Hough transform
You are encouraged to solve this task according to the task description, using any language you may know.
Implement the Hough transform, which is used as part of feature extraction with digital images. It is a tool that makes it far easier to identify straight lines in the source image, whatever their orientation.
The transform maps each point in the target image, , to the average color of the pixels on the corresponding line of the source image (in -space, where the line corresponds to points of the form ). The idea is that where there is a straight line in the original image, it corresponds to a bright (or dark, depending on the color of the background field) spot; by applying a suitable filter to the results of the transform, it is possible to extract the locations of the lines in the original image.
The target space actually uses polar coordinates, but is conventionally plotted on rectangular coordinates for display. There's no specification of exactly how to map polar coordinates to a flat surface for display, but a convenient method is to use one axis for and the other for , with the center of the source image being the origin.
There is also a spherical Hough transform, which is more suited to identifying planes in 3D data.
C
(Tested only with the pentagon image given) <lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdint.h>
- include <string.h>
- include <math.h>
- include <cairo.h>
- ifndef M_PI
- define M_PI 3.1415927
- endif
- define GR(X,Y) (d[(*s)*(Y)+bpp*(X)+((2)%bpp)])
- define GG(X,Y) (d[(*s)*(Y)+bpp*(X)+((1)%bpp)])
- define GB(X,Y) (d[(*s)*(Y)+bpp*(X)+((0)%bpp)])
- define SR(X,Y) (ht[4*tw*((Y)%th)+4*((X)%tw)+2])
- define SG(X,Y) (ht[4*tw*((Y)%th)+4*((X)%tw)+1])
- define SB(X,Y) (ht[4*tw*((Y)%th)+4*((X)%tw)+0])
- define RAD(A) (M_PI*((double)(A))/180.0)
uint8_t *houghtransform(uint8_t *d, int *w, int *h, int *s, int bpp) {
int rho, theta, y, x, W = *w, H = *h;
int th = sqrt(W*W + H*H)/2.0;
int tw = 360;
uint8_t *ht = malloc(th*tw*4);
memset(ht, 0, 4*th*tw); // black bg
for(rho = 0; rho < th; rho++)
{
for(theta = 0; theta < tw/*720*/; theta++)
{
double C = cos(RAD(theta));
double S = sin(RAD(theta));
uint32_t totalred = 0;
uint32_t totalgreen = 0;
uint32_t totalblue = 0;
uint32_t totalpix = 0;
if ( theta < 45 || (theta > 135 && theta < 225) || theta > 315) {
for(y = 0; y < H; y++) { double dx = W/2.0 + (rho - (H/2.0-y)*S)/C; if ( dx < 0 || dx >= W ) continue; x = floor(dx+.5); if (x == W) continue; totalpix++; totalred += GR(x, y); totalgreen += GG(x, y); totalblue += GB(x, y); }
} else {
for(x = 0; x < W; x++) { double dy = H/2.0 - (rho - (x - W/2.0)*C)/S; if ( dy < 0 || dy >= H ) continue; y = floor(dy+.5); if (y == H) continue; totalpix++; totalred += GR(x, y); totalgreen += GG(x, y); totalblue += GB(x, y); }
}
if ( totalpix > 0 ) {
double dp = totalpix; SR(theta, rho) = (int)(totalred/dp) &0xff; SG(theta, rho) = (int)(totalgreen/dp) &0xff; SB(theta, rho) = (int)(totalblue/dp) &0xff;
} } }
*h = th; // sqrt(W*W+H*H)/2 *w = tw; // 360 *s = 4*tw; return ht;
}
int main(int argc, char **argv) {
cairo_surface_t *inputimg = NULL; cairo_surface_t *houghimg = NULL; uint8_t *houghdata = NULL, *inputdata = NULL; int w, h, s, bpp;
if ( argc < 3 ) return EXIT_FAILURE;
inputimg = cairo_image_surface_create_from_png(argv[1]);
w = cairo_image_surface_get_width(inputimg);
h = cairo_image_surface_get_height(inputimg);
s = cairo_image_surface_get_stride(inputimg);
bpp = cairo_image_surface_get_format(inputimg);
switch(bpp)
{
case CAIRO_FORMAT_ARGB32: bpp = 4; break;
case CAIRO_FORMAT_RGB24: bpp = 3; break;
case CAIRO_FORMAT_A8: bpp = 1; break;
default:
fprintf(stderr, "unsupported\n");
goto destroy;
}
inputdata = cairo_image_surface_get_data(inputimg);
houghdata = houghtransform(inputdata, &w, &h, &s, bpp);
printf("w=%d, h=%d\n", w, h);
houghimg = cairo_image_surface_create_for_data(houghdata,
CAIRO_FORMAT_RGB24, w, h, s);
cairo_surface_write_to_png(houghimg, argv[2]);
destroy:
if (inputimg != NULL) cairo_surface_destroy(inputimg); if (houghimg != NULL) cairo_surface_destroy(houghimg);
return EXIT_SUCCESS;
}</lang>
Output image (but with white background):
J
Solution: <lang j>NB.*houghTransform v Produces a density plot of image y in hough space NB. y is picture as an array with 1 at non-white points, NB. x is resolution (width,height) of resulting image houghTransform=: dyad define
'w h'=. x NB. width and height of target image theta=. o. (%~ 0.5+i.) w NB. theta in radians from 0 to π rho=. (4$.$. |.y) +/ .* 2 1 o./theta NB. rho for each pixel at each theta 'min max'=. (,~-) +/&.:*: $y NB. min/max possible rho rho=. <. 0.5+ h * (rho-min) % max-min NB. Rescale rho from 0 to h and round to int |.([: <:@(#/.~) (i.h)&,)"1&.|: rho NB. consolidate into picture
)</lang>
Example use:
<lang j> require 'viewmat media/platimg'
Img=: readimg jpath '~temp/pentagon.png' viewmat 460 360 houghTransform _1 > Img</lang>
MATLAB
This solution takes an image and the theta resolution as inputs. The image itself must be a 2-D boolean array. This array is constructed such that all of the pixels on an edge have the value "true." This can be done for a normal image using an "edge finding" algorithm to preprocess the image. In the case of the example image the pentagon "edges" are black pixels. So when the image is imported into MATLAB simply say any pixel colored black is true. The syntax is usually, cdata < 255. Where the vale 255 represents white and 0 represents black.
<lang MATLAB>function [rho,theta,houghSpace] = houghTransform(theImage,thetaSampleFrequency)
%Define the hough space theImage = flipud(theImage); [width,height] = size(theImage); rhoLimit = norm([width height]); rho = (-rhoLimit:1:rhoLimit); theta = (0:thetaSampleFrequency:pi); numThetas = numel(theta); houghSpace = zeros(numel(rho),numThetas); %Find the "edge" pixels [xIndicies,yIndicies] = find(theImage); %Preallocate space for the accumulator array numEdgePixels = numel(xIndicies); accumulator = zeros(numEdgePixels,numThetas); %Preallocate cosine and sine calculations to increase speed. In %addition to precallculating sine and cosine we are also multiplying %them by the proper pixel weights such that the rows will be indexed by %the pixel number and the columns will be indexed by the thetas. %Example: cosine(3,:) is 2*cosine(0 to pi) % cosine(:,1) is (0 to width of image)*cosine(0) cosine = (0:width-1)'*cos(theta); %Matrix Outerproduct sine = (0:height-1)'*sin(theta); %Matrix Outerproduct accumulator((1:numEdgePixels),:) = cosine(xIndicies,:) + sine(yIndicies,:);
%Scan over the thetas and bin the rhos
for i = (1:numThetas)
houghSpace(:,i) = hist(accumulator(:,i),rho);
end
pcolor(theta,rho,houghSpace);
shading flat;
title('Hough Transform');
xlabel('Theta (radians)');
ylabel('Rho (pixels)');
colormap('gray');
end</lang>
Sample Usage: <lang MATLAB>>> uiopen('C:\Documents and Settings\owner\Desktop\Chris\MATLAB\RosettaCode\180px-Pentagon.png',1) >> houghTransform(cdata(:,:,1)<255,1/200); %The image from uiopen is stored in cdata. The reason why the image is cdata<255 is because the "edge" pixels are black.</lang>
Python
This is the classical Hough transform as described in wikipedia. The code does not compute averages; it merely makes a point on the transformed image darker if a lot of points on the original image lie on the corresponding line. The output is almost identical to that of the Tcl code. The code works only with gray-scale images, but it is easy to extend to RGB. <lang python> from math import hypot, pi, cos, sin import Image
def hough(im, ntx=460, mry=360):
"Calculate Hough transform."
pim = im.load()
nimx, mimy = im.size
mry = int(mry/2)*2 #Make sure that this is even
him = Image.new("L", (ntx, mry), 255)
phim = him.load()
rmax = hypot(nimx, mimy) dr = rmax / (mry/2) dth = pi / ntx
for jx in xrange(nimx):
for iy in xrange(mimy):
col = pim[jx, iy]
if col == 255: continue
for jtx in xrange(ntx):
th = dth * jtx
r = jx*cos(th) + iy*sin(th)
iry = mry/2 + int(r/dr+0.5)
phim[jtx, iry] -= 1
return him
def test():
"Test Hough transform with pentagon."
im = Image.open("pentagon.png").convert("L")
him = hough(im)
him.save("ho5.bmp")
if __name__ == "__main__": test()
</lang>
Tcl
<lang tcl>package require Tk
set PI 3.1415927 proc HoughTransform {src trg {fieldColor "#000000"}} {
global PI
set w [image width $src]
set h [image height $src]
set targetH [expr {int(hypot($w, $h)/2)}]
# Configure the target buffer
$trg configure -width 360 -height $targetH
$trg put $fieldColor -to 0 0 359 [expr {$targetH-1}]
# Iterate over the target's space of pixels.
for {set rho 0} {$rho < $targetH} {incr rho} {
set row {} for {set theta 0} {$theta < 360} {incr theta} { set cos [expr {cos($theta/180.0*$PI)}] set sin [expr {sin($theta/180.0*$PI)}] set totalRed 0 set totalGreen 0 set totalBlue 0 set totalPix 0
# Sum the colors of the line with equation x*cos(θ) + y*sin(θ) = ρ if {$theta<45 || ($theta>135 && $theta<225) || $theta>315} { # For these half-quadrants, it's better to iterate by 'y' for {set y 0} {$y<$h} {incr y} { set x [expr { $w/2 + ($rho - ($h/2-$y)*$sin)/$cos }] if {$x < 0 || $x >= $w} continue set x [expr {round($x)}] if {$x == $w} continue incr totalPix lassign [$src get $x $y] r g b incr totalRed $r incr totalGreen $g incr totalBlue $b } } else { # For the other half-quadrants, it's better to iterate by 'x' for {set x 0} {$x<$w} {incr x} { set y [expr { $h/2 - ($rho - ($x-$w/2)*$cos)/$sin }] if {$y < 0 || $y >= $h} continue set y [expr {round($y)}] if {$y == $h} continue incr totalPix lassign [$src get $x $y] r g b incr totalRed $r incr totalGreen $g incr totalBlue $b } }
# Convert the summed colors back into a pixel for insertion into # the target buffer. if {$totalPix > 0} { set totalPix [expr {double($totalPix)}] set col [format "#%02x%02x%02x" \ [expr {round($totalRed/$totalPix)}] \ [expr {round($totalGreen/$totalPix)}] \ [expr {round($totalBlue/$totalPix)}]] } else { set col $fieldColor } lappend row $col } $trg put [list $row] -to 0 $rho
}
}</lang>
Demonstration Code
Takes the name of the image to apply the transform to as an argument. If using PNG images,
or
<lang tcl># Demonstration code if {[catch {
package require Tk 8.6; # Just for PNG format handler
}] == 1} then {catch {
package require Img
}}
- If neither Tk8.6 nor Img, then only GIF and PPM images can be loaded
set f [lindex $argv 0] image create photo srcImg -file $f image create photo targetImg pack [labelframe .l1 -text Source] [labelframe .l2 -text Target] pack [label .l1.i -image srcImg] pack [label .l2.i -image targetImg]
- Postpone until after we've drawn ourselves
after idle HoughTransform srcImg targetImg [lrange $argv 1 end]</lang>
