# Multidimensional image processing (`scipy.ndimage`)¶

## 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.
`numpy` is suited very well for this type of applications due
its inherent multidimensional nature. The `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.

## Filter functions¶

The functions described in this section all perform some type of spatial
filtering of the 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:

```
>>> footprint = 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:

```
>>> 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:

```
>>> 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:

```
>>> 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:

```
>>> 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. 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

correlate1dfunction calculates a one-dimensional correlation along the given axis. The lines of the array along the given axis are correlated with the givenweights. Theweightsparameter must be a one-dimensional sequences of numbers.The function

correlateimplements multidimensional correlation of the input array with a given kernel.The

convolve1dfunction calculates a one-dimensional convolution along the given axis. The lines of the array along the given axis are convoluted with the givenweights. Theweightsparameter must be a one-dimensional sequences of numbers.Note

A convolution is essentially a correlation after mirroring the kernel. As a result, the

originparameter behaves differently than in the case of a correlation: the result is shifted in the opposite directions.The function

convolveimplements 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

originparameter behaves differently than in the case of a correlation: the results is shifted in the opposite direction.

### Smoothing filters¶

The

gaussian_filter1dfunction implements a one-dimensional Gaussian filter. The standard-deviation of the Gaussian filter is passed through the parametersigma. Settingorder= 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

gaussian_filterfunction implements a multidimensional Gaussian filter. The standard-deviations of the Gaussian filter along each axis are passed through the parametersigmaas a sequence or numbers. Ifsigmais 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. Theorderparameter 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

uniform_filter1dfunction calculates a one-dimensional uniform filter of the givensizealong the given axis.The

uniform_filterimplements a multidimensional uniform filter. The sizes of the uniform filter are given for each axis as a sequence of integers by thesizeparameter. Ifsizeis 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

minimum_filter1dfunction calculates a one-dimensional minimum filter of givensizealong the given axis.The

maximum_filter1dfunction calculates a one-dimensional maximum filter of givensizealong the given axis.The

minimum_filterfunction calculates a multidimensional minimum filter. Either the sizes of a rectangular kernel or the footprint of the kernel must be provided. Thesizeparameter, 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. Thefootprint, if provided, must be an array that defines the shape of the kernel by its non-zero elements.The

maximum_filterfunction calculates a multidimensional maximum filter. Either the sizes of a rectangular kernel or the footprint of the kernel must be provided. Thesizeparameter, 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. Thefootprint, if provided, must be an array that defines the shape of the kernel by its non-zero elements.The

rank_filterfunction calculates a multidimensional rank filter. Therankmay 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. Thesizeparameter, 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. Thefootprint, if provided, must be an array that defines the shape of the kernel by its non-zero elements.The

percentile_filterfunction calculates a multidimensional percentile filter. Thepercentilemay be less then zero, i.e.,percentile= -20 equalspercentile= 80. Either the sizes of a rectangular kernel or the footprint of the kernel must be provided. Thesizeparameter, 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. Thefootprint, if provided, must be an array that defines the shape of the kernel by its non-zero elements.The

median_filterfunction calculates a multidimensional median filter. Either the sizes of a rectangular kernel or the footprint of the kernel must be provided. Thesizeparameter, 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. Thefootprintif 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
`gaussian_filter1d` described in
*Smoothing filters* can be used to calculate
derivatives along a given axis using the *order* parameter. Other
derivative filters are the Prewitt and Sobel filters:

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 and to construct the Laplace filter:

The function

generic_laplacecalculates a laplace filter using the function passed throughderivative2to calculate second derivatives. The functionderivative2should have the following signature:derivative2(input, axis, output, mode, cval, *extra_arguments, **extra_keywords)It should calculate the second derivative along the dimension

axis. Ifoutputis not None it should use that for the output and return None, otherwise it should return the result.mode,cvalhave the usual meaning.The

extra_argumentsandextra_keywordsarguments can be used to pass a tuple of extra arguments and a dictionary of named arguments that are passed toderivative2at each call.For example:

>>> def d2(input, axis, output, mode, cval): ... return correlate1d(input, [1, -2, 1], axis, output, mode, cval, 0) ... >>> a = zeros((5, 5)) >>> a[2, 2] = 1 >>> 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_argumentsargument we could do:>>> def d2(input, axis, output, mode, cval, weights): ... return correlate1d(input, weights, axis, output, mode, cval, 0,) ... >>> a = 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:

>>> 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
`generic_laplace` by providing appropriate functions for the
second derivative function:

The function

laplacecalculates the Laplace using discrete differentiation for the second derivative (i.e. convolution with[1, -2, 1]).The function

gaussian_laplacecalculates the Laplace usinggaussian_filterto calculate the second derivatives. The standard-deviations of the Gaussian filter along each axis are passed through the parametersigmaas a sequence or numbers. Ifsigmais 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 `generic_gradient_magnitude`
function that calculated the gradient magnitude of an array:

The function

generic_gradient_magnitudecalculates a gradient magnitude using the function passed throughderivativeto calculate first derivatives. The functionderivativeshould have the following signature:derivative(input, axis, output, mode, cval, *extra_arguments, **extra_keywords)It should calculate the derivative along the dimension

axis. Ifoutputis not None it should use that for the output and return None, otherwise it should return the result.mode,cvalhave the usual meaning.The

extra_argumentsandextra_keywordsarguments can be used to pass a tuple of extra arguments and a dictionary of named arguments that are passed toderivativeat each call.For example, the

sobelfunction fits the required signature:>>> a = zeros((5, 5)) >>> a[2, 2] = 1 >>> 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

generic_laplacefor examples of using theextra_argumentsandextra_keywordsarguments.

The `sobel` and `prewitt` functions fit the required signature and
can therefore directly be used with `generic_gradient_magnitude`.
The following function implements the gradient magnitude using
Gaussian derivatives:

The functiongaussian_gradient_magnitudecalculates the gradient magnitude usinggaussian_filterto calculate the first derivatives. The standard-deviations of the Gaussian filter along each axis are passed through the parametersigmaas a sequence or numbers. Ifsigmais not a sequence but a single number, the standard deviation of the filter is equal along all directions.

### 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 `PyCObject` (see
*Extending ndimage in C* for more information).

The

generic_filter1dfunction implements a generic one-dimensional filter function, where the actual filtering operation must be supplied as a python function (or other callable object). Thegeneric_filter1dfunction iterates over the lines of an array and callsfunctionat each line. The arguments that are passed tofunctionare one-dimensional arrays of thetFloat64type. The first contains the values of the current line. It is extended at the beginning end the end, according to thefilter_sizeandoriginarguments. The second array should be modified in-place to provide the output values of the line. For example consider a correlation along one dimension:>>> a = 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

generic_filter1das follows:>>> def fnc(iline, oline): ... oline[...] = iline[:-2] + 2 * iline[1:-1] + 3 * iline[2:] ... >>> 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_argumentsandextra_keywordsarguments 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:>>> 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:

>>> 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

generic_filterfunction implements a generic filter function, where the actual filtering operation must be supplied as a python function (or other callable object). Thegeneric_filterfunction iterates over the array and callsfunctionat each element. The argument offunctionis a one-dimensional array of thetFloat64type, 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:>>> a = 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_filteras follows:>>> def fnc(buffer): ... return (buffer * array([1, 3])).sum() ... >>> 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

generic_filter, either the sizes of a rectangular kernel or the footprint of the kernel must be provided. Thesizeparameter, 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. Thefootprint, 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_argumentsandextra_keywordsarguments 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:>>> def fnc(buffer, weights): ... weights = 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:

>>> 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 `generic_filter`, but
additionally prints the current coordinates:

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

```
>>> a = 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 `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

fourier_shiftfunction multiplies the input array with the multidimensional Fourier transform of a shift operation for the given shift. Theshiftparameter is a sequences of shifts for each dimension, or a single value for all dimensions.The

fourier_gaussianfunction multiplies the input array with the multidimensional Fourier transform of a Gaussian filter with given standard-deviationssigma. Thesigmaparameter is a sequences of values for each dimension, or a single value for all dimensions.The

fourier_uniformfunction multiplies the input array with the multidimensional Fourier transform of a uniform filter with given sizessize. Thesizeparameter is a sequences of values for each dimension, or a single value for all dimensions.The

fourier_ellipsoidfunction multiplies the input array with the multidimensional Fourier transform of a elliptically shaped filter with given sizessize. Thesizeparameter 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.

## 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: M. Unser, “Splines: A Perfect Fit for Signal and Image Processing,” IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 22-38, November 1999.

### Spline pre-filters¶

Interpolation using
splines of an order larger than 1 requires a pre- filtering step.
The interpolation functions described in section
*Interpolation functions* apply pre-filtering by calling
`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

spline_filter1dfunction 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

spline_filterfunction 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 *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

geometric_transformfunction applies an arbitrary geometric transform to the input. The givenmappingfunction is called at each point in the output to find the corresponding coordinates in the input.mappingmust 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:

>>> a = arange(12).reshape(4,3).astype(np.float64) >>> def shift_func(output_coordinates): ... return (output_coordinates[0] - 0.5, output_coordinates[1] - 0.5) ... >>> 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_argumentsandextra_keywordsarguments 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:>>> 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:

>>> 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

PyCObject. SeeExtending ndimage in Cfor more information.The function

map_coordinatesapplies 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 parametercoordinatesis used to find for each point in the output the corresponding coordinates in the input. The values ofcoordinatesalong the first axis are the coordinates in the input array at which the output value is found. (See also the numarraycoordinatesfunction.) 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):>>> a = arange(12).reshape(4,3).astype(np.float64) >>> a array([[ 0., 1., 2.], [ 3., 4., 5.], [ 6., 7., 8.], [ 9., 10., 11.]]) >>> map_coordinates(a, [[0.5, 2], [0.5, 1]]) array([ 1.3625 7. ])The

affine_transformfunction applies an affine transformation to the input array. The given transformationmatrixandoffsetare 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 transformationmatrixmust 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

shiftfunction returns a shifted version of the input, using spline interpolation of the requestedorder.The

zoomfunction returns a rescaled version of the input, using spline interpolation of the requestedorder.The

rotatefunction returns the input array rotated in the plane defined by the two axes given by the parameteraxes, using spline interpolation of the requestedorder. The angle must be given in degrees. Ifreshapeis true, then the size of the output array is adapted to contain the rotated input.

## Morphology¶

### Binary morphology¶

Binary morphology (need something to put here).

The

generate_binary_structurefunctions generates a binary structuring element for use in binary morphology operations. Therankof 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 toconnectivity. For instance, two dimensional 4-connected and 8-connected structures are generated as follows:>>> 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

binary_erosionfunction 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 inFilter functions. If no structuring element is provided, an element with connectivity equal to one is generated usinggenerate_binary_structure. Theborder_valueparameter gives the value of the array outside boundaries. The erosion is repeatediterationstimes. Ifiterationsis less than one, the erosion is repeated until the result does not change anymore. If amaskarray is given, only those elements with a true value at the corresponding mask element are modified at each iteration.The

binary_dilationfunction 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 inFilter functions. If no structuring element is provided, an element with connectivity equal to one is generated usinggenerate_binary_structure. Theborder_valueparameter gives the value of the array outside boundaries. The dilation is repeatediterationstimes. Ifiterationsis less than one, the dilation is repeated until the result does not change anymore. If amaskarray 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

binary_dilationto find all elements that touch the border, by repeatedly dilating an empty array from the border using the data array as the mask:>>> struct = array([[0, 1, 0], [1, 1, 1], [0, 1, 0]]) >>> a = 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]]) >>> binary_dilation(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 `binary_erosion` and `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

iterate_structurefunction returns a structure by dilation of the input structureiteration- 1 times with itself. For instance:>>> struct = generate_binary_structure(2, 1) >>> struct array([[False, True, False], [ True, True, True], [False, True, False]], dtype=bool) >>> 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

iterationserosions or dilations must be achieved with the iterated structure. The adapted origin is simply obtained by multiplying with the number of iterations. For convenience theiterate_structurealso returns the adapted origin if theoriginparameter is not None:>>> 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. Following functions provide a few of these operations for convenience:

The

binary_openingfunction 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 inFilter functions. If no structuring element is provided, an element with connectivity equal to one is generated usinggenerate_binary_structure. Theiterationsparameter gives the number of erosions that is performed followed by the same number of dilations.The

binary_closingfunction 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 inFilter functions. If no structuring element is provided, an element with connectivity equal to one is generated usinggenerate_binary_structure. Theiterationsparameter gives the number of dilations that is performed followed by the same number of erosions.The

binary_fill_holesfunction 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 inFilter functions. If no structuring element is provided, an element with connectivity equal to one is generated usinggenerate_binary_structure.The

binary_hit_or_missfunction 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 logicalnotof the input with the second structure, followed by the logicalandof these two erosions. The origin parameters control the placement of the structuring elements as described inFilter functions. Iforigin2equals None it is set equal to theorigin1parameter. If the first structuring element is not provided, a structuring element with connectivity equal to one is generated usinggenerate_binary_structure, ifstructure2is not provided, it is set equal to the logicalnotofstructure1.

### 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
*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

grey_erosionfunction calculates a multidimensional grey- scale erosion.The

grey_dilationfunction calculates a multidimensional grey- scale dilation.

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

The

grey_openingfunction 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

grey_closingfunction 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

morphological_gradientfunction 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

morphological_laplacefunction 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

white_tophatfunction 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

black_tophatfunction implements a black top-hat filter of arrays of arbitrary rank. The black top-hat is equal to the difference of the a grey-scale closing and the input.

## 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

distance_transform_cdtuses 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 usinggenerate_binary_structurewith a squared distance equal to 1. If the structure is equal to ‘chessboard’, a structure is generated usinggenerate_binary_structurewith a squared distance equal to the rank of the array. These choices correspond to the common interpretations of the cityblock and the chessboard distancemetrics 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, andreturn_indicesflags can be used to indicate if the distance transform, the feature transform, or both must be returned.The

distancesandindicesarguments can be used to give optional output arrays that must be of the correct size and type (bothInt32).The basics of the algorithm used to implement this function is described in: G. Borgefors, “Distance transformations in arbitrary dimensions.”, Computer Vision, Graphics, and Image Processing, 27:321-345, 1984.

The function

distance_transform_edtcalculates 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, andreturn_indicesflags 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

samplingparameter 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

distancesandindicesarguments can be used to give optional output arrays that must be of the correct size and type (Float64andInt32).The algorithm used to implement this function is described in: 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.

The function

distance_transform_bfuses 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, andreturn_indicesflags 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

samplingparameter 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

distancesandindicesarguments can be used to give optional output arrays that must be of the correct size and type (Float64andInt32).Note

This function uses a slow brute-force algorithm, the function

distance_transform_cdtcan be used to more efficiently calculate cityblock and chessboard distance transforms. The functiondistance_transform_edtcan 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 `numpy` functions:

```
>>> a = 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]])
>>> 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 `label` generates
an array where each object is assigned a unique number:

The

labelfunction 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 theoutputparameter 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:>>> a = 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]] >>> 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:

>>> a = 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

generate_binary_structure(seeBinary 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:>>> l, n = label([1, 0, 1, 0, 1]) >>> l array([1 0 2 0 3]) >>> l = 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

labelis 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 `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

watershed_iftfunction applies a watershed from markers algorithm, using an Iterative Forest Transform, as described in: 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.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:

>>> input = 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 = 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) >>> 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:>>> markers = 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. Thereforewatershed_ifttreats 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:>>> markers = 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

generate_binary_structure(seeBinary 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:>>> 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

watershed_iftlimits the data types of the input toUInt8andUInt16.

## Object measurements¶

Given an array of labeled objects, the properties of the individual
objects can be measured. The `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

find_objectsfunction 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:>>> a = 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) >>> 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]])

find_objectsreturns slices for all objects, unless themax_labelparameter is larger then zero, in which case only the firstmax_labelobjects are returned. If an index is missing in thelabelarray, None is return instead of a slice. For example:>>> find_objects([1, 0, 3, 4], max_label = 3) [(slice(0, 1, None),), None, (slice(2, 3, None),)]

The list of slices generated by `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:

```
>>> image = arange(4 * 6).reshape(4, 6)
>>> mask = 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:

```
>>> 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:

```
>>> sum(image, labels, 2)
80
```

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

```
>>> 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:

```
>>> 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

sumfunction calculates the sum of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

meanfunction calculates the mean of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

variancefunction calculates the variance of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

standard_deviationfunction calculates the standard deviation of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

minimumfunction calculates the minimum of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

maximumfunction calculates the maximum of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

minimum_positionfunction calculates the position of the minimum of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

maximum_positionfunction calculates the position of the maximum of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

extremafunction calculates the minimum, the maximum, and their positions, of the elements of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation. The result is a tuple giving the minimum, the maximum, the position of the minimum and the postition of the maximum. The result is the same as a tuple formed by the results of the functionsminimum,maximum,minimum_position, andmaximum_positionthat are described above.The

center_of_massfunction calculates the center of mass of the of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare used in the calculation.The

histogramfunction calculates a histogram of the of the object with label(s) given byindex, using thelabelsarray for the object labels. Ifindexis None, all elements with a non-zero label value are treated as a single object. Iflabelis None, all elements ofinputare 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 typeInt32.

## Extending `ndimage` in C¶

A few functions in the `scipy.ndimage` take a call-back
argument. This can be a python function, but also a `PyCObject`
containing a pointer to a C function. To use this feature, you must
write your own C extension that defines the function, and define a Python function that returns a `PyCObject` containing a pointer to this function.

An example of a function that supports this is
`geometric_transform` (see *Interpolation functions*).
You can pass it a python callable object that defines a mapping
from all output coordinates to corresponding coordinates in the
input array. This mapping function can also be a C function, which
generally will be much more efficient, since the overhead of
calling a python function at each element is avoided.

For example to implement a simple shift function we define the following function:

```
static int
_shift_function(int *output_coordinates, double* input_coordinates,
int output_rank, int input_rank, void *callback_data)
{
int ii;
/* get the shift from the callback data pointer: */
double shift = *(double*)callback_data;
/* calculate the coordinates: */
for(ii = 0; ii < irank; ii++)
icoor[ii] = ocoor[ii] - shift;
/* return OK status: */
return 1;
}
```

This function is called at every element of the output array,
passing the current coordinates in the *output_coordinates* array.
On return, the *input_coordinates* array must contain the
coordinates at which the input is interpolated. The ranks of the
input and output array are passed through *output_rank* and
*input_rank*. The value of the shift is passed through the
*callback_data* argument, which is a pointer to void. The function
returns an error status, in this case always 1, since no error can
occur.

A pointer to this function and a pointer to the shift value must be
passed to `geometric_transform`. Both are passed by a single
`PyCObject` which is created by the following python extension
function:

```
static PyObject *
py_shift_function(PyObject *obj, PyObject *args)
{
double shift = 0.0;
if (!PyArg_ParseTuple(args, "d", &shift)) {
PyErr_SetString(PyExc_RuntimeError, "invalid parameters");
return NULL;
} else {
/* assign the shift to a dynamically allocated location: */
double *cdata = (double*)malloc(sizeof(double));
*cdata = shift;
/* wrap function and callback_data in a CObject: */
return PyCObject_FromVoidPtrAndDesc(_shift_function, cdata,
_destructor);
}
}
```

The value of the shift is obtained and then assigned to a
dynamically allocated memory location. Both this data pointer and
the function pointer are then wrapped in a `PyCObject`, which is
returned. Additionally, a pointer to a destructor function is
given, that will free the memory we allocated for the shift value
when the `PyCObject` is destroyed. This destructor is very simple:

```
static void
_destructor(void* cobject, void *cdata)
{
if (cdata)
free(cdata);
}
```

To use these functions, an extension module is built:

```
static PyMethodDef methods[] = {
{"shift_function", (PyCFunction)py_shift_function, METH_VARARGS, ""},
{NULL, NULL, 0, NULL}
};
void
initexample(void)
{
Py_InitModule("example", methods);
}
```

This extension can then be used in Python, for example:

```
>>> import example
>>> array = arange(12).reshape=(4, 3).astype(np.float64)
>>> fnc = example.shift_function(0.5)
>>> geometric_transform(array, fnc)
array([[ 0. 0. 0. ],
[ 0. 1.3625 2.7375],
[ 0. 4.8125 6.1875],
[ 0. 8.2625 9.6375]])
```

C callback functions for use with `ndimage` functions must all
be written according to this scheme. The next section lists the
`ndimage` functions that acccept a C callback function and
gives the prototype of the callback function.

## Functions that support C callback functions¶

The `ndimage` functions that support C callback functions are
described here. Obviously, the prototype of the function that is
provided to these functions must match exactly that what they
expect. Therefore we give here the prototypes of the callback
functions. All these callback functions accept a void
*callback_data* pointer that must be wrapped in a `PyCObject` using
the Python `PyCObject_FromVoidPtrAndDesc` function, which can also
accept a pointer to a destructor function to free any memory
allocated for *callback_data*. If *callback_data* is not needed,
`PyCObject_FromVoidPtr` may be used instead. The callback
functions must return an integer error status that is equal to zero
if something went wrong, or 1 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 `generic_filter` (see
*Generic filter functions*) accepts a callback function with the
following prototype:

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 thebufferparameter, and the number of elements within the footprint throughfilter_size. The calculated valued should be returned in thereturn_valueargument.

The function `generic_filter1d` (see
*Generic filter functions*) accepts a callback function with the
following prototype:

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 theinput_linearray. The length of the input line (after extension) is passed throughinput_length. The callback function should apply the 1D filter and store the result in the array passed throughoutput_line. The length of the output line is passed throughoutput_length.

The function `geometric_transform` (see
*Interpolation functions*) expects a function with the following
prototype:

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 throughoutput_coordinates. The callback function must return the coordinates at which the input must be interpolated ininput_coordinates. The rank of the input and output arrays are given byinput_rankandoutput_rankrespectively.