pn))
(define (A Ps)
(define P2 (vector-ref Ps 2))
(define-values (rv _)
(for/fold ((acc 0) (Pn-1 P2))
((Pn (in-sequences (in-vector Ps 3) (in-value P2))))
(values (+ acc (if (and (= 0 Pn-1) (= 1 Pn)) 1 0)) Pn)))
rv)
(define-syntax-rule (not-all-black? Pa Pb Pc) (zero? (* Pa Pb Pc))) (define (z-s-thin v h w)
; return idx when thin necessary, #f otherwise
(define (thin? Ps n/bour-check-1 n/bour-check-2)
(match-define (vector idx P1 P2 _ P4 _ P6 _ P8 _) Ps)
(and (= P1 1) (<= 2 (B Ps) 6) (= (A Ps) 1)
(n/bour-check-1 P2 P4 P6 P8)
(n/bour-check-2 P2 P4 P6 P8)
idx))
(define (has-white?-246 P2 P4 P6 P8) (not-all-black? P2 P4 P6))
(define (has-white?-468 P2 P4 P6 P8) (not-all-black? P4 P6 P8))
(define (has-white?-248 P2 P4 P6 P8) (not-all-black? P2 P4 P8))
(define (has-white?-268 P2 P4 P6 P8) (not-all-black? P2 P6 P8))
(define (step-n even-Pn-check-1 even-Pn-check-2)
(for*/list ((r (in-range 1 (- h 1)))
(c (in-range 1 (- w 1)))
(idx (in-value (thin? (Pns v w r c)
even-Pn-check-1
even-Pn-check-2)))
#:when idx) idx))
(define (step-1) (step-n has-white?-246 has-white?-468))
(define (step-2) (step-n has-white?-248 has-white?-268))
(define (inner-z-s-thin)
(define changed-list-1 (step-1))
(for ((idx (in-list changed-list-1))) (vector-set! v idx 0))
(define changed-list-2 (step-2))
(for ((idx (in-list changed-list-2))) (vector-set! v idx 0))
(unless (and (null? changed-list-1) (null? changed-list-2)) (inner-z-s-thin)))
(inner-z-s-thin))
(define (read-display-thin-display-image img-str)
(define-values (v h w) (img-01string->vector img-str)) (printf "Original image:~%") (display-img v w) (z-s-thin v h w) (printf "Thinned image:~%") (display-img v w))
(define e.g.-image #<<EOS 00000000000000000000000000000000 01111111110000000111111110000000 01110001111000001111001111000000 01110000111000001110000111000000 01110001111000001110000000000000 01111111110000001110000000000000 01110111100000001110000111000000 01110011110011101111001111011100 01110001111011100111111110011100 00000000000000000000000000000000 EOS
)
(define e.g.-image/2 #<<EOS
- ..###
- ..###
- ..###
- ..###
- ..##.
- ..##.
- ..##.
- ..##.
- ..##.
- ..##.
- ..##.
- ..##.
- .
....... EOS
)
(module+ main
; (read-display-thin-display-image e.g.-image/2) ; (newline) (read-display-thin-display-image e.g.-image))</lang>
- Output:
Only the requested image is output:
Original image: ................................ .#########.......########....... .###...####.....####..####...... .###....###.....###....###...... .###...####.....###............. .#########......###............. .###.####.......###....###...... .###..####..###.####..####.###.. .###...####.###..########..###.. ................................ Thinned image: ................................ ..#######.........######........ ..#.....#........##............. ..#......#.......#.............. ..#.....#........#.............. ..#####.#........#.............. .......##........#.............. ........#....#...##....##...#... .........#.........####......... ................................
Ruby
<lang ruby>
- Thinning RC
- Nigel_Galloway: October 18th., 2013.
s2 = [[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0],
[0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0],
[0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0],
[0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0],
[0,1,1,1,0,0,1,1,1,1,0,0,1,1,1,0,1,1,1,1,0,0,1,1,1,1,0,1,1,1,0,0],
[0,1,1,1,0,0,0,1,1,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,1,0,0,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]];
@r = 1
def zs(ng,g)
return 0 if ng[1][1] == 0 or (ng[1][2] + ng[0][1] + ng[1+g][g]) == 3 or (ng[g][1+g] + ng[2][1] + ng[1][0]) == 3
t = -1; ng.each{|n| n.each{|g| t+=g}}; return 0 unless (2 <= t and t <= 6)
t = -1; [ng[0][1],ng[0][2],ng[1][2],ng[2][2],ng[2][1],ng[2][0],ng[1][0],ng[0][0],ng[0][1]].each{|n|
if n==0 then t += (t==0 or t==2)? 0 : 1 else t += (t==0 or t==2)? 1 : 0 end}
return 0 unless t==1 or t==2
return @r=1
end
s2.each{|row| row.each{|col| print(col==1? "#": " ")}; print("\n")} while @r == 1
@r = 0
s1 = Array.new(s2.length){Array.new(s2[0].length,0)}
(1...s2.length-1).each{|n| (1...s2[0].length-1).each{|g| s1[n][g] = s2[n][g] - zs(s2[n-1..n+1].collect{|n| n[g-1..g+1]},0)}}
s2 = Array.new(s1.length){Array.new(s1[0].length,0)}
(1...s2.length-1).each{|n| (1...s2[0].length-1).each{|g| s2[n][g] = s1[n][g] - zs(s1[n-1..n+1].collect{|n| n[g-1..g+1]},1)}}
end s2.each{|row| row.each{|col| print(col==1? "#": " ")}; print("\n")} </lang>
- Output:
######### ########
### #### #### ####
### ### ### ###
### #### ###
######### ###
### #### ### ###
### #### ### #### #### ###
### #### ### ######## ###
##### ####
# # # ##
# # #
# # #
#### # #
# ## #
# # #
# # ###### #