PDL::Image2D - Miscellaneous 2D image processing functions

Miscellaneous 2D image processing functions - for want of anywhere else to put them.

 use PDL::Image2D;

  Signature: (a(m,n); kern(p,q); [o]b(m,n); int opt)

2D convolution of an array with a kernel (smoothing)

For large kernels, using a FFT routine, such as fftconvolve() in CWPDL::FFT, will be quicker.

 $new = conv2d $old, $kernel, {OPTIONS}

 $smoothed = conv2d $image, ones(3,3), {Boundary => Reflect}

 Boundary - controls what values are assumed for the image when kernel
            crosses its edge:
            => Default  - periodic boundary conditions 
                          (i.e. wrap around axis)
            => Reflect  - reflect at boundary
            => Truncate - truncate at boundary

Unlike the FFT routines, conv2d is able to process bad values.

  Signature: (a(m,n); kern(p,q); [o]b(m,n); int opt)

2D median-convolution of an array with a kernel (smoothing)

Note: only points in the kernel >0 are included in the median, other points are weighted by the kernel value (medianing lots of zeroes is rather pointless)

 $new = med2d $old, $kernel, {OPTIONS}

 $smoothed = med2d $image, ones(3,3), {Boundary => Reflect}

 Boundary - controls what values are assumed for the image when kernel
            crosses its edge:
            => Default  - periodic boundary conditions (i.e. wrap around axis)
            => Reflect  - reflect at boundary
            => Truncate - truncate at boundary

Bad values are ignored in the calculation. If all elements within the kernel are bad, the output is set bad.

  Signature: (a(m,n); [o]b(m,n); int __p_size; int __q_size; int opt)

2D median-convolution of an array in a pxq window (smoothing)

Note: this routine does the median over all points in a rectangular window and is not quite as flexible as CWmed2d in this regard but slightly faster instead

 $new = med2df $old, $xwidth, $ywidth, {OPTIONS}

 $smoothed = med2df $image, 3, 3, {Boundary => Reflect}

 Boundary - controls what values are assumed for the image when kernel
            crosses its edge:
            => Default  - periodic boundary conditions (i.e. wrap around axis)
            => Reflect  - reflect at boundary
            => Truncate - truncate at boundary

med2df does not process bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

  Signature: (a(n,m); [o] b(n,m); int wx; int wy; int edgezero)

fast 2D boxcar average

  $smoothim = $im->box2d($wx,$wy,$edgezero=1);

The edgezero argument controls if edge is set to zero (edgezero=1) or just keeps the original (unfiltered) values.

CWbox2d should be updated to support similar edge options as CWconv2d and CWmed2d etc.

Boxcar averaging is a pretty crude way of filtering. For serious stuff better filters are around (e.g., use conv2d with the appropriate kernel). On the other hand it is fast and computational cost grows only approximately linearly with window size.

box2d does not process bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

  Signature: (a(m,n); int bad(m,n); [o]b(m,n))

patch bad pixels out of 2D images using a mask

 $patched = patch2d $data, $bad;

CW$bad is a 2D mask array where 1=bad pixel 0=good pixel. Pixels are replaced by the average of their non-bad neighbours; if all neighbours are bad, the original data value is copied across.

This routine does not handle bad values - use patchbad2d instead

  Signature: (a(m,n); [o]b(m,n))

patch bad pixels out of 2D images containing bad values

 $patched = patchbad2d $data;

Pixels are replaced by the average of their non-bad neighbours; if all neighbours are bad, the output is set bad. If the input piddle contains no bad values, then a straight copy is performed (see patch2d).

patchbad2d handles bad values. The output piddle may contain bad values, depending on the pattern of bad values in the input piddle.

  Signature: (a(m,n); [o]val(); int [o]x(); int[o]y())

Return value/position of maximum value in 2D image

Contributed by Tim Jeness

Bad values are excluded from the search. If all pixels are bad then the output is set bad.

  Signature: (im(m,n); x(); y(); box(); [o]xcen(); [o]ycen())

Refine a list of object positions in 2D image by centroiding in a box

CW$box is the full-width of the box, i.e. the window is CW+/- $box/2.

Bad pixels are excluded from the centroid calculation. If all elements are bad (or the pixel sum is 0 - but why would you be centroiding something with negatives in...) then the output values are set bad.

  Signature: (a(m,n); [o]b(m,n))

Connected 8-component labeling of a binary image.

Connected 8-component labeling of 0,1 image - i.e. find seperate segmented objects and fill object pixels with object number

 $segmented = cc8compt( $image > $threshold );

cc8compt ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

  Signature: (int [o,nc] im(m,n); float ps(two=2,np); int col())

fill the area inside the given polygon with a given colour

This function works inplace, i.e. modifies CWim.

polyfill ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

return the (dataflown) area of an image within a polygon

  # increment intensity in area bounded by $poly
  $im->polyfillv($pol)++; # legal in perl >= 5.6
  # compute average intensity within area bounded by $poly
  $av = $im->polyfillv($poly)->avg;

  Signature: (im(m,n); float angle(); bg(); int aa(); [o] om(p,q))

rotate an image by given CWangle

  # rotate by 10.5 degrees with antialiasing, set missing values to 7
  $rot = $im->rot2d(10.5,7,1);

This function rotates an image through an CWangle between -90 and + 90 degrees. Uses/doesn't use antialiasing depending on the CWaa flag. Pixels outside the rotated image are set to CWbg.

Code modified from pnmrotate (Copyright Jef Poskanzer) with an algorithm based on A Fast Algorithm for General Raster Rotation by Alan Paeth, Graphics Interface '86, pp. 77-81.

Use the CWrotnewsz function to find out about the dimension of the newly created image

  ($newcols,$newrows) = rotnewsz $oldn, $oldm, $angle;

PDL::Transform offers a more general interface to distortions, including rotation, with various types of sampling; but rot2d is faster.

rot2d ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

  Signature: (I(n,m); O(q,p))

Bilinearly maps the first piddle in the second. The interpolated values are actually added to the second piddle which is supposed to be larger than the first one.

bilin2d ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

  Signature: (I(m,n); O(p,q))

The first piddle is rescaled to the dimensions of the second (expanding or meaning values as needed) and then added to it in place. Nothing useful is returned.

If you want photometric accuracy or automatic FITS header metadata tracking, consider using PDL::Transform::map instead: it does these things, at some speed penalty compared to rescale2d.

rescale2d ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

Find the best-fit 2D polynomial to describe a coordinate transformation.

  ( $px, $py ) = fitwarp2d( $x, $y, $u, $v, $nf. { options } )

Given a set of points in the output plane (CW$u,$v), find the best-fit (using singular-value decomposition) 2D polynomial to describe the mapping back to the image plane (CW$x,$y). The order of the fit is controlled by the CW$nf parameter (the maximum power of the polynomial is CW$nf - 1), and you can restrict the terms to fit using the CWFIT option.

CW$px and CW$py are CWnp by CWnp element piddles which describe a polynomial mapping (of order CWnp-1) from the output CW(u,v) image to the input CW(x,y) image:

  x = sum(j=0,np-1) sum(i=0,np-1) px(i,j) * u^i * v^j
  y = sum(j=0,np-1) sum(i=0,np-1) py(i,j) * u^i * v^j

The transformation is returned for the reverse direction (ie output to input image) since that is what is required by the warp2d() routine. The applywarp2d() routine can be used to convert a set of CW$u,$v points given CW$px and CW$py.

Options:

  FIT     - which terms to fit? default ones(byte,$nf,$nf)
  THRESH  - in svd, remove terms smaller than THRESH * max value
            default is 1.0e-5

FIT

CWFIT allows you to restrict which terms of the polynomial to fit: only those terms for which the FIT piddle evaluates to true will be evaluated. If a 2D piddle is sent in, then it is used for the x and y polynomials; otherwise CW$fit->slice(":,:,(0)") will be used for CW$px and CW$fit->slice(":,:,(1)") will be used for CW$py.

THRESH

Remove all singular values whose valus is less than CWTHRESH times the largest singular value.

The number of points must be at least equal to the number of terms to fit (CW$nf*$nf points for the default value of CWFIT).

  # points in original image
  $x = pdl( 0,   0, 100, 100 );
  $y = pdl( 0, 100, 100,   0 );
  # get warped to these positions
  $u = pdl( 10, 10, 90, 90 );
  $v = pdl( 10, 90, 90, 10 );
  # 
  # shift of origin + scale x/y axis only
  $fit = byte( [ [1,1], [0,0] ], [ [1,0], [1,0] ] );
  ( $px, $py ) = fitwarp2d( $x, $y, $u, $v, 2, { FIT => $fit } );
  print "px = ${px}py = $py";
  px = 
  [
   [-12.5  1.25]
   [    0     0]
  ]
  py = 
  [
   [-12.5     0]
   [ 1.25     0]
  ]
  #
  # Compared to allowing all 4 terms
  ( $px, $py ) = fitwarp2d( $x, $y, $u, $v, 2 );
  print "px = ${px}py = $py";
  px = 
  [
   [         -12.5           1.25]
   [  1.110223e-16 -1.1275703e-17]
  ]
  py = 
  [
   [         -12.5  1.6653345e-16]
   [          1.25 -5.8546917e-18]
  ]

Transform a set of points using a 2-D polynomial mapping

  ( $x, $y ) = applywarp2d( $px, $py, $u, $v )

Convert a set of points (stored in 1D piddles CW$u,$v) to CW$x,$y using the 2-D polynomial with coefficients stored in CW$px and CW$py. See fitwarp2d() for more information on the format of CW$px and CW$py.

  Signature: (img(m,n); double px(np,np); double py(np,np); [o] warp(m,n); { options })

Warp a 2D image given a polynomial describing the reverse mapping.

  $out = warp2d( $img, $px, $py, { options } );

Apply the polynomial transformation encoded in the CW$px and CW$py piddles to warp the input image CW$img into the output image CW$out.

The format for the polynomial transformation is described in the documentation for the fitwarp2d() routine.

At each point CWx,y, the closest 16 pixel values are combined with an interpolation kernel to calculate the value at CWu,v. The interpolation is therefore done in the image, rather than Fourier, domain. By default, a CWtanh kernel is used, but this can be changed using the CWKERNEL option discussed below (the choice of kernel depends on the frequency content of the input image).

The routine is based on the CWwarping command from the Eclipse data-reduction package - see http://www.eso.org/eclipse/ - and for further details on image resampling see Wolberg, G., Digital Image Warping, 1990, IEEE Computer Society Press ISBN 0-8186-8944-7).

Currently the output image is the same size as the input one, which means data will be lost if the transformation reduces the pixel scale. This will (hopefully) be changed soon.

  $img = rvals(byte,501,501);
  imag $img, { JUSTIFY => 1 };
  # 
  # use a not-particularly-obvious transformation:
  #   x = -10 + 0.5 * $u - 0.1 * $v 
  #   y = -20 + $v - 0.002 * $u * $v
  #
  $px  = pdl( [ -10, 0.5 ], [ -0.1, 0 ] );
  $py  = pdl( [ -20, 0 ], [ 1, 0.002 ] );
  $wrp = warp2d( $img, $px, $py );
  #
  # see the warped image
  imag $warp, { JUSTIFY => 1 };

The options are:

  KERNEL - default value is tanh
  NOVAL  - default value is 0

CWKERNEL is used to specify which interpolation kernel to use (to see what these kernels look like, use the warp2d_kernel() routine). The options are:

tanh

Hyperbolic tangent: the approximation of an ideal box filter by the product of symmetric tanh functions.

sinc

For a correctly sampled signal, the ideal filter in the fourier domain is a rectangle, which produces a CWsinc interpolation kernel in the spatial domain:

  sinc(x) = sin(pi * x) / (pi * x)

However, it is not ideal for the CW4x4 pixel region used here.

sinc2

This is the square of the sinc function.

lanczos

Although defined differently to the CWtanh kernel, the result is very similar in the spatial domain. The Lanczos function is defined as

  L(x) = sinc(x) * sinc(x/2)  if abs(x) < 2
       = 0                       otherwise

hann

This kernel is derived from the following function:

  H(x) = a + (1-a) * cos(2*pi*x/(N-1))  if abs(x) < 0.5*(N-1)
       = 0                                 otherwise

with CWa = 0.5 and N currently equal to 2001.

hamming

This kernel uses the same CWH(x) as the Hann filter, but with CWa = 0.54.

CWNOVAL gives the value used to indicate that a pixel in the output image does not map onto one in the input image.

Return the specified kernel, as used by warp2d

  ( $x, $k ) = warp2d_kernel( $name )

The valid values for CW$name are the same as the CWKERNEL option of warp2d().

  line warp2d_kernel( "hamming" );

Copyright (C) Karl Glazebrook 1997 with additions by Robin Williams (rjrw@ast.leeds.ac.uk), Tim Jeness (timj@jach.hawaii.edu), and Doug Burke (burke@ifa.hawaii.edu).

All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For details, see the file COPYING in the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be included in the file.