PDL::Image2D - Miscellaneous 2D image processing functions

- NAME
- DESCRIPTION
- SYNOPSIS
- FUNCTIONS

- conv2d
- med2d
- med2df
- patch2d
- patchbad2d
- max2d_ind
- centroid2d
- cc8compt
- polyfill
- polyfillv
- rot2d
- bilin2d
- rescale2d
- fitwarp2d
- applywarp2d
- warp2d
- warp2d_kernel

- AUTHORS

# NAME

PDL::Image2D - Miscellaneous 2D image processing functions

# DESCRIPTION

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

# SYNOPSIS

     use PDL::Image2D;

# FUNCTIONS

## 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

## 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

## 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

## 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.

## 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).

## 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

## 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`.

## cc8compt

      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 );

## polyfill

      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 `im`.

## polyfillv

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;

## 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;

## bilin2d

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

Bilineary 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.

## rescale2d

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

The first piddle is rescaled to the dimensions of the second (expandind or meaning values as needed) and then added to it.

## fitwarp2d

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 (`$u,$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)
      THRESH  - in svd, remove terms smaller than THRESH * max value
                default is 1.0e-5

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`.

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

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]
      ]

## applywarp2d

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 `$u,$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

Return the specified kernel, as used by warp2d

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

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

      line warp2d_kernel( "hamming" );

# AUTHORS

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.