Multidimensional image processing (`scipy.ndimage`)
====================================================

.. moduleauthor:: Peter Verveer <verveer@users.sourceforge.net>

.. currentmodule:: scipy.ndimage

.. _ndimage-introduction:

Introduction
------------

Image processing and analysis are generally seen as operations on
2-D arrays of values. There are, however, a number of
fields where images of higher dimensionality must be analyzed. Good
examples of these are medical imaging and biological imaging.
:mod:`numpy` is suited very well for this type of applications due to
its inherent multidimensional nature. The :mod:`scipy.ndimage`
packages provides a number of general image processing and analysis
functions that are designed to operate with arrays of arbitrary
dimensionality. The packages currently includes: functions for
linear and non-linear filtering, binary morphology, B-spline
interpolation, and object measurements.

.. _ndimage-properties-shared-by-all-functions:

Properties shared by all functions
----------------------------------

All functions share some common properties. Notably, all functions
allow the specification of an output array with the *output*
argument. With this argument, you can specify an array that will be
changed in-place with the result with the operation. In this case,
the result is not returned. Usually, using the *output* argument is
more efficient, since an existing array is used to store the
result.

The type of arrays returned is dependent on the type of operation,
but it is, in most cases, equal to the type of the input. If,
however, the *output* argument is used, the type of the result is
equal to the type of the specified output argument. If no output
argument is given, it is still possible to specify what the result
of the output should be. This is done by simply assigning the
desired `numpy` type object to the output argument. For example:

.. code:: python

    >>> from scipy.ndimage import correlate
    >>> import numpy as np
    >>> correlate(np.arange(10), [1, 2.5])
    array([ 0,  2,  6,  9, 13, 16, 20, 23, 27, 30])
    >>> correlate(np.arange(10), [1, 2.5], output=np.float64)
    array([  0. ,   2.5,   6. ,   9.5,  13. ,  16.5,  20. ,  23.5,  27. ,  30.5])

.. _ndimage-filter-functions:

Filter functions
----------------

The functions described in this section all perform some type of spatial
filtering of the input array: the elements in the output are some function
of the values in the neighborhood of the corresponding input element. We refer
to this neighborhood of elements as the filter kernel, which is often
rectangular in shape but may also have an arbitrary footprint. Many
of the functions described below allow you to define the footprint
of the kernel by passing a mask through the *footprint* parameter.
For example, a cross-shaped kernel can be defined as follows:

.. code:: python

    >>> footprint = np.array([[0, 1, 0], [1, 1, 1], [0, 1, 0]])
    >>> footprint
    array([[0, 1, 0],
           [1, 1, 1],
           [0, 1, 0]])

Usually, the origin of the kernel is at the center calculated by
dividing the dimensions of the kernel shape by two. For instance,
the origin of a 1-D kernel of length three is at the
second element. Take, for example, the correlation of a
1-D array with a filter of length 3 consisting of
ones:

.. code:: python

    >>> from scipy.ndimage import correlate1d
    >>> a = [0, 0, 0, 1, 0, 0, 0]
    >>> correlate1d(a, [1, 1, 1])
    array([0, 0, 1, 1, 1, 0, 0])

Sometimes, it is convenient to choose a different origin for the
kernel. For this reason, most functions support the *origin*
parameter, which gives the origin of the filter relative to its
center. For example:

.. code:: python

    >>> a = [0, 0, 0, 1, 0, 0, 0]
    >>> correlate1d(a, [1, 1, 1], origin = -1)
    array([0, 1, 1, 1, 0, 0, 0])

The effect is a shift of the result towards the left. This feature
will not be needed very often, but it may be useful, especially for
filters that have an even size. A good example is the calculation
of backward and forward differences:

.. code:: python

    >>> a = [0, 0, 1, 1, 1, 0, 0]
    >>> correlate1d(a, [-1, 1])               # backward difference
    array([ 0,  0,  1,  0,  0, -1,  0])
    >>> correlate1d(a, [-1, 1], origin = -1)  # forward difference
    array([ 0,  1,  0,  0, -1,  0,  0])

We could also have calculated the forward difference as follows:

.. code:: python

    >>> correlate1d(a, [0, -1, 1])
    array([ 0,  1,  0,  0, -1,  0,  0])

However, using the origin parameter instead of a larger kernel is
more efficient. For multidimensional kernels, *origin* can be a
number, in which case the origin is assumed to be equal along all
axes, or a sequence giving the origin along each axis.

Since the output elements are a function of elements in the
neighborhood of the input elements, the borders of the array need to
be dealt with appropriately by providing the values outside the
borders. This is done by assuming that the arrays are extended beyond
their boundaries according to certain boundary conditions. In the
functions described below, the boundary conditions can be selected
using the *mode* parameter, which must be a string with the name of the
boundary condition. The following boundary conditions are currently
supported:

 ==========   ====================================   ====================
  **mode**              **description**                  **example**
 ==========   ====================================   ====================
 "nearest"    use the value at the boundary          [1 2 3]->[1 1 2 3 3]
 "wrap"       periodically replicate the array       [1 2 3]->[3 1 2 3 1]
 "reflect"    reflect the array at the boundary      [1 2 3]->[1 1 2 3 3]
 "mirror"     mirror the array at the boundary       [1 2 3]->[2 1 2 3 2]
 "constant"   use a constant value, default is 0.0   [1 2 3]->[0 1 2 3 0]
 ==========   ====================================   ====================

The following synonyms are also supported for consistency with the
interpolation routines:

 ===============   =========================
     **mode**           **description**
 ===============   =========================
 "grid-constant"   equivalent to "constant"*
 "grid-mirror"     equivalent to "reflect"
 "grid-wrap"       equivalent to "wrap"
 ===============   =========================

\* "grid-constant" and "constant" are equivalent for filtering operations, but
have different behavior in interpolation functions. For API consistency, the
filtering functions accept either name.

The "constant" mode is special since it needs an additional parameter to
specify the constant value that should be used.

Note that modes mirror and reflect differ only in whether the sample at the
boundary is repeated upon reflection. For mode mirror, the point of symmetry is
exactly at the final sample, so that value is not repeated. This mode is also
known as whole-sample symmetric since the point of symmetry falls on the final
sample. Similarly, reflect is often referred to as half-sample symmetric as the
point of symmetry is half a sample beyond the array boundary.

.. note::

   The easiest way to implement such boundary conditions would be to
   copy the data to a larger array and extend the data at the borders
   according to the boundary conditions. For large arrays and large
   filter kernels, this would be very memory consuming, and the
   functions described below, therefore, use a different approach that
   does not require allocating large temporary buffers.

Correlation and convolution
^^^^^^^^^^^^^^^^^^^^^^^^^^^

- The :func:`correlate1d` function calculates a 1-D
  correlation along the given axis. The lines of the array along the
  given axis are correlated with the given *weights*. The *weights*
  parameter must be a 1-D sequence of numbers.

- The function :func:`correlate` implements multidimensional
  correlation of the input array with a given kernel.

- The :func:`convolve1d` function calculates a 1-D
  convolution along the given axis. The lines of the array along the
  given axis are convoluted with the given *weights*. The *weights*
  parameter must be a 1-D sequence of numbers.

- The function :func:`convolve` implements multidimensional
  convolution of the input array with a given kernel.

  .. note::

     A convolution is essentially a correlation after mirroring the
     kernel. As a result, the *origin* parameter behaves differently
     than in the case of a correlation: the results is shifted in the
     opposite direction.

.. _ndimage-filter-functions-smoothing:

Smoothing filters
^^^^^^^^^^^^^^^^^

- The :func:`gaussian_filter1d` function implements a 1-D
  Gaussian filter. The standard deviation of the Gaussian filter is
  passed through the parameter *sigma*. Setting *order* = 0
  corresponds to convolution with a Gaussian kernel. An order of 1, 2,
  or 3 corresponds to convolution with the first, second, or third
  derivatives of a Gaussian. Higher-order derivatives are not
  implemented. 



- The :func:`gaussian_filter` function implements a multidimensional
  Gaussian filter. The standard deviations of the Gaussian filter
  along each axis are passed through the parameter *sigma* as a
  sequence or numbers. If *sigma* is not a sequence but a single
  number, the standard deviation of the filter is equal along all
  directions. The order of the filter can be specified separately for
  each axis. An order of 0 corresponds to convolution with a Gaussian
  kernel. An order of 1, 2, or 3 corresponds to convolution with the
  first, second, or third derivatives of a Gaussian. Higher-order
  derivatives are not implemented. The *order* parameter must be a
  number, to specify the same order for all axes, or a sequence of
  numbers to specify a different order for each axis. The example below
  shows the filter applied on test data with different values of *sigma*.
  The *order* parameter is kept at 0. 

  .. plot:: tutorial/examples/gaussian_filter_plot1.py
      :align: center
      :alt: " "
      :include-source: 0

  .. note::

     The multidimensional filter is implemented as a sequence of
     1-D Gaussian filters. The intermediate arrays are
     stored in the same data type as the output. Therefore, for
     output types with a lower precision, the results may be imprecise
     because intermediate results may be stored with insufficient
     precision. This can be prevented by specifying a more precise
     output type.

- The :func:`uniform_filter1d` function calculates a 1-D
  uniform filter of the given *size* along the given axis.

- The :func:`uniform_filter` implements a multidimensional uniform
  filter. The sizes of the uniform filter are given for each axis as a
  sequence of integers by the *size* parameter. If *size* is not a
  sequence, but a single number, the sizes along all axes are assumed
  to be equal.

  .. note::

     The multidimensional filter is implemented as a sequence of
     1-D uniform filters. The intermediate arrays are
     stored in the same data type as the output. Therefore, for output
     types with a lower precision, the results may be imprecise
     because intermediate results may be stored with insufficient
     precision. This can be prevented by specifying a more precise
     output type.

Filters based on order statistics
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

- The :func:`minimum_filter1d` function calculates a 1-D
  minimum filter of the given *size* along the given axis.

- The :func:`maximum_filter1d` function calculates a 1-D
  maximum filter of the given *size* along the given axis.

- The :func:`minimum_filter` function calculates a multidimensional
  minimum filter. Either the sizes of a rectangular kernel or the
  footprint of the kernel must be provided. The *size* parameter, if
  provided, must be a sequence of sizes or a single number, in which
  case the size of the filter is assumed to be equal along each axis.
  The *footprint*, if provided, must be an array that defines the
  shape of the kernel by its non-zero elements.

- The :func:`maximum_filter` function calculates a multidimensional
  maximum filter. Either the sizes of a rectangular kernel or the
  footprint of the kernel must be provided. The *size* parameter, if
  provided, must be a sequence of sizes or a single number, in which
  case the size of the filter is assumed to be equal along each axis.
  The *footprint*, if provided, must be an array that defines the
  shape of the kernel by its non-zero elements.

- The :func:`rank_filter` function calculates a multidimensional rank
  filter. The *rank* may be less than zero, i.e., *rank* = -1
  indicates the largest element. Either the sizes of a rectangular
  kernel or the footprint of the kernel must be provided. The *size*
  parameter, if provided, must be a sequence of sizes or a single
  number, in which case the size of the filter is assumed to be equal
  along each axis. The *footprint*, if provided, must be an array that
  defines the shape of the kernel by its non-zero elements.

- The :func:`percentile_filter` function calculates a multidimensional
  percentile filter. The *percentile* may be less than zero, i.e.,
  *percentile* = -20 equals *percentile* = 80. Either the sizes of a
  rectangular kernel or the footprint of the kernel must be provided.
  The *size* parameter, if provided, must be a sequence of sizes or a
  single number, in which case the size of the filter is assumed to be
  equal along each axis. The *footprint*, if provided, must be an
  array that defines the shape of the kernel by its non-zero elements.

- The :func:`median_filter` function calculates a multidimensional
  median filter. Either the sizes of a rectangular kernel or the
  footprint of the kernel must be provided. The *size* parameter, if
  provided, must be a sequence of sizes or a single number, in which
  case the size of the filter is assumed to be equal along each
  axis. The *footprint* if provided, must be an array that defines the
  shape of the kernel by its non-zero elements.

Derivatives
^^^^^^^^^^^

Derivative filters can be constructed in several ways. The function
:func:`gaussian_filter1d`, described in
:ref:`ndimage-filter-functions-smoothing`, can be used to calculate
derivatives along a given axis using the *order* parameter. Other
derivative filters are the Prewitt and Sobel filters:

- The :func:`prewitt` function calculates a derivative along the given
  axis.
- The :func:`sobel` function calculates a derivative along the given
  axis.

The Laplace filter is calculated by the sum of the second derivatives
along all axes. Thus, different Laplace filters can be constructed
using different second-derivative functions. Therefore, we provide a
general function that takes a function argument to calculate the
second derivative along a given direction.

- The function :func:`generic_laplace` calculates a Laplace filter
  using the function passed through ``derivative2`` to calculate
  second derivatives. The function ``derivative2`` should have the
  following signature

  .. code:: python

     derivative2(input, axis, output, mode, cval, *extra_arguments, **extra_keywords)

  It should calculate the second derivative along the dimension
  *axis*. If *output* is not ``None``, it should use that for the
  output and return ``None``, otherwise it should return the
  result. *mode*, *cval* have the usual meaning.

  The *extra_arguments* and *extra_keywords* arguments can be used
  to pass a tuple of extra arguments and a dictionary of named
  arguments that are passed to ``derivative2`` at each call.

  For example

  .. code:: python

     >>> def d2(input, axis, output, mode, cval):
     ...     return correlate1d(input, [1, -2, 1], axis, output, mode, cval, 0)
     ...
     >>> a = np.zeros((5, 5))
     >>> a[2, 2] = 1
     >>> from scipy.ndimage import generic_laplace
     >>> generic_laplace(a, d2)
     array([[ 0.,  0.,  0.,  0.,  0.],
	    [ 0.,  0.,  1.,  0.,  0.],
	    [ 0.,  1., -4.,  1.,  0.],
            [ 0.,  0.,  1.,  0.,  0.],
            [ 0.,  0.,  0.,  0.,  0.]])

  To demonstrate the use of the *extra_arguments* argument, we could do

  .. code:: python

     >>> def d2(input, axis, output, mode, cval, weights):
     ...     return correlate1d(input, weights, axis, output, mode, cval, 0,)
     ...
     >>> a = np.zeros((5, 5))
     >>> a[2, 2] = 1
     >>> generic_laplace(a, d2, extra_arguments = ([1, -2, 1],))
     array([[ 0.,  0.,  0.,  0.,  0.],
	    [ 0.,  0.,  1.,  0.,  0.],
            [ 0.,  1., -4.,  1.,  0.],
            [ 0.,  0.,  1.,  0.,  0.],
            [ 0.,  0.,  0.,  0.,  0.]])

  or

  .. code:: python

     >>> generic_laplace(a, d2, extra_keywords = {'weights': [1, -2, 1]})
     array([[ 0.,  0.,  0.,  0.,  0.],
	    [ 0.,  0.,  1.,  0.,  0.],
	    [ 0.,  1., -4.,  1.,  0.],
	    [ 0.,  0.,  1.,  0.,  0.],
	    [ 0.,  0.,  0.,  0.,  0.]])

The following two functions are implemented using
:func:`generic_laplace` by providing appropriate functions for the
second-derivative function:

- The function :func:`laplace` calculates the Laplace using discrete
  differentiation for the second derivative (i.e., convolution with
  ``[1, -2, 1]``).

- The function :func:`gaussian_laplace` calculates the Laplace filter
  using :func:`gaussian_filter` to calculate the second
  derivatives. The standard deviations of the Gaussian filter along
  each axis are passed through the parameter *sigma* as a sequence or
  numbers. If *sigma* is not a sequence but a single number, the
  standard deviation of the filter is equal along all directions.

The gradient magnitude is defined as the square root of the sum of the
squares of the gradients in all directions. Similar to the generic
Laplace function, there is a :func:`generic_gradient_magnitude`
function that calculates the gradient magnitude of an array.

- The function :func:`generic_gradient_magnitude` calculates a
  gradient magnitude using the function passed through
  ``derivative`` to calculate first derivatives. The function
  ``derivative`` should have the following signature

  .. code:: python

     derivative(input, axis, output, mode, cval, *extra_arguments, **extra_keywords)

  It should calculate the derivative along the dimension *axis*. If
  *output* is not ``None``, it should use that for the output and return
  ``None``, otherwise it should return the result. *mode*, *cval* have the
  usual meaning.

  The *extra_arguments* and *extra_keywords* arguments can be used to
  pass a tuple of extra arguments and a dictionary of named arguments
  that are passed to *derivative* at each call.

  For example, the :func:`sobel` function fits the required signature

  .. code:: python

     >>> a = np.zeros((5, 5))
     >>> a[2, 2] = 1
     >>> from scipy.ndimage import sobel, generic_gradient_magnitude
     >>> generic_gradient_magnitude(a, sobel)
     array([[ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
	    [ 0.        ,  1.41421356,  2.        ,  1.41421356,  0.        ],
            [ 0.        ,  2.        ,  0.        ,  2.        ,  0.        ],
            [ 0.        ,  1.41421356,  2.        ,  1.41421356,  0.        ],
            [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ]])

  See the documentation of :func:`generic_laplace` for examples of
  using the *extra_arguments* and *extra_keywords* arguments.

The :func:`sobel` and :func:`prewitt` functions fit the required
signature and can, therefore, be used directly with
:func:`generic_gradient_magnitude`.

- The function :func:`gaussian_gradient_magnitude` calculates the
  gradient magnitude using :func:`gaussian_filter` to calculate the
  first derivatives. The standard deviations of the Gaussian filter
  along each axis are passed through the parameter *sigma* as a
  sequence or numbers. If *sigma* is not a sequence but a single
  number, the standard deviation of the filter is equal along all
  directions.

.. _ndimage-genericfilters:

Generic filter functions
^^^^^^^^^^^^^^^^^^^^^^^^

To implement filter functions, generic functions can be used that
accept a callable object that implements the filtering operation. The
iteration over the input and output arrays is handled by these generic
functions, along with such details as the implementation of the
boundary conditions. Only a callable object implementing a callback
function that does the actual filtering work must be provided. The
callback function can also be written in C and passed using a
:c:type:`PyCapsule` (see :ref:`ndimage-ccallbacks` for more
information).

- The :func:`generic_filter1d` function implements a generic
  1-D filter function, where the actual filtering
  operation must be supplied as a python function (or other callable
  object). The :func:`generic_filter1d` function iterates over the
  lines of an array and calls ``function`` at each line. The
  arguments that are passed to ``function`` are 1-D
  arrays of the ``numpy.float64`` type. The first contains the values
  of the current line. It is extended at the beginning and the end,
  according to the *filter_size* and *origin* arguments. The second
  array should be modified in-place to provide the output values of
  the line. For example, consider a correlation along one dimension:

  .. code:: python

     >>> a = np.arange(12).reshape(3,4)
     >>> correlate1d(a, [1, 2, 3])
     array([[ 3,  8, 14, 17],
	    [27, 32, 38, 41],
            [51, 56, 62, 65]])

  The same operation can be implemented using :func:`generic_filter1d`,
  as follows:

  .. code:: python

     >>> def fnc(iline, oline):
     ...     oline[...] = iline[:-2] + 2 * iline[1:-1] + 3 * iline[2:]
     ...
     >>> from scipy.ndimage import generic_filter1d
     >>> generic_filter1d(a, fnc, 3)
     array([[ 3,  8, 14, 17],
	    [27, 32, 38, 41],
            [51, 56, 62, 65]])

  Here, the origin of the kernel was (by default) assumed to be in the
  middle of the filter of length 3. Therefore, each input line had been
  extended by one value at the beginning and at the end, before the
  function was called.

  Optionally, extra arguments can be defined and passed to the filter
  function. The *extra_arguments* and *extra_keywords* arguments can
  be used to pass a tuple of extra arguments and/or a dictionary of
  named arguments that are passed to derivative at each call. For
  example, we can pass the parameters of our filter as an argument

  .. code:: python

     >>> def fnc(iline, oline, a, b):
     ...     oline[...] = iline[:-2] + a * iline[1:-1] + b * iline[2:]
     ...
     >>> generic_filter1d(a, fnc, 3, extra_arguments = (2, 3))
     array([[ 3,  8, 14, 17],
	    [27, 32, 38, 41],
            [51, 56, 62, 65]])

  or

  .. code:: python

     >>> generic_filter1d(a, fnc, 3, extra_keywords = {'a':2, 'b':3})
     array([[ 3,  8, 14, 17],
	    [27, 32, 38, 41],
            [51, 56, 62, 65]])

- The :func:`generic_filter` function implements a generic filter
  function, where the actual filtering operation must be supplied as a
  python function (or other callable object). The
  :func:`generic_filter` function iterates over the array and calls
  ``function`` at each element. The argument of ``function``
  is a 1-D array of the ``numpy.float64`` type that
  contains the values around the current element that are within the
  footprint of the filter. The function should return a single value
  that can be converted to a double precision number. For example,
  consider a correlation:

  .. code:: python

     >>> a = np.arange(12).reshape(3,4)
     >>> correlate(a, [[1, 0], [0, 3]])
     array([[ 0,  3,  7, 11],
	    [12, 15, 19, 23],
            [28, 31, 35, 39]])

  The same operation can be implemented using *generic_filter*, as
  follows:

  .. code:: python

     >>> def fnc(buffer):
     ...     return (buffer * np.array([1, 3])).sum()
     ...
     >>> from scipy.ndimage import generic_filter
     >>> generic_filter(a, fnc, footprint = [[1, 0], [0, 1]])
     array([[ 0,  3,  7, 11],
	    [12, 15, 19, 23],
            [28, 31, 35, 39]])

  Here, a kernel footprint was specified that contains only two
  elements. Therefore, the filter function receives a buffer of length
  equal to two, which was multiplied with the proper weights and the
  result summed.

  When calling :func:`generic_filter`, either the sizes of a
  rectangular kernel or the footprint of the kernel must be
  provided. The *size* parameter, if provided, must be a sequence of
  sizes or a single number, in which case the size of the filter is
  assumed to be equal along each axis. The *footprint*, if provided,
  must be an array that defines the shape of the kernel by its
  non-zero elements.

  Optionally, extra arguments can be defined and passed to the filter
  function. The *extra_arguments* and *extra_keywords* arguments can
  be used to pass a tuple of extra arguments and/or a dictionary of
  named arguments that are passed to derivative at each call. For
  example, we can pass the parameters of our filter as an argument

  .. code:: python

     >>> def fnc(buffer, weights):
     ...     weights = np.asarray(weights)
     ...     return (buffer * weights).sum()
     ...
     >>> generic_filter(a, fnc, footprint = [[1, 0], [0, 1]], extra_arguments = ([1, 3],))
     array([[ 0,  3,  7, 11],
	    [12, 15, 19, 23],
            [28, 31, 35, 39]])

  or

  .. code:: python

     >>> generic_filter(a, fnc, footprint = [[1, 0], [0, 1]], extra_keywords= {'weights': [1, 3]})
     array([[ 0,  3,  7, 11],
	    [12, 15, 19, 23],
	    [28, 31, 35, 39]])

These functions iterate over the lines or elements starting at the
last axis, i.e., the last index changes the fastest. This order of
iteration is guaranteed for the case that it is important to adapt the
filter depending on spatial location. Here is an example of using a
class that implements the filter and keeps track of the current
coordinates while iterating. It performs the same filter operation as
described above for :func:`generic_filter`, but additionally prints
the current coordinates:

.. code:: python

   >>> a = np.arange(12).reshape(3,4)
   >>>
   >>> class fnc_class:
   ...     def __init__(self, shape):
   ...         # store the shape:
   ...         self.shape = shape
   ...         # initialize the coordinates:
   ...         self.coordinates = [0] * len(shape)
   ...
   ...     def filter(self, buffer):
   ...         result = (buffer * np.array([1, 3])).sum()
   ...         print(self.coordinates)
   ...         # calculate the next coordinates:
   ...         axes = list(range(len(self.shape)))
   ...         axes.reverse()
   ...         for jj in axes:
   ...             if self.coordinates[jj] < self.shape[jj] - 1:
   ...                 self.coordinates[jj] += 1
   ...                 break
   ...             else:
   ...                 self.coordinates[jj] = 0
   ...         return result
   ...
   >>> fnc = fnc_class(shape = (3,4))
   >>> generic_filter(a, fnc.filter, footprint = [[1, 0], [0, 1]])
   [0, 0]
   [0, 1]
   [0, 2]
   [0, 3]
   [1, 0]
   [1, 1]
   [1, 2]
   [1, 3]
   [2, 0]
   [2, 1]
   [2, 2]
   [2, 3]
   array([[ 0,  3,  7, 11],
	  [12, 15, 19, 23],
	  [28, 31, 35, 39]])

For the :func:`generic_filter1d` function, the same approach works,
except that this function does not iterate over the axis that is being
filtered. The example for :func:`generic_filter1d` then becomes this:

.. code:: python

   >>> a = np.arange(12).reshape(3,4)
   >>>
   >>> class fnc1d_class:
   ...     def __init__(self, shape, axis = -1):
   ...         # store the filter axis:
   ...         self.axis = axis
   ...         # store the shape:
   ...         self.shape = shape
   ...         # initialize the coordinates:
   ...         self.coordinates = [0] * len(shape)
   ...
   ...     def filter(self, iline, oline):
   ...         oline[...] = iline[:-2] + 2 * iline[1:-1] + 3 * iline[2:]
   ...         print(self.coordinates)
   ...         # calculate the next coordinates:
   ...         axes = list(range(len(self.shape)))
   ...         # skip the filter axis:
   ...         del axes[self.axis]
   ...         axes.reverse()
   ...         for jj in axes:
   ...             if self.coordinates[jj] < self.shape[jj] - 1:
   ...                 self.coordinates[jj] += 1
   ...                 break
   ...             else:
   ...                 self.coordinates[jj] = 0
   ...
   >>> fnc = fnc1d_class(shape = (3,4))
   >>> generic_filter1d(a, fnc.filter, 3)
   [0, 0]
   [1, 0]
   [2, 0]
   array([[ 3,  8, 14, 17],
	  [27, 32, 38, 41],
          [51, 56, 62, 65]])

Fourier domain filters
^^^^^^^^^^^^^^^^^^^^^^

The functions described in this section perform filtering
operations in the Fourier domain. Thus, the input array of such a
function should be compatible with an inverse Fourier transform
function, such as the functions from the :mod:`numpy.fft` module. We,
therefore, have to deal with arrays that may be the result of a real
or a complex Fourier transform. In the case of a real Fourier
transform, only half of the of the symmetric complex transform is
stored. Additionally, it needs to be known what the length of the
axis was that was transformed by the real fft. The functions
described here provide a parameter *n* that, in the case of a real
transform, must be equal to the length of the real transform axis
before transformation. If this parameter is less than zero, it is
assumed that the input array was the result of a complex Fourier
transform. The parameter *axis* can be used to indicate along which
axis the real transform was executed.

- The :func:`fourier_shift` function multiplies the input array with
  the multidimensional Fourier transform of a shift operation for the
  given shift. The *shift* parameter is a sequence of shifts for each
  dimension or a single value for all dimensions.

- The :func:`fourier_gaussian` function multiplies the input array
  with the multidimensional Fourier transform of a Gaussian filter
  with given standard deviations *sigma*. The *sigma* parameter is a
  sequence of values for each dimension or a single value for all
  dimensions.

- The :func:`fourier_uniform` function multiplies the input array with
  the multidimensional Fourier transform of a uniform filter with
  given sizes *size*. The *size* parameter is a sequence of values
  for each dimension or a single value for all dimensions.

- The :func:`fourier_ellipsoid` function multiplies the input array
  with the multidimensional Fourier transform of an elliptically-shaped
  filter with given sizes *size*. The *size* parameter is a sequence
  of values for each dimension or a single value for all dimensions.
  This function is only implemented for dimensions 1, 2, and 3.

.. _ndimage-interpolation:

Interpolation functions
-----------------------

This section describes various interpolation functions that are based
on B-spline theory. A good introduction to B-splines can be found
in [1]_ with detailed algorithms for image interpolation given in [5]_.

Spline pre-filters
^^^^^^^^^^^^^^^^^^

Interpolation using splines of an order larger than 1 requires a
pre-filtering step. The interpolation functions described in section
:ref:`ndimage-interpolation` apply pre-filtering by calling
:func:`spline_filter`, but they can be instructed not to do this by
setting the *prefilter* keyword equal to False. This is useful if more
than one interpolation operation is done on the same array. In this
case, it is more efficient to do the pre-filtering only once and use a
pre-filtered array as the input of the interpolation functions. The
following two functions implement the pre-filtering:

- The :func:`spline_filter1d` function calculates a 1-D
  spline filter along the given axis. An output array can optionally
  be provided. The order of the spline must be larger than 1 and less
  than 6.

- The :func:`spline_filter` function calculates a multidimensional
  spline filter.

  .. note::

     The multidimensional filter is implemented as a sequence of
     1-D spline filters. The intermediate arrays are
     stored in the same data type as the output. Therefore, if an
     output with a limited precision is requested, the results may be
     imprecise because intermediate results may be stored with
     insufficient precision. This can be prevented by specifying a
     output type of high precision.

.. _ndimage-interpolation-modes:

Interpolation boundary handling
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The interpolation functions all employ spline interpolation to effect some
type of geometric transformation of the input array. This requires a
mapping of the output coordinates to the input coordinates, and
therefore, the possibility arises that input values outside the
boundaries may be needed. This problem is solved in the same way as
described in :ref:`ndimage-filter-functions` for the multidimensional
filter functions. Therefore, these functions all support a *mode*
parameter that determines how the boundaries are handled, and a *cval*
parameter that gives a constant value in case that the 'constant' mode
is used. The behavior of all modes, including at non-integer locations is
illustrated below. Note the boundaries are not handled the same for all modes;
`reflect` (aka `grid-mirror`) and `grid-wrap` involve symmetry or repetition
about a point that is half way between image samples (dashed vertical lines)
while modes `mirror` and `wrap` treat the image as if it's extent ends exactly
at the first and last sample point rather than 0.5 samples past it.

.. plot:: tutorial/examples/plot_boundary_modes.py
   :alt: " "
   :include-source: False

The coordinates of image samples fall on integer sampling locations
in the range from 0 to ``shape[i] - 1`` along each axis, ``i``. The figure
below illustrates the interpolation of a point at location ``(3.7, 3.3)``
within an image of shape ``(7, 7)``. For an interpolation of order ``n``,
``n + 1`` samples are involved along each axis. The filled circles
illustrate the sampling locations involved in the interpolation of the value at
the location of the red x.

.. plot:: tutorial/examples/plot_interp_grid.py
   :alt: " "
   :include-source: False


Interpolation functions
^^^^^^^^^^^^^^^^^^^^^^^

- The :func:`geometric_transform` function applies an arbitrary
  geometric transform to the input. The given *mapping* function is
  called at each point in the output to find the corresponding
  coordinates in the input. *mapping* must be a callable object that
  accepts a tuple of length equal to the output array rank and returns
  the corresponding input coordinates as a tuple of length equal to
  the input array rank. The output shape and output type can
  optionally be provided. If not given, they are equal to the input
  shape and type.

  For example:

  .. code:: python

     >>> a = np.arange(12).reshape(4,3).astype(np.float64)
     >>> def shift_func(output_coordinates):
     ...     return (output_coordinates[0] - 0.5, output_coordinates[1] - 0.5)
     ...
     >>> from scipy.ndimage import geometric_transform
     >>> geometric_transform(a, shift_func)
     array([[ 0.    ,  0.    ,  0.    ],
	    [ 0.    ,  1.3625,  2.7375],
            [ 0.    ,  4.8125,  6.1875],
            [ 0.    ,  8.2625,  9.6375]])

  Optionally, extra arguments can be defined and passed to the filter
  function. The *extra_arguments* and *extra_keywords* arguments can
  be used to pass a tuple of extra arguments and/or a dictionary of
  named arguments that are passed to derivative at each call. For
  example, we can pass the shifts in our example as arguments

  .. code:: python

     >>> def shift_func(output_coordinates, s0, s1):
     ...     return (output_coordinates[0] - s0, output_coordinates[1] - s1)
     ...
     >>> geometric_transform(a, shift_func, extra_arguments = (0.5, 0.5))
     array([[ 0.    ,  0.    ,  0.    ],
	    [ 0.    ,  1.3625,  2.7375],
            [ 0.    ,  4.8125,  6.1875],
            [ 0.    ,  8.2625,  9.6375]])

  or

  .. code:: python

     >>> geometric_transform(a, shift_func, extra_keywords = {'s0': 0.5, 's1': 0.5})
     array([[ 0.    ,  0.    ,  0.    ],
	    [ 0.    ,  1.3625,  2.7375],
	    [ 0.    ,  4.8125,  6.1875],
	    [ 0.    ,  8.2625,  9.6375]])

  .. note::

     The mapping function can also be written in C and passed using a
     `scipy.LowLevelCallable`. See :ref:`ndimage-ccallbacks` for more
     information.

- The function :func:`map_coordinates` applies an arbitrary coordinate
  transformation using the given array of coordinates. The shape of
  the output is derived from that of the coordinate array by dropping
  the first axis. The parameter *coordinates* is used to find for each
  point in the output the corresponding coordinates in the input. The
  values of *coordinates* along the first axis are the coordinates in
  the input array at which the output value is found. (See also the
  numarray `coordinates` function.) Since the coordinates may be non-
  integer coordinates, the value of the input at these coordinates is
  determined by spline interpolation of the requested order.

  Here is an example that interpolates a 2D array at ``(0.5, 0.5)`` and
  ``(1, 2)``:

  .. code:: python

     >>> a = np.arange(12).reshape(4,3).astype(np.float64)
     >>> a
     array([[  0.,   1.,   2.],
	    [  3.,   4.,   5.],
	    [  6.,   7.,   8.],
	    [  9.,  10.,  11.]])
     >>> from scipy.ndimage import map_coordinates
     >>> map_coordinates(a, [[0.5, 2], [0.5, 1]])
     array([ 1.3625,  7.])

- The :func:`affine_transform` function applies an affine
  transformation to the input array. The given transformation *matrix*
  and *offset* are used to find for each point in the output the
  corresponding coordinates in the input. The value of the input at
  the calculated coordinates is determined by spline interpolation of
  the requested order. The transformation *matrix* must be
  2-D or can also be given as a 1-D sequence
  or array. In the latter case, it is assumed that the matrix is
  diagonal. A more efficient interpolation algorithm is then applied
  that exploits the separability of the problem. The output shape and
  output type can optionally be provided. If not given, they are equal
  to the input shape and type.

- The :func:`shift` function returns a shifted version of the input,
  using spline interpolation of the requested *order*.

- The :func:`zoom` function returns a rescaled version of the input,
  using spline interpolation of the requested *order*.

- The :func:`rotate` function returns the input array rotated in the
  plane defined by the two axes given by the parameter *axes*, using
  spline interpolation of the requested *order*. The angle must be
  given in degrees. If *reshape* is true, then the size of the output
  array is adapted to contain the rotated input.

.. _ndimage-morphology:

Morphology
----------

.. _ndimage-binary-morphology:

Binary morphology
^^^^^^^^^^^^^^^^^

- The :func:`generate_binary_structure` functions generates a binary
  structuring element for use in binary morphology operations. The
  *rank* of the structure must be provided. The size of the structure
  that is returned is equal to three in each direction. The value of
  each element is equal to one if the square of the Euclidean distance
  from the element to the center is less than or equal to
  *connectivity*. For instance, 2-D 4-connected and
  8-connected structures are generated as follows:

  .. code:: python

     >>> from scipy.ndimage import generate_binary_structure
     >>> generate_binary_structure(2, 1)
     array([[False,  True, False],
	    [ True,  True,  True],
            [False,  True, False]], dtype=bool)
     >>> generate_binary_structure(2, 2)
     array([[ True,  True,  True],
	    [ True,  True,  True],
            [ True,  True,  True]], dtype=bool)

This is a visual presentation of `generate_binary_structure` in 3D:

  .. plot:: tutorial/examples/ndimage/3D_binary_structure.py
      :align: center
      :alt: " "
      :include-source: 0

Most binary morphology functions can be expressed in terms of the
basic operations erosion and dilation, which can be seen here:

  .. plot:: tutorial/examples/morphology_binary_dilation_erosion.py
      :align: center
      :alt: " "
      :include-source: 0

- The :func:`binary_erosion` function implements binary erosion of
  arrays of arbitrary rank with the given structuring element. The
  origin parameter controls the placement of the structuring element,
  as described in :ref:`ndimage-filter-functions`. If no structuring
  element is provided, an element with connectivity equal to one is
  generated using :func:`generate_binary_structure`. The
  *border_value* parameter gives the value of the array outside
  boundaries. The erosion is repeated *iterations* times. If
  *iterations* is less than one, the erosion is repeated until the
  result does not change anymore. If a *mask* array is given, only
  those elements with a true value at the corresponding mask element
  are modified at each iteration.

- The :func:`binary_dilation` function implements binary dilation of
  arrays of arbitrary rank with the given structuring element. The
  origin parameter controls the placement of the structuring element,
  as described in :ref:`ndimage-filter-functions`. If no structuring
  element is provided, an element with connectivity equal to one is
  generated using :func:`generate_binary_structure`. The
  *border_value* parameter gives the value of the array outside
  boundaries. The dilation is repeated *iterations* times. If
  *iterations* is less than one, the dilation is repeated until the
  result does not change anymore. If a *mask* array is given, only
  those elements with a true value at the corresponding mask element
  are modified at each iteration.

Here is an example of using :func:`binary_dilation` to find all elements
that touch the border, by repeatedly dilating an empty array from
the border using the data array as the mask:

.. code:: python

   >>> struct = np.array([[0, 1, 0], [1, 1, 1], [0, 1, 0]])
   >>> a = np.array([[1,0,0,0,0], [1,1,0,1,0], [0,0,1,1,0], [0,0,0,0,0]])
   >>> a
   array([[1, 0, 0, 0, 0],
	  [1, 1, 0, 1, 0],
          [0, 0, 1, 1, 0],
          [0, 0, 0, 0, 0]])
   >>> from scipy.ndimage import binary_dilation
   >>> binary_dilation(np.zeros(a.shape), struct, -1, a, border_value=1)
   array([[ True, False, False, False, False],
	  [ True,  True, False, False, False],
          [False, False, False, False, False],
          [False, False, False, False, False]], dtype=bool)

The :func:`binary_erosion` and :func:`binary_dilation` functions both
have an *iterations* parameter, which allows the erosion or dilation to
be repeated a number of times. Repeating an erosion or a dilation with
a given structure *n* times is equivalent to an erosion or a dilation
with a structure that is *n-1* times dilated with itself. A function
is provided that allows the calculation of a structure that is dilated
a number of times with itself:

- The :func:`iterate_structure` function returns a structure by dilation
  of the input structure *iteration* - 1 times with itself.

  For instance:

  .. code:: python

     >>> struct = generate_binary_structure(2, 1)
     >>> struct
     array([[False,  True, False],
	    [ True,  True,  True],
	    [False,  True, False]], dtype=bool)
     >>> from scipy.ndimage import iterate_structure
     >>> iterate_structure(struct, 2)
     array([[False, False,  True, False, False],
	    [False,  True,  True,  True, False],
	    [ True,  True,  True,  True,  True],
	    [False,  True,  True,  True, False],
            [False, False,  True, False, False]], dtype=bool)

     If the origin of the original structure is equal to 0, then it is
     also equal to 0 for the iterated structure. If not, the origin
     must also be adapted if the equivalent of the *iterations*
     erosions or dilations must be achieved with the iterated
     structure. The adapted origin is simply obtained by multiplying
     with the number of iterations. For convenience, the
     :func:`iterate_structure` also returns the adapted origin if the
     *origin* parameter is not ``None``:

     .. code:: python

	>>> iterate_structure(struct, 2, -1)
	(array([[False, False,  True, False, False],
	        [False,  True,  True,  True, False],
		[ True,  True,  True,  True,  True],
		[False,  True,  True,  True, False],
		[False, False,  True, False, False]], dtype=bool), [-2, -2])

Other morphology operations can be defined in terms of erosion and
dilation. The following functions provide a few of these operations
for convenience:

- The :func:`binary_opening` function implements binary opening of
  arrays of arbitrary rank with the given structuring element. Binary
  opening is equivalent to a binary erosion followed by a binary
  dilation with the same structuring element. The origin parameter
  controls the placement of the structuring element, as described in
  :ref:`ndimage-filter-functions`. If no structuring element is
  provided, an element with connectivity equal to one is generated
  using :func:`generate_binary_structure`. The *iterations* parameter
  gives the number of erosions that is performed followed by the same
  number of dilations.

- The :func:`binary_closing` function implements binary closing of
  arrays of arbitrary rank with the given structuring element. Binary
  closing is equivalent to a binary dilation followed by a binary
  erosion with the same structuring element. The origin parameter
  controls the placement of the structuring element, as described in
  :ref:`ndimage-filter-functions`. If no structuring element is
  provided, an element with connectivity equal to one is generated
  using :func:`generate_binary_structure`. The *iterations* parameter
  gives the number of dilations that is performed followed by the same
  number of erosions.

- The :func:`binary_fill_holes` function is used to close holes in
  objects in a binary image, where the structure defines the
  connectivity of the holes. The origin parameter controls the
  placement of the structuring element, as described in
  :ref:`ndimage-filter-functions`. If no structuring element is
  provided, an element with connectivity equal to one is generated
  using :func:`generate_binary_structure`.

- The :func:`binary_hit_or_miss` function implements a binary
  hit-or-miss transform of arrays of arbitrary rank with the given
  structuring elements. The hit-or-miss transform is calculated by
  erosion of the input with the first structure, erosion of the
  logical *not* of the input with the second structure, followed by
  the logical *and* of these two erosions. The origin parameters
  control the placement of the structuring elements, as described in
  :ref:`ndimage-filter-functions`. If *origin2* equals ``None``, it is set
  equal to the *origin1* parameter. If the first structuring element
  is not provided, a structuring element with connectivity equal to
  one is generated using :func:`generate_binary_structure`. If
  *structure2* is not provided, it is set equal to the logical *not*
  of *structure1*.

.. _ndimage-grey-morphology:

Grey-scale morphology
^^^^^^^^^^^^^^^^^^^^^

Grey-scale morphology operations are the equivalents of binary
morphology operations that operate on arrays with arbitrary values.
Below, we describe the grey-scale equivalents of erosion, dilation,
opening and closing. These operations are implemented in a similar
fashion as the filters described in :ref:`ndimage-filter-functions`,
and we refer to this section for the description of filter kernels and
footprints, and the handling of array borders. The grey-scale
morphology operations optionally take a *structure* parameter that
gives the values of the structuring element. If this parameter is not
given, the structuring element is assumed to be flat with a value equal
to zero. The shape of the structure can optionally be defined by the
*footprint* parameter. If this parameter is not given, the structure
is assumed to be rectangular, with sizes equal to the dimensions of
the *structure* array, or by the *size* parameter if *structure* is
not given. The *size* parameter is only used if both *structure* and
*footprint* are not given, in which case the structuring element is
assumed to be rectangular and flat with the dimensions given by
*size*. The *size* parameter, if provided, must be a sequence of sizes
or a single number in which case the size of the filter is assumed to
be equal along each axis. The *footprint* parameter, if provided, must
be an array that defines the shape of the kernel by its non-zero
elements.

Similarly to binary erosion and dilation, there are operations for
grey-scale erosion and dilation:

- The :func:`grey_erosion` function calculates a multidimensional
  grey-scale erosion.

- The :func:`grey_dilation` function calculates a multidimensional
  grey-scale dilation.

Grey-scale opening and closing operations can be defined similarly to
their binary counterparts:

- The :func:`grey_opening` function implements grey-scale opening of
  arrays of arbitrary rank. Grey-scale opening is equivalent to a
  grey-scale erosion followed by a grey-scale dilation.

- The :func:`grey_closing` function implements grey-scale closing of
  arrays of arbitrary rank. Grey-scale opening is equivalent to a
  grey-scale dilation followed by a grey-scale erosion.

- The :func:`morphological_gradient` function implements a grey-scale
  morphological gradient of arrays of arbitrary rank. The grey-scale
  morphological gradient is equal to the difference of a grey-scale
  dilation and a grey-scale erosion.

- The :func:`morphological_laplace` function implements a grey-scale
  morphological laplace of arrays of arbitrary rank. The grey-scale
  morphological laplace is equal to the sum of a grey-scale dilation
  and a grey-scale erosion minus twice the input.

- The :func:`white_tophat` function implements a white top-hat filter
  of arrays of arbitrary rank. The white top-hat is equal to the
  difference of the input and a grey-scale opening.

- The :func:`black_tophat` function implements a black top-hat filter
  of arrays of arbitrary rank. The black top-hat is equal to the
  difference of a grey-scale closing and the input.

.. _ndimage-distance-transforms:

Distance transforms
-------------------

Distance transforms are used to calculate the minimum distance from
each element of an object to the background. The following functions
implement distance transforms for three different distance metrics:
Euclidean, city block, and chessboard distances.

- The function :func:`distance_transform_cdt` uses a chamfer type
  algorithm to calculate the distance transform of the input, by
  replacing each object element (defined by values larger than zero)
  with the shortest distance to the background (all non-object
  elements). The structure determines the type of chamfering that is
  done. If the structure is equal to 'cityblock', a structure is
  generated using :func:`generate_binary_structure` with a squared
  distance equal to 1. If the structure is equal to 'chessboard', a
  structure is generated using :func:`generate_binary_structure` with
  a squared distance equal to the rank of the array. These choices
  correspond to the common interpretations of the city block and the
  chessboard distance metrics in two dimensions.

  In addition to the distance transform, the feature transform can be
  calculated. In this case, the index of the closest background element
  is returned along the first axis of the result. The
  *return_distances*, and *return_indices* flags can be used to
  indicate if the distance transform, the feature transform, or both
  must be returned.

  The *distances* and *indices* arguments can be used to give optional
  output arrays that must be of the correct size and type (both
  ``numpy.int32``). The basics of the algorithm used to implement this
  function are described in [2]_.

- The function :func:`distance_transform_edt` calculates the exact
  Euclidean distance transform of the input, by replacing each object
  element (defined by values larger than zero) with the shortest
  Euclidean distance to the background (all non-object elements).

  In addition to the distance transform, the feature transform can be
  calculated. In this case, the index of the closest background element
  is returned along the first axis of the result. The
  *return_distances* and *return_indices* flags can be used to
  indicate if the distance transform, the feature transform, or both
  must be returned.

  Optionally, the sampling along each axis can be given by the
  *sampling* parameter, which should be a sequence of length equal to
  the input rank, or a single number in which the sampling is assumed
  to be equal along all axes.

  The *distances* and *indices* arguments can be used to give optional
  output arrays that must be of the correct size and type
  (``numpy.float64`` and ``numpy.int32``).The algorithm used to
  implement this function is described in [3]_.

- The function :func:`distance_transform_bf` uses a brute-force
  algorithm to calculate the distance transform of the input, by
  replacing each object element (defined by values larger than zero)
  with the shortest distance to the background (all non-object
  elements). The metric must be one of "euclidean", "cityblock", or
  "chessboard".

  In addition to the distance transform, the feature transform can be
  calculated. In this case, the index of the closest background element
  is returned along the first axis of the result. The
  *return_distances* and *return_indices* flags can be used to
  indicate if the distance transform, the feature transform, or both
  must be returned.

  Optionally, the sampling along each axis can be given by the
  *sampling* parameter, which should be a sequence of length equal to
  the input rank, or a single number in which the sampling is assumed
  to be equal along all axes. This parameter is only used in the case
  of the Euclidean distance transform.

  The *distances* and *indices* arguments can be used to give optional
  output arrays that must be of the correct size and type
  (``numpy.float64`` and ``numpy.int32``).

  .. note::

     This function uses a slow brute-force algorithm, the function
     :func:`distance_transform_cdt` can be used to more efficiently
     calculate city block and chessboard distance transforms. The
     function :func:`distance_transform_edt` can be used to more
     efficiently calculate the exact Euclidean distance transform.

Segmentation and labeling
-------------------------

Segmentation is the process of separating objects of interest from
the background. The most simple approach is, probably, intensity
thresholding, which is easily done with :mod:`numpy` functions:

.. code:: python

   >>> a = np.array([[1,2,2,1,1,0],
   ...               [0,2,3,1,2,0],
   ...               [1,1,1,3,3,2],
   ...               [1,1,1,1,2,1]])
   >>> np.where(a > 1, 1, 0)
   array([[0, 1, 1, 0, 0, 0],
	  [0, 1, 1, 0, 1, 0],
	  [0, 0, 0, 1, 1, 1],
	  [0, 0, 0, 0, 1, 0]])

The result is a binary image, in which the individual objects still
need to be identified and labeled. The function :func:`label`
generates an array where each object is assigned a unique number:

- The :func:`label` function generates an array where the objects in
  the input are labeled with an integer index. It returns a tuple
  consisting of the array of object labels and the number of objects
  found, unless the *output* parameter is given, in which case only
  the number of objects is returned. The connectivity of the objects
  is defined by a structuring element. For instance, in 2D
  using a 4-connected structuring element gives:

  .. code:: python

     >>> a = np.array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
     >>> s = [[0, 1, 0], [1,1,1], [0,1,0]]
     >>> from scipy.ndimage import label
     >>> label(a, s)
     (array([[0, 1, 1, 0, 0, 0],
	     [0, 1, 1, 0, 2, 0],
	     [0, 0, 0, 2, 2, 2],
	     [0, 0, 0, 0, 2, 0]]), 2)

  These two objects are not connected because there is no way in which
  we can place the structuring element, such that it overlaps with both
  objects. However, an 8-connected structuring element results in only
  a single object:

  .. code:: python

     >>> a = np.array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
     >>> s = [[1,1,1], [1,1,1], [1,1,1]]
     >>> label(a, s)[0]
     array([[0, 1, 1, 0, 0, 0],
	    [0, 1, 1, 0, 1, 0],
            [0, 0, 0, 1, 1, 1],
            [0, 0, 0, 0, 1, 0]])

  If no structuring element is provided, one is generated by calling
  :func:`generate_binary_structure` (see
  :ref:`ndimage-binary-morphology`) using a connectivity of one (which
  in 2D is the 4-connected structure of the first example). The input
  can be of any type, any value not equal to zero is taken to be part
  of an object. This is useful if you need to 're-label' an array of
  object indices, for instance, after removing unwanted objects. Just
  apply the label function again to the index array. For instance:

  .. code:: python

     >>> l, n = label([1, 0, 1, 0, 1])
     >>> l
     array([1, 0, 2, 0, 3])
     >>> l = np.where(l != 2, l, 0)
     >>> l
     array([1, 0, 0, 0, 3])
     >>> label(l)[0]
     array([1, 0, 0, 0, 2])

  .. note::

     The structuring element used by :func:`label` is assumed to be
     symmetric.

There is a large number of other approaches for segmentation, for
instance, from an estimation of the borders of the objects that can be
obtained by derivative filters. One such approach is
watershed segmentation. The function :func:`watershed_ift` generates
an array where each object is assigned a unique label, from an array
that localizes the object borders, generated, for instance, by a
gradient magnitude filter. It uses an array containing initial markers
for the objects:

- The :func:`watershed_ift` function applies a watershed from markers
  algorithm, using Image Foresting Transform, as described in
  [4]_.

- The inputs of this function are the array to which the transform is
  applied, and an array of markers that designate the objects by a
  unique label, where any non-zero value is a marker. For instance:

  .. code:: python

     >>> input = np.array([[0, 0, 0, 0, 0, 0, 0],
     ...                   [0, 1, 1, 1, 1, 1, 0],
     ...                   [0, 1, 0, 0, 0, 1, 0],
     ...                   [0, 1, 0, 0, 0, 1, 0],
     ...                   [0, 1, 0, 0, 0, 1, 0],
     ...                   [0, 1, 1, 1, 1, 1, 0],
     ...                   [0, 0, 0, 0, 0, 0, 0]], np.uint8)
     >>> markers = np.array([[1, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 2, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0]], np.int8)
     >>> from scipy.ndimage import watershed_ift
     >>> watershed_ift(input, markers)
     array([[1, 1, 1, 1, 1, 1, 1],
	    [1, 1, 2, 2, 2, 1, 1],
            [1, 2, 2, 2, 2, 2, 1],
            [1, 2, 2, 2, 2, 2, 1],
            [1, 2, 2, 2, 2, 2, 1],
            [1, 1, 2, 2, 2, 1, 1],
            [1, 1, 1, 1, 1, 1, 1]], dtype=int8)

  Here, two markers were used to designate an object (*marker* = 2) and
  the background (*marker* = 1). The order in which these are
  processed is arbitrary: moving the marker for the background to the
  lower-right corner of the array yields a different result:

  .. code:: python

     >>> markers = np.array([[0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 2, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 1]], np.int8)
     >>> watershed_ift(input, markers)
     array([[1, 1, 1, 1, 1, 1, 1],
	    [1, 1, 1, 1, 1, 1, 1],
	    [1, 1, 2, 2, 2, 1, 1],
	    [1, 1, 2, 2, 2, 1, 1],
            [1, 1, 2, 2, 2, 1, 1],
            [1, 1, 1, 1, 1, 1, 1],
            [1, 1, 1, 1, 1, 1, 1]], dtype=int8)

  The result is that the object (*marker* = 2) is smaller because the
  second marker was processed earlier. This may not be the desired
  effect if the first marker was supposed to designate a background
  object. Therefore, :func:`watershed_ift` treats markers with a
  negative value explicitly as background markers and processes them
  after the normal markers. For instance, replacing the first marker
  by a negative marker gives a result similar to the first example:

  .. code:: python

     >>> markers = np.array([[0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 2, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, 0],
     ...                     [0, 0, 0, 0, 0, 0, -1]], np.int8)
     >>> watershed_ift(input, markers)
     array([[-1, -1, -1, -1, -1, -1, -1],
	    [-1, -1,  2,  2,  2, -1, -1],
	    [-1,  2,  2,  2,  2,  2, -1],
	    [-1,  2,  2,  2,  2,  2, -1],
            [-1,  2,  2,  2,  2,  2, -1],
            [-1, -1,  2,  2,  2, -1, -1],
            [-1, -1, -1, -1, -1, -1, -1]], dtype=int8)

  The connectivity of the objects is defined by a structuring
  element. If no structuring element is provided, one is generated by
  calling :func:`generate_binary_structure` (see
  :ref:`ndimage-binary-morphology`) using a connectivity of one (which
  in 2D is a 4-connected structure.) For example, using an 8-connected
  structure with the last example yields a different object:

  .. code:: python

     >>> watershed_ift(input, markers,
     ...               structure = [[1,1,1], [1,1,1], [1,1,1]])
     array([[-1, -1, -1, -1, -1, -1, -1],
	    [-1,  2,  2,  2,  2,  2, -1],
            [-1,  2,  2,  2,  2,  2, -1],
            [-1,  2,  2,  2,  2,  2, -1],
            [-1,  2,  2,  2,  2,  2, -1],
            [-1,  2,  2,  2,  2,  2, -1],
            [-1, -1, -1, -1, -1, -1, -1]], dtype=int8)

  .. note::

     The implementation of :func:`watershed_ift` limits the data types
     of the input to ``numpy.uint8`` and ``numpy.uint16``.

.. _ndimage-object-measurements:

Object measurements
-------------------

Given an array of labeled objects, the properties of the individual
objects can be measured. The :func:`find_objects` function can be used
to generate a list of slices that for each object, give the
smallest sub-array that fully contains the object:

- The :func:`find_objects` function finds all objects in a labeled
  array and returns a list of slices that correspond to the smallest
  regions in the array that contains the object.

  For instance:

  .. code:: python

     >>> a = np.array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
     >>> l, n = label(a)
     >>> from scipy.ndimage import find_objects
     >>> f = find_objects(l)
     >>> a[f[0]]
     array([[1, 1],
	    [1, 1]])
     >>> a[f[1]]
     array([[0, 1, 0],
	    [1, 1, 1],
	    [0, 1, 0]])

  The function :func:`find_objects` returns slices for all objects,
  unless the *max_label* parameter is larger then zero, in which case
  only the first *max_label* objects are returned. If an index is
  missing in the *label* array, ``None`` is return instead of a
  slice. For example:

  .. code:: python

     >>> from scipy.ndimage import find_objects
     >>> find_objects([1, 0, 3, 4], max_label = 3)
     [(slice(0, 1, None),), None, (slice(2, 3, None),)]

The list of slices generated by :func:`find_objects` is useful to find
the position and dimensions of the objects in the array, but can also
be used to perform measurements on the individual objects. Say, we want
to find the sum of the intensities of an object in image:

.. code:: python

   >>> image = np.arange(4 * 6).reshape(4, 6)
   >>> mask = np.array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
   >>> labels = label(mask)[0]
   >>> slices = find_objects(labels)

Then we can calculate the sum of the elements in the second object:

.. code:: python

   >>> np.where(labels[slices[1]] == 2, image[slices[1]], 0).sum()
   80

That is, however, not particularly efficient and may also be more
complicated for other types of measurements. Therefore, a few
measurements functions are defined that accept the array of object
labels and the index of the object to be measured. For instance,
calculating the sum of the intensities can be done by:

.. code:: python

   >>> from scipy.ndimage import sum as ndi_sum
   >>> ndi_sum(image, labels, 2)
   80

For large arrays and small objects, it is more efficient to call the
measurement functions after slicing the array:

.. code:: python

   >>> ndi_sum(image[slices[1]], labels[slices[1]], 2)
   80

Alternatively, we can do the measurements for a number of labels with
a single function call, returning a list of results. For instance, to
measure the sum of the values of the background and the second object
in our example, we give a list of labels:

.. code:: python

   >>> ndi_sum(image, labels, [0, 2])
   array([178.0, 80.0])

The measurement functions described below all support the *index*
parameter to indicate which object(s) should be measured. The default
value of *index* is ``None``. This indicates that all elements where the
label is larger than zero should be treated as a single object and
measured. Thus, in this case the *labels* array is treated as a mask
defined by the elements that are larger than zero. If *index* is a
number or a sequence of numbers it gives the labels of the objects
that are measured. If *index* is a sequence, a list of the results is
returned. Functions that return more than one result return their
result as a tuple if *index* is a single number, or as a tuple of
lists if *index* is a sequence.

- The :func:`sum` function calculates the sum of the elements of the
  object with label(s) given by *index*, using the *labels* array for
  the object labels. If *index* is ``None``, all elements with a
  non-zero label value are treated as a single object. If *label* is
  ``None``, all elements of *input* are used in the calculation.

- The :func:`mean` function calculates the mean of the elements of the
  object with label(s) given by *index*, using the *labels* array for
  the object labels. If *index* is ``None``, all elements with a
  non-zero label value are treated as a single object. If *label* is
  ``None``, all elements of *input* are used in the calculation.

- The :func:`variance` function calculates the variance of the
  elements of the object with label(s) given by *index*, using the
  *labels* array for the object labels. If *index* is ``None``, all
  elements with a non-zero label value are treated as a single
  object. If *label* is ``None``, all elements of *input* are used in
  the calculation.

- The :func:`standard_deviation` function calculates the standard
  deviation of the elements of the object with label(s) given by
  *index*, using the *labels* array for the object labels. If *index*
  is ``None``, all elements with a non-zero label value are treated as
  a single object. If *label* is ``None``, all elements of *input* are
  used in the calculation.

- The :func:`minimum` function calculates the minimum of the elements
  of the object with label(s) given by *index*, using the *labels*
  array for the object labels. If *index* is ``None``, all elements
  with a non-zero label value are treated as a single object. If
  *label* is ``None``, all elements of *input* are used in the
  calculation.

- The :func:`maximum` function calculates the maximum of the elements
  of the object with label(s) given by *index*, using the *labels*
  array for the object labels. If *index* is ``None``, all elements
  with a non-zero label value are treated as a single object. If
  *label* is ``None``, all elements of *input* are used in the
  calculation.

- The :func:`minimum_position` function calculates the position of the
  minimum of the elements of the object with label(s) given by
  *index*, using the *labels* array for the object labels. If *index*
  is ``None``, all elements with a non-zero label value are treated as
  a single object. If *label* is ``None``, all elements of *input* are
  used in the calculation.

- The :func:`maximum_position` function calculates the position of the
  maximum of the elements of the object with label(s) given by
  *index*, using the *labels* array for the object labels. If *index*
  is ``None``, all elements with a non-zero label value are treated as
  a single object. If *label* is ``None``, all elements of *input* are
  used in the calculation.

- The :func:`extrema` function calculates the minimum, the maximum,
  and their positions, of the elements of the object with label(s)
  given by *index*, using the *labels* array for the object labels. If
  *index* is ``None``, all elements with a non-zero label value are
  treated as a single object. If *label* is ``None``, all elements of
  *input* are used in the calculation. The result is a tuple giving
  the minimum, the maximum, the position of the minimum, and the
  position of the maximum. The result is the same as a tuple formed by
  the results of the functions *minimum*, *maximum*,
  *minimum_position*, and *maximum_position* that are described above.

- The :func:`center_of_mass` function calculates the center of mass of
  the object with label(s) given by *index*, using the *labels*
  array for the object labels. If *index* is ``None``, all elements
  with a non-zero label value are treated as a single object. If
  *label* is ``None``, all elements of *input* are used in the
  calculation.

- The :func:`histogram` function calculates a histogram of the
  object with label(s) given by *index*, using the *labels* array for
  the object labels. If *index* is ``None``, all elements with a
  non-zero label value are treated as a single object. If *label* is
  ``None``, all elements of *input* are used in the calculation.
  Histograms are defined by their minimum (*min*), maximum (*max*), and
  the number of bins (*bins*). They are returned as 1-D
  arrays of type ``numpy.int32``.

.. _ndimage-ccallbacks:

Extending :mod:`scipy.ndimage` in C
-----------------------------------

A few functions in :mod:`scipy.ndimage` take a callback argument. This
can be either a python function or a `scipy.LowLevelCallable` containing a
pointer to a C function. Using a C function will generally be more
efficient, since it avoids the overhead of calling a python function on
many elements of an array. To use a C function, you must write a C
extension that contains the callback function and a Python function
that returns a `scipy.LowLevelCallable` containing a pointer to the
callback.

An example of a function that supports callbacks is
:func:`geometric_transform`, which accepts a callback function that
defines a mapping from all output coordinates to corresponding
coordinates in the input array. Consider the following python example,
which uses :func:`geometric_transform` to implement a shift function.

.. code:: python

   from scipy import ndimage

   def transform(output_coordinates, shift):
       input_coordinates = output_coordinates[0] - shift, output_coordinates[1] - shift
       return input_coordinates

   im = np.arange(12).reshape(4, 3).astype(np.float64)
   shift = 0.5
   print(ndimage.geometric_transform(im, transform, extra_arguments=(shift,)))

We can also implement the callback function with the following C code:

.. code:: c

   /* example.c */

   #include <Python.h>
   #include <numpy/npy_common.h>

   static int
   _transform(npy_intp *output_coordinates, double *input_coordinates,
              int output_rank, int input_rank, void *user_data)
   {
       npy_intp i;
       double shift = *(double *)user_data;

       for (i = 0; i < input_rank; i++) {
           input_coordinates[i] = output_coordinates[i] - shift;
       }
       return 1;
   }

   static char *transform_signature = "int (npy_intp *, double *, int, int, void *)";

   static PyObject *
   py_get_transform(PyObject *obj, PyObject *args)
   {
       if (!PyArg_ParseTuple(args, "")) return NULL;
       return PyCapsule_New(_transform, transform_signature, NULL);
   }

   static PyMethodDef ExampleMethods[] = {
       {"get_transform", (PyCFunction)py_get_transform, METH_VARARGS, ""},
       {NULL, NULL, 0, NULL}
   };

   /* Initialize the module */
   static struct PyModuleDef example = {
       PyModuleDef_HEAD_INIT,
       "example",
       NULL,
       -1,
       ExampleMethods,
       NULL,
       NULL,
       NULL,
       NULL
   };

   PyMODINIT_FUNC
   PyInit_example(void)
   {
       return PyModule_Create(&example);
   }

More information on writing Python extension modules can be found
`here`__. If the C code is in the file ``example.c``, then it can be
compiled after adding it to ``meson.build`` (see examples inside
``meson.build`` files) and follow what's there. After that is done,
running the script:

__ https://docs.python.org/3/extending/index.html

.. code:: python

   import ctypes
   import numpy as np
   from scipy import ndimage, LowLevelCallable

   from example import get_transform

   shift = 0.5

   user_data = ctypes.c_double(shift)
   ptr = ctypes.cast(ctypes.pointer(user_data), ctypes.c_void_p)
   callback = LowLevelCallable(get_transform(), ptr)
   im = np.arange(12).reshape(4, 3).astype(np.float64)
   print(ndimage.geometric_transform(im, callback))

produces the same result as the original python script.

In the C version, ``_transform`` is the callback function and the
parameters ``output_coordinates`` and ``input_coordinates`` play the
same role as they do in the python version, while ``output_rank`` and
``input_rank`` provide the equivalents of ``len(output_coordinates)``
and ``len(input_coordinates)``. The variable ``shift`` is passed
through ``user_data`` instead of
``extra_arguments``. Finally, the C callback function returns an integer
status, which is one upon success and zero otherwise.

The function ``py_transform`` wraps the callback function in a
:c:type:`PyCapsule`. The main steps are:

- Initialize a :c:type:`PyCapsule`. The first argument is a pointer to
  the callback function.

- The second argument is the function signature, which must match exactly
  the one expected by :mod:`~scipy.ndimage`.

- Above, we used  `scipy.LowLevelCallable` to specify ``user_data``
  that we generated with `ctypes`.

  A different approach would be to supply the data in the capsule context,
  that can be set by `PyCapsule_SetContext` and omit specifying
  ``user_data`` in `scipy.LowLevelCallable`. However, in this approach we would
  need to deal with allocation/freeing of the data --- freeing the data
  after the capsule has been destroyed can be done by specifying a non-NULL
  callback function in the third argument of `PyCapsule_New`.

C callback functions for :mod:`~scipy.ndimage` all follow this scheme. The
next section lists the :mod:`~scipy.ndimage` functions that accept a C
callback function and gives the prototype of the function.

.. seealso::

   The functions that support low-level callback arguments are:

   `generic_filter`, `generic_filter1d`, `geometric_transform`

Below, we show alternative ways to write the code, using Numba_, Cython_,
ctypes_, or cffi_ instead of writing wrapper code in C.

.. _Numba: https://numba.pydata.org/
.. _Cython: https://cython.org/
.. _ctypes: https://docs.python.org/3/library/ctypes.html
.. _cffi: https://cffi.readthedocs.io/

.. rubric:: Numba

Numba_ provides a way to write low-level functions easily in Python.
We can write the above using Numba as:

.. code:: python

   # example.py
   import numpy as np
   import ctypes
   from scipy import ndimage, LowLevelCallable
   from numba import cfunc, types, carray

   @cfunc(types.intc(types.CPointer(types.intp),
                     types.CPointer(types.double),
                     types.intc,
                     types.intc,
                     types.voidptr))
   def transform(output_coordinates_ptr, input_coordinates_ptr,
                 output_rank, input_rank, user_data):
       input_coordinates = carray(input_coordinates_ptr, (input_rank,))
       output_coordinates = carray(output_coordinates_ptr, (output_rank,))
       shift = carray(user_data, (1,), types.double)[0]

       for i in range(input_rank):
           input_coordinates[i] = output_coordinates[i] - shift

       return 1

   shift = 0.5

   # Then call the function
   user_data = ctypes.c_double(shift)
   ptr = ctypes.cast(ctypes.pointer(user_data), ctypes.c_void_p)
   callback = LowLevelCallable(transform.ctypes, ptr)

   im = np.arange(12).reshape(4, 3).astype(np.float64)
   print(ndimage.geometric_transform(im, callback))


.. rubric:: Cython

Functionally the same code as above can be written in Cython with
somewhat less boilerplate as follows:

.. code:: cython

   # example.pyx

   from numpy cimport npy_intp as intp

   cdef api int transform(intp *output_coordinates, double *input_coordinates,
                          int output_rank, int input_rank, void *user_data):
       cdef intp i
       cdef double shift = (<double *>user_data)[0]

       for i in range(input_rank):
           input_coordinates[i] = output_coordinates[i] - shift
       return 1

.. code:: python

   # script.py

   import ctypes
   import numpy as np
   from scipy import ndimage, LowLevelCallable

   import example

   shift = 0.5

   user_data = ctypes.c_double(shift)
   ptr = ctypes.cast(ctypes.pointer(user_data), ctypes.c_void_p)
   callback = LowLevelCallable.from_cython(example, "transform", ptr)
   im = np.arange(12).reshape(4, 3).astype(np.float64)
   print(ndimage.geometric_transform(im, callback))


.. rubric:: cffi

With cffi_, you can interface with a C function residing in a shared
library (DLL). First, we need to write the shared library, which we do
in C --- this example is for Linux/OSX:

.. code:: c

   /*
     example.c
     Needs to be compiled with "gcc -std=c99 -shared -fPIC -o example.so example.c"
     or similar
    */

   #include <stdint.h>

   int
   _transform(intptr_t *output_coordinates, double *input_coordinates,
              int output_rank, int input_rank, void *user_data)
   {
       int i;
       double shift = *(double *)user_data;

       for (i = 0; i < input_rank; i++) {
           input_coordinates[i] = output_coordinates[i] - shift;
       }
       return 1;
   }

The Python code calling the library is:

.. code:: python

   import os
   import numpy as np
   from scipy import ndimage, LowLevelCallable
   import cffi

   # Construct the FFI object, and copypaste the function declaration
   ffi = cffi.FFI()
   ffi.cdef("""
   int _transform(intptr_t *output_coordinates, double *input_coordinates,
                  int output_rank, int input_rank, void *user_data);
   """)

   # Open library
   lib = ffi.dlopen(os.path.abspath("example.so"))

   # Do the function call
   user_data = ffi.new('double *', 0.5)
   callback = LowLevelCallable(lib._transform, user_data)
   im = np.arange(12).reshape(4, 3).astype(np.float64)
   print(ndimage.geometric_transform(im, callback))

You can find more information in the cffi_ documentation.

.. rubric:: ctypes

With *ctypes*, the C code and the compilation of the so/DLL is as for
cffi above.  The Python code is different:

.. code:: python

   # script.py

   import os
   import ctypes
   import numpy as np
   from scipy import ndimage, LowLevelCallable

   lib = ctypes.CDLL(os.path.abspath('example.so'))

   shift = 0.5

   user_data = ctypes.c_double(shift)
   ptr = ctypes.cast(ctypes.pointer(user_data), ctypes.c_void_p)

   # Ctypes has no built-in intptr type, so override the signature
   # instead of trying to get it via ctypes
   callback = LowLevelCallable(lib._transform, ptr,
       "int _transform(intptr_t *, double *, int, int, void *)")

   # Perform the call
   im = np.arange(12).reshape(4, 3).astype(np.float64)
   print(ndimage.geometric_transform(im, callback))

You can find more information in the ctypes_ documentation.


References
----------

.. [1] M. Unser, "Splines: A Perfect Fit for Signal and Image
       Processing," IEEE Signal Processing Magazine, vol. 16, no. 6, pp.
       22-38, November 1999.

.. [2] G. Borgefors, "Distance transformations in arbitrary
       dimensions.", Computer Vision, Graphics, and Image Processing,
       27:321-345, 1984.

.. [3] C. R. Maurer, Jr., R. Qi, and V. Raghavan, "A linear time
       algorithm for computing exact euclidean distance transforms of
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