Median filter

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Task
Median filter
You are encouraged to solve this task according to the task description, using any language you may know.

The median filter takes in the neighbourhood the median color (see Median filter)

(to test the function below, you can use these input and output solutions)

Contents

[edit] Ada

function Median (Picture : Image; Radius : Positive) return Image is
type Extended_Luminance is range 0..10_000_000;
type VRGB is record
Color : Pixel;
Value : Luminance;
end record;
Width : constant Positive := 2*Radius*(Radius+1);
type Window is array (-Width..Width) of VRGB;
Sorted : Window;
Next  : Integer;
 
procedure Put (Color : Pixel) is -- Sort using binary search
pragma Inline (Put);
This  : constant Luminance :=
Luminance
( ( 2_126 * Extended_Luminance (Color.R)
+ 7_152 * Extended_Luminance (Color.G)
+ 722 * Extended_Luminance (Color.B)
)
/ 10_000
);
That  : Luminance;
Low  : Integer := Window'First;
High  : Integer := Next - 1;
Middle : Integer := (Low + High) / 2;
begin
while Low <= High loop
That  := Sorted (Middle).Value;
if That > This then
High := Middle - 1;
elsif That < This then
Low := Middle + 1;
else
exit;
end if;
Middle := (Low + High) / 2;
end loop;
Sorted (Middle + 1..Next) := Sorted (Middle..Next - 1);
Sorted (Middle) := (Color, This);
Next := Next + 1;
end Put;
Result : Image (Picture'Range (1), Picture'Range (2));
begin
for I in Picture'Range (1) loop
for J in Picture'Range (2) loop
Next := Window'First;
for X in I - Radius .. I + Radius loop
for Y in J - Radius .. J + Radius loop
Put
( Picture
( Integer'Min (Picture'Last (1), Integer'Max (Picture'First (1), X)),
Integer'Min (Picture'Last (2), Integer'Max (Picture'First (2), Y))
) );
end loop;
end loop;
Result (I, J) := Sorted (0).Color;
end loop;
end loop;
return Result;
end Median;

The implementation works with an arbitrary window width. The window is specified by its radius R>0. The resulting width is 2R+1. The filter uses the original pixels of the image from the median of the window sorted according to the luminance. The image edges are extrapolated using the nearest pixel on the border. Sorting uses binary search. (For practical use, note that median filter is extremely slow.)

The following sample code illustrates use:

   F1, F2 : File_Type;
begin
Open (F1, In_File, "city.ppm");
Create (F2, Out_File, "city_median.ppm");
Put_PPM (F2, Median (Get_PPM (F1), 1)); -- Window 3x3
Close (F1);
Close (F2);

[edit] BBC BASIC

This example is a 5 x 5 median filter:

Greyscale bbc.jpg
Median bbc.jpg
      INSTALL @lib$+"SORTLIB"
Sort% = FN_sortinit(0,0)
 
Width% = 200
Height% = 200
 
DIM out&(Width%-1, Height%-1)
 
VDU 23,22,Width%;Height%;8,16,16,128
*DISPLAY Lenagrey
OFF
 
REM Do the median filtering:
DIM pix&(24)
C% = 25
FOR Y% = 2 TO Height%-3
FOR X% = 2 TO Width%-3
P% = 0
FOR I% = -2 TO 2
FOR J% = -2 TO 2
pix&(P%) = TINT((X%+I%)*2, (Y%+J%)*2) AND &FF
P% += 1
NEXT
NEXT
CALL Sort%, pix&(0)
out&(X%, Y%) = pix&(12)
NEXT
NEXT Y%
 
REM Display:
GCOL 1
FOR Y% = 0 TO Height%-1
FOR X% = 0 TO Width%-1
COLOUR 1, out&(X%,Y%), out&(X%,Y%), out&(X%,Y%)
LINE X%*2,Y%*2,X%*2,Y%*2
NEXT
NEXT Y%
 
REPEAT
WAIT 1
UNTIL FALSE

[edit] C

O(n) filter with histogram.

#include <stdio.h>
#include <stdlib.h>
#include <fcntl.h>
#include <unistd.h>
#include <ctype.h>
#include <string.h>
 
typedef struct { unsigned char r, g, b; } rgb_t;
typedef struct {
int w, h;
rgb_t **pix;
} image_t, *image;
 
typedef struct {
int r[256], g[256], b[256];
int n;
} color_histo_t;
 
int write_ppm(image im, char *fn)
{
FILE *fp = fopen(fn, "w");
if (!fp) return 0;
fprintf(fp, "P6\n%d %d\n255\n", im->w, im->h);
fwrite(im->pix[0], 1, sizeof(rgb_t) * im->w * im->h, fp);
fclose(fp);
return 1;
}
 
image img_new(int w, int h)
{
int i;
image im = malloc(sizeof(image_t) + h * sizeof(rgb_t*)
+ sizeof(rgb_t) * w * h);
im->w = w; im->h = h;
im->pix = (rgb_t**)(im + 1);
for (im->pix[0] = (rgb_t*)(im->pix + h), i = 1; i < h; i++)
im->pix[i] = im->pix[i - 1] + w;
return im;
}
 
int read_num(FILE *f)
{
int n;
while (!fscanf(f, "%d ", &n)) {
if ((n = fgetc(f)) == '#') {
while ((n = fgetc(f)) != '\n')
if (n == EOF) break;
if (n == '\n') continue;
} else return 0;
}
return n;
}
 
image read_ppm(char *fn)
{
FILE *fp = fopen(fn, "r");
int w, h, maxval;
image im = 0;
if (!fp) return 0;
 
if (fgetc(fp) != 'P' || fgetc(fp) != '6' || !isspace(fgetc(fp)))
goto bail;
 
w = read_num(fp);
h = read_num(fp);
maxval = read_num(fp);
if (!w || !h || !maxval) goto bail;
 
im = img_new(w, h);
fread(im->pix[0], 1, sizeof(rgb_t) * w * h, fp);
bail:
if (fp) fclose(fp);
return im;
}
 
void del_pixels(image im, int row, int col, int size, color_histo_t *h)
{
int i;
rgb_t *pix;
 
if (col < 0 || col >= im->w) return;
for (i = row - size; i <= row + size && i < im->h; i++) {
if (i < 0) continue;
pix = im->pix[i] + col;
h->r[pix->r]--;
h->g[pix->g]--;
h->b[pix->b]--;
h->n--;
}
}
 
void add_pixels(image im, int row, int col, int size, color_histo_t *h)
{
int i;
rgb_t *pix;
 
if (col < 0 || col >= im->w) return;
for (i = row - size; i <= row + size && i < im->h; i++) {
if (i < 0) continue;
pix = im->pix[i] + col;
h->r[pix->r]++;
h->g[pix->g]++;
h->b[pix->b]++;
h->n++;
}
}
 
void init_histo(image im, int row, int size, color_histo_t*h)
{
int j;
 
memset(h, 0, sizeof(color_histo_t));
 
for (j = 0; j < size && j < im->w; j++)
add_pixels(im, row, j, size, h);
}
 
int median(const int *x, int n)
{
int i;
for (n /= 2, i = 0; i < 256 && (n -= x[i]) > 0; i++);
return i;
}
 
void median_color(rgb_t *pix, const color_histo_t *h)
{
pix->r = median(h->r, h->n);
pix->g = median(h->g, h->n);
pix->b = median(h->b, h->n);
}
 
image median_filter(image in, int size)
{
int row, col;
image out = img_new(in->w, in->h);
color_histo_t h;
 
for (row = 0; row < in->h; row ++) {
for (col = 0; col < in->w; col++) {
if (!col) init_histo(in, row, size, &h);
else {
del_pixels(in, row, col - size, size, &h);
add_pixels(in, row, col + size, size, &h);
}
median_color(out->pix[row] + col, &h);
}
}
 
return out;
}
 
int main(int c, char **v)
{
int size;
image in, out;
if (c <= 3) {
printf("Usage: %s size ppm_in ppm_out\n", v[0]);
return 0;
}
size = atoi(v[1]);
printf("filter size %d\n", size);
if (size < 0) size = 1;
 
in = read_ppm(v[2]);
out = median_filter(in, size);
write_ppm(out, v[3]);
free(in);
free(out);
 
return 0;
}

[edit] D

This uses modules of the Bitmap and Grayscale image Tasks.

The implementation uses algorithm described in Median Filtering in Constant Time paper with some slight differences, that shouldn't have impact on complexity.

Currently this code works only on greyscale images.

import grayscale_image;
 
Image!Color medianFilter(uint radius=10, Color)(in Image!Color img)
pure nothrow if (radius > 0) {
alias Hist = uint[256];
 
static ubyte median(uint no)(in ref Hist cumulative) pure nothrow {
size_t localSum = 0;
foreach (immutable ubyte k, immutable v; cumulative)
if (v) {
localSum += v;
if (localSum > no / 2)
return k;
}
return 0;
}
 
// Copy image borders in the result image.
auto result = new Image!Color(img.nx, img.ny);
foreach (immutable y; 0 .. img.ny)
foreach (immutable x; 0 .. img.nx)
if (x < radius || x > img.nx - radius ||
y < radius || y > img.ny - radius)
result[x, y] = img[x, y];
 
enum edge = 2 * radius + 1;
auto hCol = new Hist[img.nx];
 
// Create histogram columns.
foreach (immutable y; 0 .. edge - 1)
foreach (immutable x, ref hx; hCol)
hx[img[x, y]]++;
 
foreach (immutable y; radius .. img.ny - radius) {
// Add to each histogram column lower pixel.
foreach (immutable x, ref hx; hCol)
hx[img[x, y + radius]]++;
 
// Calculate main Histogram using first edge-1 columns.
Hist H;
foreach (immutable x; 0 .. edge - 1)
foreach (immutable k, immutable v; hCol[x])
if (v)
H[k] += v;
 
foreach (immutable x; radius .. img.nx - radius) {
// Add right-most column.
foreach (immutable k, immutable v; hCol[x + radius])
if (v)
H[k] += v;
 
result[x, y] = Color(median!(edge ^^ 2)(H));
 
// Drop left-most column.
foreach (immutable k, immutable v; hCol[x - radius])
if (v)
H[k] -= v;
}
 
// Substract the upper pixels.
foreach (immutable x, ref hx; hCol)
hx[img[x, y - radius]]--;
}
 
return result;
}
 
version (median_filter_main) {
void main() { // Demo.
loadPGM!Gray(null, "lena.pgm")
.medianFilter!10()
.savePGM("lena_median_r10.pgm");
}
}

Compile with -version=median_filter_main to run the demo.

[edit] GDL

GDL has no inbuilt median filter function, which is native in IDL. This example is based on pseudocode here: http://en.wikipedia.org/wiki/Median_filter#2D_median_filter_pseudo_code, however, it works with 1D arrays only. It does not process boundaries.

 
FUNCTION MEDIANF,ARRAY,WINDOW
RET=fltarr(N_ELEMENTS(ARRAY),1)
EDGEX=WINDOW/2
FOR X=EDGEX, N_ELEMENTS(ARRAY)-EDGEX DO BEGIN
PRINT, "X", X
COLARRAY=fltarr(WINDOW,1)
FOR FX=0, WINDOW-1 DO BEGIN
COLARRAY[FX]=ARRAY[X + FX - EDGEX]
END
T=COLARRAY[SORT(COLARRAY)]
RET[X]=T[WINDOW/2]
END
RETURN, RET
END
 

Usage:

Result = MEDIANF(ARRAY, WINDOW)

[edit] Go

Implemented with existing GetPx/SetPx functions at Grayscale image task. It could be sped up by putting code in the raster package, but if you're concerned about speed, you should implement one of the O(n) algorithms available.

package main
 
// Files required to build supporting package raster are found in:
// * Bitmap
// * Grayscale image
// * Read a PPM file
// * Write a PPM file
 
import (
"fmt"
"raster"
)
 
var g0, g1 *raster.Grmap
var ko [][]int
var kc []uint16
var mid int
 
func init() {
// hard code box of 9 pixels
ko = [][]int{
{-1, -1}, {0, -1}, {1, -1},
{-1, 0}, {0, 0}, {1, 0},
{-1, 1}, {0, 1}, {1, 1}}
kc = make([]uint16, len(ko))
mid = len(ko) / 2
}
 
func main() {
// Example file used here is Lenna50.jpg from the task "Percentage
// difference between images" converted with with the command
// convert Lenna50.jpg -colorspace gray Lenna50.ppm
// It shows very obvious compression artifacts when viewed at higher
// zoom factors.
b, err := raster.ReadPpmFile("Lenna50.ppm")
if err != nil {
fmt.Println(err)
return
}
g0 = b.Grmap()
w, h := g0.Extent()
g1 = raster.NewGrmap(w, h)
for y := 0; y < h; y++ {
for x := 0; x < w; x++ {
g1.SetPx(x, y, median(x, y))
}
}
// side by side comparison with input file shows compression artifacts
// greatly smoothed over, although at some loss of contrast.
err = g1.Bitmap().WritePpmFile("median.ppm")
if err != nil {
fmt.Println(err)
}
}
 
func median(x, y int) uint16 {
var n int
// construct sorted list as pixels are read. insertion sort can't be
// beat for a small number of items, plus there would be lots of overhead
// just to get numbers in and out of a library sort routine.
for _, o := range ko {
// read a pixel of the kernel
c, ok := g0.GetPx(x+o[0], y+o[1])
if !ok {
continue
}
// insert it in sorted order
var i int
for ; i < n; i++ {
if c < kc[i] {
for j := n; j > i; j-- {
kc[j] = kc[j-1]
}
break
}
}
kc[i] = c
n++
}
// compute median from sorted list
switch {
case n == len(kc): // the usual case, pixel with complete neighborhood
return kc[mid]
case n%2 == 1: // edge case, odd number of pixels
return kc[n/2]
}
// else edge case, even number of pixels
m := n / 2
return (kc[m-1] + kc[m]) / 2
}

[edit] J

The task could be solved the following way. First, for each pixel of input, collect pixels which fall into the corresponding window, where median value will be calculated. Then, for each window - the set of pixels - find the median value. To compare 3-channel pixels we first convert them into 1-channel gray values.

The following verbs are used to work with bitmaps:

 
makeRGB=: 0&$: : (($,)~ ,&3)
toGray=: <. @: (+/) @: (0.2126 0.7152 0.0722 & *)"1
 

We'll determine the window as a square zone around each pixel, with the given pixel in the center of the zone. Such a window always have odd height and width. We'll say the window radius is 0 if the window contain only the given pixel - in this case the resulting picture will be identical to the input. The radius is 1 if the window is 3x3 pixels, with given pixel in the center. Radius is 2 if the window is 5x5 pixels, with given pixel in the center, etc.

To get all pixels in the window, first calculate coordinates - or indexes - of those pixels. For the pixels on the edges of the input bitmap, include only those indexes which correspond to actually existing pixels - no negative indexes and no indexes outside of the bitmap boundaries.

 
median_filter =: dyad define
win =. y -~ i. >: +: y
height =. {: }: $ x
width =. {. }: $ x
h_indexes =. < @ (#~ >:&0 * <&height) @ (win&+)"0 i. height
w_indexes =. < @ (#~ >:&0 * <&width) @ (win&+)"0 i. width
sets =. w_indexes < @ ({&x) @ < @ ,"0 0/ h_indexes
medians =. ({~ <. @ -: @ {. @ $) @ ({~ /: @: toGray) @ (,/) @ > sets
)
 

Example:

 
] bmp =. ?. 256 + makeRGB 4 5
34 39 168
133 133 40
210 137 244
66 183 114
211 241 75
 
212 68 13
91 246 128
203 236 213
162 92 165
90 203 161
 
104 124 113
199 61 60
135 179 241
142 156 125
64 77 61
 
130 70 200
114 32 55
94 211 182
29 49 252
116 139 217
bmp median_filter 1
133 133 40
210 137 244
210 137 244
90 203 161
90 203 161
 
104 124 113
133 133 40
66 183 114
210 137 244
66 183 114
 
212 68 13
104 124 113
142 156 125
142 156 125
116 139 217
 
130 70 200
104 124 113
142 156 125
142 156 125
116 139 217
 

[edit] Mathematica

 
MedianFilter[img,n]
 


[edit] OCaml

let color_add (r1,g1,b1) (r2,g2,b2) =
( (r1 + r2),
(g1 + g2),
(b1 + b2) )
 
let color_div (r,g,b) d =
( (r / d),
(g / d),
(b / d) )
 
let compare_as_grayscale (r1,g1,b1) (r2,g2,b2) =
let v1 = (2_126 * r1 + 7_152 * g1 + 722 * b1)
and v2 = (2_126 * r2 + 7_152 * g2 + 722 * b2) in
(Pervasives.compare v1 v2)
 
let get_rgb img x y =
let _, r_channel,_,_ = img in
let width = Bigarray.Array2.dim1 r_channel
and height = Bigarray.Array2.dim2 r_channel in
if (x < 0) || (x >= width) then (0,0,0) else
if (y < 0) || (y >= height) then (0,0,0) else (* feed borders with black *)
(get_pixel img x y)
 
 
let median_value img radius =
let samples = (radius*2+1) * (radius*2+1) in
fun x y ->
let sample = ref [] in
 
for _x = (x - radius) to (x + radius) do
for _y = (y - radius) to (y + radius) do
 
let v = get_rgb img _x _y in
 
sample := v :: !sample;
done;
done;
 
let ssample = List.sort compare_as_grayscale !sample in
let mid = (samples / 2) in
 
if (samples mod 2) = 1
then List.nth ssample (mid+1)
else
let median1 = List.nth ssample (mid)
and median2 = List.nth ssample (mid+1) in
(color_div (color_add median1 median2) 2)
 
 
let median img radius =
let _, r_channel,_,_ = img in
let width = Bigarray.Array2.dim1 r_channel
and height = Bigarray.Array2.dim2 r_channel in
 
let _median_value = median_value img radius in
 
let res = new_img ~width ~height in
for y = 0 to pred height do
for x = 0 to pred width do
let color = _median_value x y in
put_pixel res color x y;
done;
done;
(res)

an alternate version of the function median_value using arrays instead of lists:

let median_value img radius =
let samples = (radius*2+1) * (radius*2+1) in
let sample = Array.make samples (0,0,0) in
fun x y ->
let i = ref 0 in
for _x = (x - radius) to (x + radius) do
for _y = (y - radius) to (y + radius) do
let v = get_rgb img _x _y in
sample.(!i) <- v;
incr i;
done;
done;
 
Array.sort compare_as_grayscale sample;
let mid = (samples / 2) in
 
if (samples mod 2) = 1
then sample.(mid+1)
else (color_div (color_add sample.(mid)
sample.(mid+1)) 2)

[edit] PicoLisp

(de ppmMedianFilter (Radius Ppm)
(let Len (inc (* 2 Radius))
(make
(chain (head Radius Ppm))
(for (Y Ppm T (cdr Y))
(NIL (nth Y Len)
(chain (tail Radius Y)) )
(link
(make
(chain (head Radius (get Y (inc Radius))))
(for (X (head Len Y) T)
(NIL (nth X 1 Len)
(chain (tail Radius (get X (inc Radius)))) )
(link
(cdr
(get
(sort
(mapcan
'((Y)
(mapcar
'((C)
(cons
(+
(* (car C) 2126) # Red
(* (cadr C) 7152) # Green
(* (caddr C) 722) ) # Blue
C ) )
(head Len Y) ) )
X ) )
(inc Radius) ) ) )
(map pop X) ) ) ) ) ) ) )

Test using 'ppmRead' from Bitmap/Read a PPM file#PicoLisp and 'ppmWrite' from Bitmap/Write a PPM file#PicoLisp:

(ppmWrite (ppmMedianFilter 2 (ppmRead "Lenna100.ppm")) "a.ppm")

[edit] Python

Works with: Python version 2.6
Library: PIL
import Image, ImageFilter
im = Image.open('image.ppm')
 
median = im.filter(ImageFilter.MedianFilter(3))
median.save('image2.ppm')

[edit] Racket

Due to the use of flomaps the solution below works for all types of images.

 
#lang racket
(require images/flomap math)
 
(define lena <<paste image of Lena here>> )
(define bm (send lena get-bitmap))
(define fm (bitmap->flomap bm))
 
(flomap->bitmap
(build-flomap
4 (send bm get-width) (send bm get-height)
(λ (k x y)
(define (f x y) (flomap-ref fm k x y))
(median < (list (f (- x 1) (- y 1))
(f (- x 1) y)
(f (- x 1) (+ y 1))
(f x (- y 1))
(f x y)
(f x (+ y 1))
(f (+ x 1) (- y 1))
(f (+ x 1) y)
(f (+ x 1) (+ y 1)))))))
 

[edit] Ruby

Translation of: Tcl
class Pixmap
def median_filter(radius=3)
radius += 1 if radius.even?
filtered = self.class.new(@width, @height)
pb = ProgressBar.new(@height) if $DEBUG
@height.times do |y|
@width.times do |x|
window = []
(x - radius).upto(x + radius).each do |win_x|
(y - radius).upto(y + radius).each do |win_y|
win_x = 0 if win_x < 0
win_y = 0 if win_y < 0
win_x = @width-1 if win_x >= @width
win_y = @height-1 if win_y >= @height
window << self[win_x, win_y]
end
end
# median
filtered[x, y] = window.sort[window.length / 2]
end
pb.update(y) if $DEBUG
end
pb.close if $DEBUG
filtered
end
end
 
class RGBColour
# refactoring
def luminosity
Integer(0.2126*@red + 0.7152*@green + 0.0722*@blue)
end
def to_grayscale
l = luminosity
self.class.new(l, l, l)
end
 
# defines how to compare (and hence, sort)
def <=>(other)
self.luminosity <=> other.luminosity
end
end
 
class ProgressBar
def initialize(max)
$stdout.sync = true
@progress_max = max
@progress_pos = 0
@progress_view = 68
$stdout.print "[#{'-'*@progress_view}]\r["
end
 
def update(n)
new_pos = n * @progress_view/@progress_max
if new_pos > @progress_pos
@progress_pos = new_pos
$stdout.print '='
end
end
 
def close
$stdout.puts '=]'
end
end
 
bitmap = Pixmap.open('file')
filtered = bitmap.median_filter

[edit] Tcl

Works with: Tcl version 8.5
Library: Tk
package require Tk
 
# Set the color of a pixel
proc applyMedian {srcImage x y -> dstImage} {
set x0 [expr {$x==0 ? 0 : $x-1}]
set y0 [expr {$y==0 ? 0 : $y-1}]
set x1 $x
set y1 $y
set x2 [expr {$x+1==[image width $srcImage] ? $x : $x+1}]
set y2 [expr {$y+1==[image height $srcImage] ? $y : $y+1}]
 
set r [set g [set b {}]]
foreach X [list $x0 $x1 $x2] {
foreach Y [list $y0 $y1 $y2] {
lassign [$srcImage get $X $Y] rPix gPix bPix
lappend r $rPix
lappend g $gPix
lappend b $bPix
}
}
set r [lindex [lsort -integer $r] 4]
set g [lindex [lsort -integer $g] 4]
set b [lindex [lsort -integer $b] 4]
$dstImage put [format "#%02x%02x%02x" $r $g $b] -to $x $y
}
# Apply the filter to the whole image
proc medianFilter {srcImage {dstImage ""}} {
if {$dstImage eq ""} {
set dstImage [image create photo]
}
set w [image width $srcImage]
set h [image height $srcImage]
for {set x 0} {$x < $w} {incr x} {
for {set y 0} {$y < $h} {incr y} {
applyMedian $srcImage $x $y -> $dstImage
}
}
return $dstImage
}
 
# Demonstration code using the Tk widget demo's teapot image
image create photo teapot -file $tk_library/demos/images/teapot.ppm
pack [labelframe .src -text Source] -side left
pack [label .src.l -image teapot]
update
pack [labelframe .dst -text Median] -side left
pack [label .dst.l -image [medianFilter teapot]]
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