conv2d

  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 "PDL::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
            => Replicate - repeat boundary pixel values

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

med2d

  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
            => Replicate - repeat boundary pixel values

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

med2df

  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 "med2d" 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
            => Replicate - repeat boundary pixel values

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.

box2d

  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.

"box2d" should be updated to support similar edge options as "conv2d" and "med2d" 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.

patch2d

  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;

$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

patchbad2d

  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.

max2d_ind

  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.

centroid2d

  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

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

cc8compt

Connected 8-component labeling of 0,1 image - i.e. find separate segmented objects and fill object pixels with object number. 8-component labeling includes all neighboring pixels. This is just a front-end to ccNcompt. See also cc4compt.

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

cc4compt

Connected 4-component labeling of 0,1 image - i.e. find separate segmented objects and fill object pixels with object number. 4-component labling does not include the diagonal neighbors. This is just a front-end to ccNcompt. See also cc8compt.

 $segmented = cc4compt( $image > $threshold );

ccNcompt

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

Connected component labeling of a binary image.

Connected component labeling of 0,1 image - i.e. find separate segmented objects and fill object pixels with object number. See also cc4compt and cc8compt.

The connectivity parameter must be 4 or 8.

 $segmented = ccNcompt( $image > $threshold, 4);

 $segmented2 = ccNcompt( $image > $threshold, 8);

where the second parameter specifies the connectivity (4 or 8) of the labeling.

ccNcompt 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.

polyfill

This function works inplace, i.e. modifies "im".

  polyfill($im,$ps,$colour,[\%options]);

The default method of determining which points lie inside of the polygon used is not as strict as the method used in pnpoly. Often, it includes vertices and edge points. Set the "Method" option to change this behaviour.

Method - Set the method used to determine which points lie in the polygon. => Default - internal PDL algorithm => pnpoly - use the pnpoly algorithm

  # Make a convex 3x3 square of 1s in an image using the pnpoly algorithm
  $ps = pdl([3,3],[3,6],[6,6],[6,3]);
  polyfill($im,$ps,1,{'Method' =>'pnpoly'});

pnpoly

  $mask = $img->pnpoly($ps);

  # Old style, do not use
  $mask = pnpoly($x, $y, $px, $py);

For a closed polygon determined by the sequence of points in {$px,$py} the output of pnpoly is a mask corresponding to whether or not each coordinate (x,y) in the set of test points, {$x,$y}, is in the interior of the polygon. This is the 'points in a polygon' algorithm from <http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html> and vectorized for PDL by Karl Glazebrook.

  # define a 3-sided polygon (a triangle)
  $ps = pdl([3, 3], [20, 20], [34, 3]);

  # $tri is 0 everywhere except for points in polygon interior
  $tri = $img->pnpoly($ps);

  With the second form, the x and y coordinates must also be specified.
  B< I<THIS IS MAINTAINED FOR BACKWARD COMPATIBILITY ONLY> >.

  $px = pdl( 3, 20, 34 );
  $py = pdl( 3, 20,  3 );
  $x = $img->xvals;      # get x pixel coords
  $y = $img->yvals;      # get y pixel coords

  # $tri is 0 everywhere except for points in polygon interior
  $tri = pnpoly($x,$y,$px,$py);

polyfillv

  polyfillv($im,$ps,[\%options]);

The default method of determining which points lie inside of the polygon used is not as strict as the method used in pnpoly. Often, it includes vertices and edge points. Set the "Method" option to change this behaviour.

Method - Set the method used to determine which points lie in the polygon. => Default - internal PDL algorithm => pnpoly - use the pnpoly algorithm

  # increment intensity in area bounded by $poly using the pnpoly algorithm
  $im->polyfillv($poly,{'Method'=>'pnpoly'})++; # legal in perl >= 5.6

  # compute average intensity within area bounded by $poly using the default algorithm
  $av = $im->polyfillv($poly)->avg;

rot2d

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

rotate an image by given "angle"

  # 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 "angle" between -90 and + 90 degrees. Uses/doesn't use antialiasing depending on the "aa" flag. Pixels outside the rotated image are set to "bg".

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 "rotnewsz" 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.

bilin2d

  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.

rescale2d

  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.

fitwarp2d

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

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

$px and $py are "np" by "np" element piddles which describe a polynomial mapping (of order "np-1") from the output "(u,v)" image to the input "(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 "$u,$v" points given $px and $py.

Options:

  FIT     - which terms to fit? default ones(byte,$nf,$nf)

FIT

"FIT" 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 "$fit->slice(":,:,(0)")" will be used for $px and "$fit->slice(":,:,(1)")" will be used for $py.

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

  # 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]
  ]

  # A higher-degree polynomial should not affect the answer much, but
  # will require more control points

  $x = $x->glue(0,pdl(50,12.5, 37.5, 12.5, 37.5));
  $y = $y->glue(0,pdl(50,12.5, 37.5, 37.5, 12.5));
  $u = $u->glue(0,pdl(73,20,40,20,40));
  $v = $v->glue(0,pdl(29,20,40,40,20));
  ( $px3, $py3 ) = fitwarp2d( $x, $y, $u, $v, 3 );
  print "px3 =${px3}py3 =$py3";
  px3 =
  [
   [-6.4981162e+08       71034917     -726498.95]
   [      49902244     -5415096.7      55945.388]
   [    -807778.46      88457.191     -902.51612]
  ]
  py3 =
  [
   [-6.2732159e+08       68576392     -701354.77]
   [      48175125     -5227679.8      54009.114]
   [    -779821.18      85395.681     -871.27997]
  ]

  #This illustrates an important point about singular value
  #decompositions that are used in fitwarp2d: like all SVDs, the
  #rotation matrices are not unique, and so the $px and $py returned
  #by fitwarp2d are not guaranteed to be the "simplest" solution.
  #They do still work, though:

  ($x3,$y3) = applywarp2d($px3,$py3,$u,$v);
  print approx $x3,$x,1e-4;
  [1 1 1 1 1 1 1 1 1]
  print approx $y3,$y;
  [1 1 1 1 1 1 1 1 1]

applywarp2d

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

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

warp2d

  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 $px and $py piddles to warp the input image $img into the output image $out.

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

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

The routine is based on the "warping" 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

"KERNEL" 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 "sinc" interpolation kernel in the spatial domain:

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

However, it is not ideal for the "4x4" pixel region used here.

sinc2

This is the square of the sinc function.

lanczos

Although defined differently to the "tanh" 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 "a = 0.5" and N currently equal to 2001.

hamming

This kernel uses the same H(x) as the Hann filter, but with "a = 0.54".

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

warp2d_kernel

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

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

  line warp2d_kernel( "hamming" );