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
two-dimensional 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
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
>>> 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 one-dimensional kernel of length three is at the
second element. Take for example the correlation of a
one-dimensional 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 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:
========== ==================================== ====================
"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]
"constant" Use a constant value, default is 0.0 [1 2 3]->[0 1 2 3 0]
========== ==================================== ====================
The "constant" mode is special since it needs an additional
parameter to specify the constant value that should be used.
.. 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 one-dimensional
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 one-dimensional sequences of numbers.
- The function :func:`correlate` implements multidimensional
correlation of the input array with a given kernel.
- The :func:`convolve1d` function calculates a one-dimensional
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 one-dimensional sequences of numbers.
.. 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 result is shifted in the
opposite directions.
- 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 one-dimensional
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.
.. note::
The multidimensional filter is implemented as a sequence of
one-dimensional 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 one-dimensional
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 axis are assumed
to be equal.
.. note::
The multidimensional filter is implemented as a sequence of
one-dimensional 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 one-dimensional
minimum filter of given *size* along the given axis.
- The :func:`maximum_filter1d` function calculates a one-dimensional
maximum filter of 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 then 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 then 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 :func:`derivative2` to calculate
second derivatives. The function :func:`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 :func:`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 calculats the gradient magnitude of an array.
- The function :func:`generic_gradient_magnitude` calculates a
gradient magnitude using the function passed through
:func:`derivative` to calculate first derivatives. The function
:func:`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 directly be used 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
:ctype:`PyCapsule` (see :ref:`ndimage-ccallbacks` for more
information).
- The :func:`generic_filter1d` function implements a generic
one-dimensional 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 :func:`function` at each line. The
arguments that are passed to :func:`function` are one-dimensional
arrays of the :ctype:`tFloat64` type. The first contains the values
of the current line. It is extended at the beginning end 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 was
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
:func:`function` at each element. The argument of :func:`function`
is a one-dimensional array of the :ctype:`tFloat64` 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 = 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 = 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 sequences 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
sequences 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 sequences 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 a elliptically shaped
filter with given sizes *size*. The *size* parameter is a sequences
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]_.
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
prefiltered array as the input of the interpolation functions. The
following two functions implement the pre-filtering:
- The :func:`spline_filter1d` function calculates a one-dimensional
spline filter along the given axis. An output array can optionally
be provided. The order of the spline must be larger then 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
one-dimensional 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.
Interpolation functions
^^^^^^^^^^^^^^^^^^^^^^^
Following 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 are 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 :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
:ctype:`PyCapsule`. 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
two-dimensional or can also be given as a one-dimensional 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 or equal to
*connectivity*. For instance, two dimensional 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)
Most binary morphology functions can be expressed in terms of the
basic operations erosion and dilation.
- 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 d
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.
Similar 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 similar 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 cityblock 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
:ctype:`Int32`). The basics of the algorithm used to implement this
function is 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
(:ctype:`Float64` and :ctype:`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
(:ctype:`Float64` and :ctype:`Int32`).
.. note::
This function uses a slow brute-force algorithm, the function
:func:`distance_transform_cdt` can be used to more efficiently
calculate cityblock 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 two dimensions
using a four-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 for instance by derivative filters. One such an 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 an Iterative Forest 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 :ctype:`UInt8` and :ctype:`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 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 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 one-dimensional
arrays of type :ctype:`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 :ctype:`PyCapsule` 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 :ctype:`PyCapsule` 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
#include <Python.h>
#include <numpy/npy_common.h>
static void _destructor(PyObject *obj)
{
void *callback_data = PyCapsule_GetContext(obj);
PyMem_Free(callback_data);
}
static int
_transform(npy_intp *output_coordinates, double *input_coordinates,
npy_intp output_rank, npy_intp input_rank, void *callback_data)
{
npy_intp i;
double shift = *(double *)callback_data;
for (i = 0; i < input_rank; i++) {
input_coordinates[i] = output_coordinates[i] - shift;
}
return 1;
}
static PyObject *
py_transform(PyObject *obj, PyObject *args)
{
double *callback_data = PyMem_Malloc(sizeof(double));
PyObject *capsule = NULL;
if (!callback_data) {
PyErr_NoMemory();
goto error;
}
if (!PyArg_ParseTuple(args, "d", callback_data)) goto error;
capsule = PyCapsule_New(_transform, NULL, _destructor);
if (!capsule) goto error;
if (PyCapsule_SetContext(capsule, callback_data) != 0) goto error;
return capsule;
error:
PyMem_Free(callback_data);
Py_XDECREF(capsule);
return NULL;
}
static PyMethodDef ExampleMethods[] = {
{"transform", (PyCFunction)py_transform, METH_VARARGS, ""},
{NULL, NULL, 0, NULL}
};
/* Initialize the module */
#if PY_VERSION_HEX >= 0x03000000
static struct PyModuleDef example = {
PyModuleDef_HEAD_INIT,
"example",
NULL,
-1,
ExampleMethods,
NULL,
NULL,
NULL,
NULL
};
PyMODINIT_FUNC
PyInit_example(void)
{
return PyModule_Create(&example);
}
#else
PyMODINIT_FUNC
initexample(void)
{
Py_InitModule("example", ExampleMethods);
}
#endif
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 with the following ``setup.py``,
__ https://docs.python.org/2/extending/extending.html
.. code:: python
from distutils.core import setup, Extension
import numpy
shift = Extension('example',
['example.c'],
include_dirs=[numpy.get_include()]
)
setup(name='example',
ext_modules=[shift]
)
and now running the script
.. code:: python
import numpy as np
from scipy import ndimage
from example import transform
im = np.arange(12).reshape(4, 3).astype(np.float64)
shift = 0.5
print(ndimage.geometric_transform(im, transform(shift)))
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 ``callback_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
:ctype:`PyCapsule`. The main steps are:
- Initialize a :ctype:`PyCapsule`. The first argument is a pointer to
the callback function and the third argument is a destructor
function which knows how to clean up memory associated with the
capsule's context pointer (see next step). The second argument is
the name of the capsule, but as :mod:`ndimage` has no way of knowing
that it is set to ``NULL``.
- Use :c:func:`PyCapsule_SetContext` to add any necessary data to the
capsule. This data is passed to the callback function through
``callback_data``, and in our example is used to set ``shift``.
C callback functions for :mod:`ndimage` all follow this scheme. The
next section lists the :mod:`ndimage` functions that accept a C
callback function and gives the prototype of the function.
Finally, the same code can be written in Cython with less
overhead as follows.
.. code:: cython
from cpython.mem cimport PyMem_Malloc, PyMem_Free
from cpython.pycapsule cimport (
PyCapsule_New, PyCapsule_SetContext, PyCapsule_GetContext
)
cimport numpy as np
from numpy cimport npy_intp as intp
cdef void _destructor(obj):
cdef void *callback_data = PyCapsule_GetContext(obj)
PyMem_Free(callback_data)
cdef int _transform(intp *output_coordinates, double *input_coordinates,
intp output_rank, intp input_rank, void *callback_data):
cdef intp i
cdef double shift = (<double *>callback_data)[0]
for i in range(input_rank):
input_coordinates[i] = output_coordinates[i] - shift
return 1
def transform(double shift):
cdef double *callback_data = <double *>PyMem_Malloc(sizeof(double))
if not callback_data:
raise MemoryError()
callback_data[0] = shift
try:
capsule = PyCapsule_New(<void *>_transform, NULL, _destructor)
PyCapsule_SetContext(capsule, callback_data)
except:
PyMem_Free(callback_data)
raise
return capsule
Functions that support C callback functions
-------------------------------------------
The :mod:`ndimage` functions that support C callback functions are
described here along with prototypes of the callback functions they
expect. All callback functions must be wrapped in a :ctype:`PyCapsule`
and accept a ``callback_data`` parameter which is set using
:c:func:`PyCapsule_SetContext`. The capsule's destructor must know how
to free memory associated with its context pointer. The callback
functions must return an integer error status that is zero if
something went wrong and one otherwise. If an error occurs, you should
normally set the python error status with an informative message
before returning, otherwise a default error message is set by the
calling function.
The function :func:`generic_filter` accepts a callback function with
the following prototype:
.. code:: c
int callback(double *buffer, npy_intp filter_size, double *res, void *callback_data)
The calling function iterates over the elements of the input and
output arrays, calling the callback function at each element. The
elements within the footprint of the filter at the current element are
passed through the ``buffer`` parameter, and the number of elements
within the footprint through ``filter_size``. The calculated value is
returned in ``return_value``.
The function :func:`generic_filter1d` accepts a callback function with
the following prototype:
.. code:: c
int callback(double *input_line, npy_intp input_length, double *output_line, npy_intp output_length, void *callback_data)
The calling function iterates over the lines of the input and output
arrays, calling the callback function at each line. The current line
is extended according to the border conditions set by the calling
function, and the result is copied into the array that is passed
through ``input_line``. The length of the input line (after extension)
is passed through ``input_length``. The callback function should apply
the filter and store the result in the array passed through
``output_line``. The length of the output line is passed through
``output_length``.
The function :func:`geometric_transform` expects a function with the
following prototype:
.. code:: c
int callback(npy_intp *output_coordinates, double *input_coordinates, npy_intp output_rank, npy_intp input_rank, void *callback_data)
The calling function iterates over the elements of the output array,
calling the callback function at each element. The coordinates of the
current output element are passed through ``output_coordinates``. The
callback function must return the coordinates at which the input must
be interpolated in ``input_coordinates``. The rank of the input and
output arrays are given by ``input_rank`` and ``output_rank``
respectively.
.. rubric:: 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
binary images in arbitrary dimensions. IEEE Trans. PAMI 25,
265-270, 2003.
.. [4] P. Felkel, R. Wegenkittl, and M. Bruckschwaiger,
"Implementation and Complexity of the Watershed-from-Markers Algorithm
Computed as a Minimal Cost Forest.", Eurographics 2001, pp. C:26-35.