# Tools: the Signal class¶

## The Signal class and its subclasses¶

Warning

This subsection can be a bit confusing for beginners. Do not worry if you do not understand it all.

HyperSpy stores the data in the `BaseSignal`

class, that is
the object that you get when e.g. you load a single file using
`load()`

. Most of the data analysis functions are also contained in
this class or its specialized subclasses. The `BaseSignal`

class contains general functionality that is available to all the subclasses.
The subclasses provide functionality that is normally specific to a particular
type of data, e.g. the `Signal1D`

class provides
common functionality to deal with one-dimensional (e.g. spectral) data and
`EELSSpectrum`

(which is a subclass of
`Signal1D`

) adds extra functionality to the
`Signal1D`

class for electron energy-loss
spectroscopy data analysis.

The table below summarises all the
currently available specialised `BaseSignal`

subclasses.

Changed in version 1.0: The `Signal1D`

,
`Signal2D`

and `BaseSignal`

classes deprecated the old Spectrum Image and Signal classes.

New in version 1.0: New `ComplexSignal`

,
`ComplexSignal1D`

and
`Complex2D`

`BaseSignal`

subclasses specialised in complex data.

The `signals`

module, which contains all available signal subclasses,
is imported in the user namespace when loading HyperSpy. In the following
example we create a Signal2D instance from a 2D numpy array:

```
>>> im = hs.signals.Signal2D(np.random.random((64,64)))
>>> im
<Signal2D, title: , dimensions: (|64, 64)>
```

The different signals store other objects in what are called attributes. For
examples, the data is stored in a numpy array in the
`data`

attribute, the original parameters in the
`original_metadata`

attribute, the mapped parameters
in the `metadata`

attribute and the axes
information (including calibration) can be accessed (and modified) in the
`AxesManager`

attribute.

## Signal initialization¶

Many of the values in the `AxesManager`

can be
set when making the `BaseSignal`

object.

```
>>> dict0 = {'size': 10, 'name':'Axis0', 'units':'A', 'scale':0.2, 'offset':1}
>>> s = hs.signals.BaseSignal(np.random.random((10,20)), axes=[dict0, dict1])
>>> s.axes_manager
<Axes manager, axes: (|20, 10)>
Name | size | index | offset | scale | units
================ | ====== | ====== | ======= | ======= | ======
---------------- | ------ | ------ | ------- | ------- | ------
Axis1 | 20 | | 2 | 0.1 | B
Axis0 | 10 | | 1 | 0.2 | A
```

This also applies to the `metadata`

.

```
>>> metadata_dict = {'General':{'name':'A BaseSignal'}}
>>> metadata_dict['General']['title'] = 'A BaseSignal title'
>>> s = hs.signals.BaseSignal(np.arange(10), metadata=metadata_dict)
>>> s.metadata
├── General
│ ├── name = A BaseSignal
│ └── title = A BaseSignal title
└── Signal
├── binned = False
└── signal_type =
```

## Binned and unbinned signals¶

New in version 0.7.

Signals that are a histogram of a probability density function (pdf) should
have the `signal.metadata.Signal.binned`

attribute set to
`True`

. This is because some methods operate differently in signals that are
*binned*.

Changed in version 1.0: `Simulation`

, `SpectrumSimulation`

and `ImageSimulation`

classes removed.

The default value of the `binned`

attribute is shown in the
following table:

BaseSignal subclass | binned |
---|---|

`BaseSignal` |
False |

`Signal1D` |
False |

`EELSSpectrum` |
True |

`EDSSEMSpectrum` |
True |

`EDSTEM` |
True |

`Signal2D` |
False |

`ComplexSignal` |
False |

`ComplexSignal1D` |
False |

`Complex2Dmixin` |
False |

To change the default value:

```
>>> s.metadata.Signal.binned = True
```

## Generic tools¶

Below we briefly introduce some of the most commonly used tools (methods). For
more details about a particular method click on its name. For a detailed list
of all the methods available see the `BaseSignal`

documentation.

The methods of this section are available to all the signals. In other chapters methods that are only available in specialized subclasses.

### Mathematical operations¶

Changed in version 1.0.

A number of mathematical operations are available
in `BaseSignal`

. Most of them are just wrapped numpy
functions.

The methods that perform mathematical operation over one or more axis at a time are:

`sum()`

`max()`

`min()`

`mean()`

`std()`

`var()`

`nansum()`

`nanmax()`

`nanmin()`

`nanmean()`

`nanstd()`

`nanvar()`

Note that by default all this methods perform the operation over *all*
navigation axes.

Example:

```
>>> s = hs.signals.BaseSignal(np.random.random((2,4,6)))
>>> s.axes_manager[0].name = 'E'
>>> s
<BaseSignal, title: , dimensions: (|6, 4, 2)>
>>> # by default perform operation over all navigation axes
>>> s.sum()
<BaseSignal, title: , dimensions: (|6, 4, 2)>
>>> # can also pass axes individually
>>> s.sum('E')
<Signal2D, title: , dimensions: (|4, 2)>
>>> # or a tuple of axes to operate on, with duplicates, by index or directly
>>> ans = s.sum((-1, s.axes_manager[1], 'E', 0))
>>> ans
<BaseSignal, title: , dimensions: (|1)>
>>> ans.axes_manager[0]
<Scalar axis, size: 1>
```

The following methods operate only on one axis at a time:

New in version 1.2: `valuemin()`

, `indexmin()`

`diff()`

`derivative()`

`integrate_simpson()`

`integrate1D()`

`valuemax()`

`indexmax()`

`valuemin()`

`indexmin()`

New in version 1.0: numpy ufunc operate on HyperSpy signals

All numpy ufunc can operate on `BaseSignal`

instances, for example:

```
>>> s = hs.signals.Signal1D([0, 1])
>>> s.metadata.General.title = "A"
>>> s
<Signal1D, title: A, dimensions: (|2)>
>>> np.exp(s)
<Signal1D, title: exp(A), dimensions: (|2)>
>>> np.exp(s).data
array([ 1. , 2.71828183])
>>> np.power(s, 2)
<Signal1D, title: power(A, 2), dimensions: (|2)>
>>> np.add(s, s)
<Signal1D, title: add(A, A), dimensions: (|2)>
>>> np.add(hs.signals.Signal1D([0, 1]), hs.signals.Signal1D([0, 1]))
<Signal1D, title: add(Untitled Signal 1, Untitled Signal 2), dimensions: (|2)>
```

Notice that the title is automatically updated. When the signal has no title a new title is automatically generated:

```
>>> np.add(hs.signals.Signal1D([0, 1]), hs.signals.Signal1D([0, 1]))
<Signal1D, title: add(Untitled Signal 1, Untitled Signal 2), dimensions: (|2)>
```

Functions (other than unfucs) that operate on numpy arrays can also operate
on `BaseSignal`

instances, however they return a numpy
array instead of a `BaseSignal`

instance e.g.:

```
>>> np.angle(s)
array([ 0., 0.])
```

### Indexing¶

New in version 0.6.

Changed in version 0.8.1.

Indexing a `BaseSignal`

provides a powerful, convenient and
Pythonic way to access and modify its data. In HyperSpy indexing is achieved
using `isig`

and `inav`

, which allow the navigation and signal dimensions
to be indexed independently. The idea is essentially to specify a subset of the
data based on its position in the array and it is therefore essential to know
the convention adopted for specifying that position, which is described here.

Those new to Python may find indexing a somewhat esoteric concept but once mastered it is one of the most powerful features of Python based code and greatly simplifies many common tasks. HyperSpy’s Signal indexing is similar to numpy array indexing and those new to Python are encouraged to read the associated numpy documentation on the subject.

Key features of indexing in HyperSpy are as follows (note that some of these features differ from numpy):

- HyperSpy indexing does:
- Allow independent indexing of signal and navigation dimensions
- Support indexing with decimal numbers.
- Use the image order for indexing i.e. [x, y, z,…] (HyperSpy) vs […,z,y,x] (numpy)

- HyperSpy indexing does not:
- Support indexing using arrays.
- Allow the addition of new axes using the newaxis object.

The examples below illustrate a range of common indexing tasks.

First consider indexing a single spectrum, which has only one signal dimension
(and no navigation dimensions) so we use `isig`

:

```
>>> s = hs.signals.Signal1D(np.arange(10))
>>> s
<Signal1D, title: , dimensions: (|10)>
>>> s.data
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> s.isig[0]
<Signal1D, title: , dimensions: (|1)>
>>> s.isig[0].data
array([0])
>>> s.isig[9].data
array([9])
>>> s.isig[-1].data
array([9])
>>> s.isig[:5]
<Signal1D, title: , dimensions: (|5)>
>>> s.isig[:5].data
array([0, 1, 2, 3, 4])
>>> s.isig[5::-1]
<Signal1D, title: , dimensions: (|6)>
>>> s.isig[5::-1]
<Signal1D, title: , dimensions: (|6)>
>>> s.isig[5::2]
<Signal1D, title: , dimensions: (|3)>
>>> s.isig[5::2].data
array([5, 7, 9])
```

Unlike numpy, HyperSpy supports indexing using decimal numbers, in which case HyperSpy indexes using the axis scales instead of the indices.

```
>>> s = hs.signals.Signal1D(np.arange(10))
>>> s
<Signal1D, title: , dimensions: (|10)>
>>> s.data
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> s.axes_manager[0].scale = 0.5
>>> s.axes_manager[0].axis
array([ 0. , 0.5, 1. , 1.5, 2. , 2.5, 3. , 3.5, 4. , 4.5])
>>> s.isig[0.5:4.].data
array([1, 2, 3, 4, 5, 6, 7])
>>> s.isig[0.5:4].data
array([1, 2, 3])
>>> s.isig[0.5:4:2].data
array([1, 3])
```

Importantly the original `BaseSignal`

and its “indexed self”
share their data and, therefore, modifying the value of the data in one
modifies the same value in the other. Note also that in the example below
s.data is used to access the data as a numpy array directly and this array is
then indexed using numpy indexing.

```
>>> s = hs.signals.Signal1D(np.arange(10))
>>> s
<Signal1D, title: , dimensions: (10,)>
>>> s.data
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> si = s.isig[::2]
>>> si.data
array([0, 2, 4, 6, 8])
>>> si.data[:] = 10
>>> si.data
array([10, 10, 10, 10, 10])
>>> s.data
array([10, 1, 10, 3, 10, 5, 10, 7, 10, 9])
>>> s.data[:] = 0
>>> si.data
array([0, 0, 0, 0, 0])
```

Of course it is also possible to use the same syntax to index multidimensional
data treating navigation axes using `inav`

and signal axes using `isig`

.

```
>>> s = hs.signals.Signal1D(np.arange(2*3*4).reshape((2,3,4)))
>>> s
<Signal1D, title: , dimensions: (3, 2|4)>
>>> s.data
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
>>> s.axes_manager[0].name = 'x'
>>> s.axes_manager[1].name = 'y'
>>> s.axes_manager[2].name = 't'
>>> s.axes_manager.signal_axes
(<t axis, size: 4>,)
>>> s.axes_manager.navigation_axes
(<x axis, size: 3, index: 0>, <y axis, size: 2, index: 0>)
>>> s.inav[0,0].data
array([0, 1, 2, 3])
>>> s.inav[0,0].axes_manager
<Axes manager, axes: (|4)>
Name | size | index | offset | scale | units
================ | ====== | ====== | ======= | ======= | ======
---------------- | ------ | ------ | ------- | ------- | ------
t | 4 | | 0 | 1 | <undefined>
>>> s.inav[0,0].isig[::-1].data
array([3, 2, 1, 0])
>>> s.isig[0]
<BaseSignal, title: , dimensions: (3, 2)>
>>> s.isig[0].axes_manager
<Axes manager, axes: (3, 2|)>
Name | size | index | offset | scale | units
================ | ====== | ====== | ======= | ======= | ======
x | 3 | 0 | 0 | 1 | <undefined>
y | 2 | 0 | 0 | 1 | <undefined>
---------------- | ------ | ------ | ------- | ------- | ------
>>> s.isig[0].data
array([[ 0, 4, 8],
[12, 16, 20]])
```

Independent indexation of the signal and navigation dimensions is demonstrated further in the following:

```
>>> s = hs.signals.Signal1D(np.arange(2*3*4).reshape((2,3,4)))
>>> s
<Signal1D, title: , dimensions: (3, 2|4)>
>>> s.data
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
>>> s.axes_manager[0].name = 'x'
>>> s.axes_manager[1].name = 'y'
>>> s.axes_manager[2].name = 't'
>>> s.axes_manager.signal_axes
(<t axis, size: 4>,)
>>> s.axes_manager.navigation_axes
(<x axis, size: 3, index: 0>, <y axis, size: 2, index: 0>)
>>> s.inav[0,0].data
array([0, 1, 2, 3])
>>> s.inav[0,0].axes_manager
<Axes manager, axes: (|4)>
Name | size | index | offset | scale | units
================ | ====== | ====== | ======= | ======= | ======
---------------- | ------ | ------ | ------- | ------- | ------
t | 4 | | 0 | 1 | <undefined>
>>> s.isig[0]
<BaseSignal, title: , dimensions: (2, 3)>
>>> s.isig[0].axes_manager
<Axes manager, axes: (3, 2|)>
Name | size | index | offset | scale | units
================ | ====== | ====== | ======= | ======= | ======
x | 3 | 0 | 0 | 1 | <undefined>
y | 2 | 0 | 0 | 1 | <undefined>
---------------- | ------ | ------ | ------- | ------- | ------
>>> s.isig[0].data
array([[ 0, 4, 8],
[12, 16, 20]])
```

The same syntax can be used to set the data values in signal and navigation dimensions respectively:

```
>>> s = hs.signals.Signal1D(np.arange(2*3*4).reshape((2,3,4)))
>>> s
<Signal1D, title: , dimensions: (3, 2|4)>
>>> s.data
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
>>> s.inav[0,0].data
array([0, 1, 2, 3])
>>> s.inav[0,0] = 1
>>> s.inav[0,0].data
array([1, 1, 1, 1])
>>> s.inav[0,0] = s.inav[1,1]
>>> s.inav[0,0].data
array([16, 17, 18, 19])
```

### Signal operations¶

New in version 0.6.

New in version 0.8.3.

`BaseSignal`

supports all the Python binary arithmetic
operations (+, -, *, //, %, divmod(), pow(), **, <<, >>, &, ^, |),
augmented binary assignments (+=, -=, *=, /=, //=, %=, **=, <<=, >>=, &=,
^=, |=), unary operations (-, +, abs() and ~) and rich comparisons operations
(<, <=, ==, x!=y, <>, >, >=).

These operations are performed element-wise. When the dimensions of the signals are not equal numpy broadcasting rules apply independently for the navigation and signal axes.

Warning

Hyperspy does not check if the calibration of the signals matches.

In the following example s2 has only one navigation axis while s has two. However, because the size of their first navigation axis is the same, their dimensions are compatible and s2 is broadcasted to match s’s dimensions.

```
>>> s = hs.signals.Signal2D(np.ones((3,2,5,4)))
>>> s2 = hs.signals.Signal2D(np.ones((2,5,4)))
>>> s
<Signal2D, title: , dimensions: (2, 3|4, 5)>
>>> s2
<Signal2D, title: , dimensions: (2|4, 5)>
>>> s + s2
<Signal2D, title: , dimensions: (2, 3|4, 5)>
```

In the following example the dimensions are not compatible and an exception is raised.

```
>>> s = hs.signals.Signal2D(np.ones((3,2,5,4)))
>>> s2 = hs.signals.Signal2D(np.ones((3,5,4)))
>>> s
<Signal2D, title: , dimensions: (2, 3|4, 5)>
>>> s2
<Signal2D, title: , dimensions: (3|4, 5)>
>>> s + s2
Traceback (most recent call last):
File "<ipython-input-55-044bb11a0bd9>", line 1, in <module>
s + s2
File "<string>", line 2, in __add__
File "/home/fjd29/Python/hyperspy/hyperspy/signal.py", line 2686, in _binary_operator_ruler
raise ValueError(exception_message)
ValueError: Invalid dimensions for this operation
```

Broadcasting operates exactly in the same way for the signal axes:

```
>>> s = hs.signals.Signal2D(np.ones((3,2,5,4)))
>>> s2 = hs.signals.Signal1D(np.ones((3, 2, 4)))
>>> s
<Signal2D, title: , dimensions: (2, 3|4, 5)>
>>> s2
<Signal1D, title: , dimensions: (2, 3|4)>
>>> s + s2
<Signal2D, title: , dimensions: (2, 3|4, 5)>
```

In-place operators also support broadcasting, but only when broadcasting would not change the left most signal dimensions:

```
>>> s += s2
>>> s
<Signal2D, title: , dimensions: (2, 3|4, 5)>
>>> s2 += s
Traceback (most recent call last):
File "<ipython-input-64-fdb9d3a69771>", line 1, in <module>
s2 += s
File "<string>", line 2, in __iadd__
File "/home/fjd29/Python/hyperspy/hyperspy/signal.py", line 2737, in _binary_operator_ruler
self.data = getattr(sdata, op_name)(odata)
ValueError: non-broadcastable output operand with shape (3,2,1,4) doesn\'t match the broadcast shape (3,2,5,4)
```

### Iterating external functions with the map method¶

New in version 0.7.

Performing an operation on the data at each coordinate, as in the previous example,
using an external function can be more easily accomplished using the
`map()`

method:

```
>>> import scipy.ndimage
>>> image_stack = hs.signals.Signal2D(np.array([scipy.misc.ascent()]*4))
>>> image_stack.axes_manager[1].name = "x"
>>> image_stack.axes_manager[2].name = "y"
>>> image_stack.map(scipy.ndimage.rotate,
... angle=45,
... reshape=False)
>>> # clip data to integer range
>>> image_stack.data = np.clip(image_stack.data, 0, 255)
>>> collage = hs.stack([image for image in image_stack], axis=0)
>>> collage.plot()
```

The `map()`

method can also take variable
arguments as in the following example.

```
>>> import scipy.ndimage
>>> image_stack = hs.signals.Signal2D(np.array([scipy.misc.ascent()]*4))
>>> image_stack.axes_manager[1].name = "x"
>>> image_stack.axes_manager[2].name = "y"
>>> angles = hs.signals.BaseSignal(np.array([0, 45, 90, 135]))
>>> image_stack.map(scipy.ndimage.rotate,
... angle=angles.T,
... reshape=False)
```

New in version 1.2.0: `inplace`

keyword and non-preserved output shapes

If all function calls do not return identically-shaped results, only navigation information is preserved, and the final result is an array where each element corresponds to the result of the function (or arbitrary object type). As such, most HyperSpy functions cannot operate on such Signal, and the data should be accessed directly.

`inplace`

keyword (by default `True`

) of the
`map()`

method allows either overwriting the current
data (default, `True`

) or storing it to a new signal (`False`

).

```
>>> import scipy.ndimage
>>> image_stack = hs.signals.Signal2D(np.array([scipy.misc.ascent()]*4))
>>> angles = hs.signals.BaseSignal(np.array([0, 45, 90, 135]))
>>> result = image_stack.map(scipy.ndimage.rotate,
... angle=angles.T,
... inplace=False,
... reshape=True)
100%|████████████████████████████████████████████| 4/4 [00:00<00:00, 18.42it/s]
>>> result
<BaseSignal, title: , dimensions: (4|)>
>>> image_stack.data.dtype
dtype('O')
>>> for d in result.data.flat:
... print(d.shape)
(512, 512)
(724, 724)
(512, 512)
(724, 724)
```

New in version 1.2.0: `parallel`

keyword.

The execution can be sped up by passing `parallel`

keyword to the
`map()`

method:

```
>>> import time
>>> def slow_func(data):
... time.sleep(1.)
... return data + 1
>>> s = hs.signals.Signal1D(np.arange(20).reshape((20,1)))
>>> s
<Signal1D, title: , dimensions: (20|1)>
>>> s.map(slow_func, parallel=False)
100%|██████████████████████████████████████| 20/20 [00:20<00:00, 1.00s/it]
>>> # some operations will be done in parallel:
>>> s.map(slow_func, parallel=True)
100%|██████████████████████████████████████| 20/20 [00:02<00:00, 6.73it/s]
```

### Cropping¶

Cropping can be performed in a very compact and powerful way using
Indexing . In addition it can be performed using the following
method or GUIs if cropping signal1D or signal2D. There is also a general `crop()`

method that operates *in place*.

### Rebinning¶

New in version 1.3: `rebin()`

generalized to remove the constrain
of the `new_shape`

needing to be a divisor of `data.shape`

.

The `rebin()`

methods supports rebinning the data to
arbitrary new shapes as long as the number of dimensions stays the same.
However, internally, it uses two different algorithms to perform the task. Only
when the new shape dimensions are divisors of the old shape’s, the operation
supports lazy-evaluation and is usually faster.
Otherwise, the operation requires linear interpolation and is generally slower if
Numba is not installed.

For example, the following two equivalent rebinning operations can be performed lazily:

```
>>> s = hs.datasets.example_signals.EDS_SEM_Spectrum().as_lazy()
>>> print(s)
<LazyEDSSEMSpectrum, title: EDS SEM Spectrum, dimensions: (|1024)>
>>> print(s.rebin(scale=[2]))
<LazyEDSSEMSpectrum, title: EDS SEM Spectrum, dimensions: (|512)>
```

```
>>> s = hs.datasets.example_signals.EDS_SEM_Spectrum().as_lazy()
>>> print(s.rebin(new_shape=[512]))
<LazyEDSSEMSpectrum, title: EDS SEM Spectrum, dimensions: (|512)>
```

On the other hand, the following rebining operation requires interpolation and cannot be performed lazily:

```
>>> spectrum = hs.signals.EDSTEMSpectrum(np.ones([4, 4, 10]))
>>> spectrum.data[1, 2, 9] = 5
>>> print(spectrum)
<EDSTEMSpectrum, title: , dimensions: (4, 4|10)>
>>> print ('Sum = ', spectrum.data.sum())
Sum = 164.0
>>> scale = [0.5, 0.5, 5]
>>> test = spectrum.rebin(scale=scale)
>>> test2 = spectrum.rebin(new_shape=(8, 8, 2)) # Equivalent to the above
>>> print(test)
<EDSTEMSpectrum, title: , dimensions: (8, 8|2)>
>>> print(test2)
<EDSTEMSpectrum, title: , dimensions: (8, 8|2)>
>>> print('Sum =', test.data.sum())
Sum = 164.0
>>> print('Sum =', test2.data.sum())
Sum = 164.0
>>> spectrum.as_lazy().rebin(scale=scale)
Traceback (most recent call last):
File "<ipython-input-26-49bca19ebf34>", line 1, in <module>
spectrum.as_lazy().rebin(scale=scale)
File "/home/fjd29/Python/hyperspy3/hyperspy/_signals/eds.py", line 184, in rebin
m = super().rebin(new_shape=new_shape, scale=scale, crop=crop, out=out)
File "/home/fjd29/Python/hyperspy3/hyperspy/_signals/lazy.py", line 246, in rebin
"Lazy rebin requires scale to be integer and divisor of the "
NotImplementedError: Lazy rebin requires scale to be integer and divisor of the original signal shape
```

### Folding and unfolding¶

When dealing with multidimensional datasets it is sometimes useful to transform the data into a two dimensional dataset. This can be accomplished using the following two methods:

`fold()`

`unfold()`

It is also possible to unfold only the navigation or only the signal space:

`unfold_navigation_space()`

`unfold_signal_space()`

### Splitting and stacking¶

Several objects can be stacked together over an existing axis or over a
new axis using the `stack()`

function, if they share axis
with same dimension.

```
>>> image = hs.signals.Signal2D(scipy.misc.ascent())
>>> image = hs.stack([hs.stack([image]*3,axis=0)]*3,axis=1)
>>> image.plot()
```

An object can be split into several objects
with the `split()`

method. This function can be used
to reverse the `stack()`

function:

```
>>> image = image.split()[0].split()[0]
>>> image.plot()
```

### FFT and iFFT¶

The Fast Fourier transform and its inverse can be applied on a signal with the `fft()`

and the `ifft()`

methods.

```
>>> import numpy as np
>>> im = hs.datasets.example_signals.object_hologram()
>>> np.log(im.fft(shifted=True).amplitude).plot()
```

- Note that for visual inspection of FFT it is common to plot logarithm of amplitude rather than FFT itself as it is done
- in the example above.

By default both methods calculate FFT and IFFT with origin at (0, 0) (not in the centre of FFT). Use shifted=True option to calculate FFT and the inverse with origin shifted in the centre.

```
>>> im_ifft = im.fft(shifted=True).ifft(shifted=True)
```

### Changing the data type¶

Even if the original data is recorded with a limited dynamic range, it is often
desirable to perform the analysis operations with a higher precision.
Conversely, if space is limited, storing in a shorter data type can decrease
the file size. The `change_dtype()`

changes the data
type in place, e.g.:

```
>>> s = hs.load('EELS Signal1D Signal2D (high-loss).dm3')
Title: EELS Signal1D Signal2D (high-loss).dm3
Signal type: EELS
Data dimensions: (21, 42, 2048)
Data representation: spectrum
Data type: float32
>>> s.change_dtype('float64')
>>> print(s)
Title: EELS Signal1D Signal2D (high-loss).dm3
Signal type: EELS
Data dimensions: (21, 42, 2048)
Data representation: spectrum
Data type: float64
```

New in version 0.7: Support for RGB signals.

In addition to all standard numpy dtypes, HyperSpy supports four extra dtypes
for RGB images **for visualization purposes only**: `rgb8`

, `rgba8`

,
`rgb16`

and `rgba16`

. This includes of course multi-dimensional RGB images.

The requirements for changing from and to any `rgbx`

dtype are more strict
than for most other dtype conversions. To change to a `rgbx`

dtype the
`signal_dimension`

must be 1 and its size 3 (4) 3(4) for `rgb`

(or
`rgba`

) dtypes and the dtype must be `uint8`

(`uint16`

) for
`rgbx8`

(`rgbx16`

). After conversion the `signal_dimension`

becomes 2.

Most operations on signals with RGB dtypes will fail. For processing simply
change their dtype to `uint8`

(`uint16`

).The dtype of images of
dtype `rgbx8`

(`rgbx16`

) can only be changed to `uint8`

(`uint16`

) and
the `signal_dimension`

becomes 1.

In the following example we create a 1D signal with signal size 3 and with
dtype `uint16`

and change its dtype to `rgb16`

for plotting.

```
>>> rgb_test = np.zeros((1024, 1024, 3))
>>> ly, lx = rgb_test.shape[:2]
>>> offset_factor = 0.16
>>> size_factor = 3
>>> Y, X = np.ogrid[0:lx, 0:ly]
>>> rgb_test[:,:,0] = (X - lx / 2 - lx*offset_factor) ** 2 + \
... (Y - ly / 2 - ly*offset_factor) ** 2 < \
... lx * ly / size_factor **2
>>> rgb_test[:,:,1] = (X - lx / 2 + lx*offset_factor) ** 2 + \
... (Y - ly / 2 - ly*offset_factor) ** 2 < \
... lx * ly / size_factor **2
>>> rgb_test[:,:,2] = (X - lx / 2) ** 2 + \
... (Y - ly / 2 + ly*offset_factor) ** 2 \
... < lx * ly / size_factor **2
>>> rgb_test *= 2**16 - 1
>>> s = hs.signals.Signal1D(rgb_test)
>>> s.change_dtype("uint16")
>>> s
<Signal1D, title: , dimensions: (1024, 1024|3)>
>>> s.change_dtype("rgb16")
>>> s
<Signal2D, title: , dimensions: (|1024, 1024)>
>>> s.plot()
```

### Transposing (changing signal spaces)¶

New in version 1.1.

`transpose()`

method changes how the dataset
dimensions are interpreted (as signal or navigation axes). By default is
swaps the signal and navigation axes. For example:

```
>>> s = hs.signals.Signal1D(np.zeros((4,5,6)))
>>> s
<Signal1D, title: , dimensions: (5, 4|6)>
>>> s.transpose()
<Signal2D, title: , dimensions: (6|4, 5)>
```

For `T()`

is a shortcut for the default behaviour:

```
>>> s = hs.signals.Signal1D(np.zeros((4,5,6))).T
<Signal2D, title: , dimensions: (6|4, 5)>
```

The method accepts both explicit axes to keep in either space, or just a number
of axes required in one space (just one number can be specified, as the other
is defined as “all other axes”). When axes order is not explicitly defined,
they are “rolled” from one space to the other as if the ```
<navigation axes |
signal axes >
```

wrap a circle. The example below should help clarifying this.

```
>>> # just create a signal with many distinct dimensions
>>> s = hs.signals.BaseSignal(np.random.rand(1,2,3,4,5,6,7,8,9))
>>> s
<BaseSignal, title: , dimensions: (|9, 8, 7, 6, 5, 4, 3, 2, 1)>
>>> s.transpose(signal_axes=5) # roll to leave 5 axes in signal space
<BaseSignal, title: , dimensions: (4, 3, 2, 1|9, 8, 7, 6, 5)>
>>> s.transpose(navigation_axes=3) # roll leave 3 axes in navigation space
<BaseSignal, title: , dimensions: (3, 2, 1|9, 8, 7, 6, 5, 4)>
>>> # 3 explicitly defined axes in signal space
>>> s.transpose(signal_axes=[0, 2, 6])
<BaseSignal, title: , dimensions: (8, 6, 5, 4, 2, 1|9, 7, 3)>
>>> # A mix of two lists, but specifying all axes explicitly
>>> # The order of axes is preserved in both lists
>>> s.transpose(navigation_axes=[1, 2, 3, 4, 5, 8], signal_axes=[0, 6, 7])
<BaseSignal, title: , dimensions: (8, 7, 6, 5, 4, 1|9, 3, 2)>
```

A convenience functions `transpose()`

is available to operate on
many signals at once, for example enabling plotting any-dimension signals
trivially:

```
>>> s2 = hs.signals.BaseSignal(np.random.rand(2, 2)) # 2D signal
>>> s3 = hs.signals.BaseSignal(np.random.rand(3, 3, 3)) # 3D signal
>>> s4 = hs.signals.BaseSignal(np.random.rand(4, 4, 4, 4)) # 4D signal
>>> hs.plot.plot_images(hs.transpose(s2, s3, s4, signal_axes=2))
```

The `transpose()`

method accepts keyword argument
`optimize`

, which is `False`

by default, meaning modifying the output
signal data **always modifies the original data** i.e. the data is just a view
of the original data. If `True`

, the method ensures the data in memory is
stored in the most efficient manner for iterating by making a copy of the data
if required, hence modifying the output signal data **not always modifies the
original data**.

The convenience methods `as_signal1D()`

and
`as_signal2D()`

internally use
`transpose()`

, but always optimize the data
for iteration over the navigation axes if required. Hence, these methods do not
always return a view of the original data. If a copy of the data is required
use
`deepcopy()`

on the output of any of these
methods e.g.:

```
>>> hs.signals.Signal1D(np.zeros((4,5,6))).T.deepcopy()
<Signal2D, title: , dimensions: (6|4, 5)>
```

## Basic statistical analysis¶

New in version 0.7.

`get_histogram()`

computes the histogram and
conveniently returns it as signal instance. It provides methods to
calculate the bins. `print_summary_statistics()`

prints the five-number summary statistics of the data.

These two methods can be combined with
`get_current_signal()`

to compute the histogram or
print the summary statistics of the signal at the current coordinates, e.g:

```
>>> s = hs.signals.EELSSpectrum(np.random.normal(size=(10,100)))
>>> s.print_summary_statistics()
Summary statistics
------------------
mean: 0.021
std: 0.957
min: -3.991
Q1: -0.608
median: 0.013
Q3: 0.652
max: 2.751
>>> s.get_current_signal().print_summary_statistics()
Summary statistics
------------------
mean: -0.019
std: 0.855
min: -2.803
Q1: -0.451
median: -0.038
Q3: 0.484
max: 1.992
```

Histogram of different objects can be compared with the functions
`plot_histograms()`

(see
visualisation for the plotting options). For example,
with histograms of several random chi-square distributions:

```
>>> img = hs.signals.Signal2D([np.random.chisquare(i+1,[100,100]) for
... i in range(5)])
>>> hs.plot.plot_histograms(img,legend='auto')
```

## Setting the noise properties¶

Some data operations require the data variance. Those methods use the
`metadata.Signal.Noise_properties.variance`

attribute if it exists. You can
set this attribute as in the following example where we set the variance to be
10:

```
s.metadata.Signal.set_item("Noise_properties.variance", 10)
```

For heterocedastic noise the `variance`

attribute must be a
`BaseSignal`

. Poissonian noise is a common case of
heterocedastic noise where the variance is equal to the expected value. The
`estimate_poissonian_noise_variance()`

`BaseSignal`

method can help setting the variance of data with
semi-poissonian noise. With the default arguments, this method simply sets the
variance attribute to the given `expected_value`

. However, more generally
(although then noise is not strictly poissonian), the variance may be
proportional to the expected value. Moreover, when the noise is a mixture of
white (gaussian) and poissonian noise, the variance is described by the
following linear model:

Where a is the `gain_factor`

, b is the `gain_offset`

(the Gaussian
noise variance) and c the `correlation_factor`

. The correlation
factor accounts for correlation of adjacent signal elements that can
be modelled as a convolution with a Gaussian point spread function.
`estimate_poissonian_noise_variance()`

can be used to
set the noise properties when the variance can be described by this linear
model, for example:

```
>>> s = hs.signals.Spectrum(np.ones(100))
>>> s.add_poissonian_noise()
>>> s.metadata
├── General
│ └── title =
└── Signal
├── binned = False
└── signal_type =
>>> s.estimate_poissonian_noise_variance()
>>> s.metadata
├── General
│ └── title =
└── Signal
├── Noise_properties
│ ├── Variance_linear_model
│ │ ├── correlation_factor = 1
│ │ ├── gain_factor = 1
│ │ └── gain_offset = 0
│ └── variance = <SpectrumSimulation, title: Variance of , dimensions: (|100)>
├── binned = False
└── signal_type =
```

## Speeding up operations¶

New in version 1.0.

### Reusing a Signal for output¶

Many signal methods create and return a new signal. For fast operations, the
new signal creation time is non-negligible. Also, when the operation is
repeated many times, for example in a loop, the cumulative creation time can
become significant. Therefore, many operations on
`BaseSignal`

accept an optional argument out. If an
existing signal is passed to out, the function output will be placed into
that signal, instead of being returned in a new signal. The following example
shows how to use this feature to slice a `BaseSignal`

. It is
important to know that the `BaseSignal`

instance passed in
the out argument must be well-suited for the purpose. Often this means that
it must have the same axes and data shape as the
`BaseSignal`

that would normally be returned by the
operation.

```
>>> s = hs.signals.Signal1D(np.arange(10))
>>> s_sum = s.sum(0)
>>> s_sum.data
array([45])
>>> s.isig[:5].sum(0, out=s_sum)
>>> s_sum.data
array([10])
>>> s_roi = s.isig[:3]
>>> s_roi
<Signal1D, title: , dimensions: (|3)>
>>> s.isig.__getitem__(slice(None, 5), out=s_roi)
>>> s_roi
<Signal1D, title: , dimensions: (|5)>
```

## Interactive operations¶

New in version 1.0.

The function `interactive()`

ease the task of defining
operations that are automatically updated when an event is triggered. By
default it recomputes the operation when data or the axes of the original
signal changes.

```
>>> s = hs.signals.Signal1D(np.arange(10.))
>>> ssum = hs.interactive(s.sum, axis=0)
>>> ssum.data
array([45.0])
>>> s.data /= 10
>>> s.events.data_changed.trigger(s)
>>> ssum.data
array([ 4.5])
```

The interactive operations can be chained.

```
>>> s = hs.signals.Signal1D(np.arange(2 * 3 * 4).reshape((2, 3, 4)))
>>> ssum = hs.interactive(s.sum, axis=0)
>>> ssum_mean = hs.interactive(ssum.mean, axis=0)
>>> ssum_mean.data
array([ 30., 33., 36., 39.])
>>> s.data
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
>>> s.data *= 10
>>> s.events.data_changed.trigger(obj=s)
>>> ssum_mean.data
array([ 300., 330., 360., 390.])
```

## Region Of Interest (ROI)¶

New in version 1.0.

A number of different ROIs are available:

`Point1DROI`

`Point2DROI`

`SpanROI`

`RectangularROI`

`CircleROI`

`Line2DROI`

Once created, a ROI can be used to return a part of any compatible signal:

```
>>> s = hs.signals.Signal1D(np.arange(2000).reshape((20,10,10)))
>>> im = hs.signals.Signal2D(np.arange(100).reshape((10,10)))
>>> roi = hs.roi.RectangularROI(left=3, right=7, top=2, bottom=5)
>>> sr = roi(s)
>>> sr
<Signal1D, title: , dimensions: (4, 3|10)>
>>> imr = roi(im)
>>> imr
<Signal2D, title: , dimensions: (|4, 3)>
```

ROIs can also be used interactively with widgets.
The following examples shows how to interactively apply ROIs to an image. Note
that *it is necessary* to plot the signal onto which the widgets will be
added before calling `interactive()`

.

```
>>> import scipy.misc
>>> im = hs.signals.Signal2D(scipy.misc.ascent())
>>> rectangular_roi = hs.roi.RectangularROI(left=30, right=500,
... top=200, bottom=400)
>>> line_roi = hs.roi.Line2DROI(0, 0, 512, 512, 1)
>>> point_roi = hs.roi.Point2DROI(256, 256)
>>> im.plot()
>>> roi2D = rectangular_roi.interactive(im, color="blue")
>>> roi1D = line_roi.interactive(im, color="yellow")
>>> roi0D = point_roi.interactive(im, color="red")
```

Notably, since ROIs are independent from the signals they sub-select, the widget can be plotted on a different signal altogether.

```
>>> import scipy.misc
>>> im = hs.signals.Signal2D(scipy.misc.ascent())
>>> s = hs.signals.Signal1D(np.random.rand(512, 512, 512))
>>> roi = hs.roi.RectangularROI(left=30, right=77, top=20, bottom=50)
>>> s.plot() # plot signal to have where to display the widget
>>> imr = roi.interactive(im, navigation_signal=s, color="red")
>>> roi(im).plot()
```

ROIs are implemented in terms of physical coordinates and not pixels, so with proper calibration will always point to the same region.

And of course, as all interactive operations, interactive ROIs are chainable. The following example shows how to display interactively the histogram of a rectangular ROI. Notice how we customise the default event connections in order to increase responsiveness.

```
>>> import scipy.misc
>>> im = hs.signals.Signal2D(scipy.misc.ascent())
>>> im.plot()
>>> roi = hs.roi.RectangularROI(left=30, right=500, top=200, bottom=400)
>>> im_roi = roi.interactive(im, color="red")
>>> roi_hist =hs.interactive(im_roi.get_histogram,
... event=im_roi.axes_manager.events.\
... any_axis_changed,
... recompute_out_event=None)
>>> roi_hist.plot()
```

New in version 1.3: ROIs can be used in place of slices when indexing and to define a
signal range in functions taken a `signal_range`

argument.

ROIs can be used in place of slices when indexing and to define a
signal range in functions taken a `signal_range`

argument. For example:

```
>>> s = hs.datasets.example_signals.EDS_TEM_Spectrum()
>>> roi = hs.roi.SpanROI(left=5, right=15)
>>> sc = s.isig[roi]
>>> s.remove_background(signal_range=roi, background_type="Polynomial")
>>> im = hs.datasets.example_signals.object_hologram()
>>> roi = hs.roi.RectangularROI(left=120, right=460., top=300, bottom=560)
>>> imc = im.isig[roi]
```

New in version 1.3: `gui()`

method.

All ROIs have a `gui()`

method that displays an user interface if
a hyperspy GUI is installed (currently only works with the
`hyperspy_gui_ipywidgets`

GUI), enabling precise control of the ROI
parameters:

```
>>> # continuing from above:
>>> roi.gui()
```

## Handling complex data¶

The HyperSpy `ComplexSignal`

signal class and its
subclasses for 1-dimensional and 2-dimensional data allow the user to access
complex properties like the real and imag parts of the data or the
amplitude (also known as the modulus) and phase (also known as angle or
argument) directly. Getting and setting those properties can be done as
follows:

```
>>> real = s.real # real is a new HS signal accessing the same data
>>> s.real = new_real # new_real can be an array or signal
>>> imag = s.imag # imag is a new HS signal accessing the same data
>>> s.imag = new_imag # new_imag can be an array or signal
```

It is important to note that data passed to the constructor of a
`ComplexSignal`

(or to a subclass), which
is not already complex, will be converted to the numpy standard of
np.complex/np.complex128. data which is already complex will be passed
as is.

To transform a real signal into a complex one use:

```
>>> s.change_dtype(complex)
```

Changing the dtype of a complex signal to something real is not clearly defined and thus not directly possible. Use the real, imag, amplitude or phase properties instead to extract the real data that is desired.

### Calculate the angle / phase / argument¶

The `angle()`

function can be used to
calculate the angle, which is equivalent to using the phase property if no
argument is used. If the data is real, the angle will be 0 for positive
values and 2$pi$ for negative values. If the deg parameter is set to
True, the result will be given in degrees, otherwise in rad (default). The
underlying function is the `angle()`

function.
`angle()`

will return an appropriate
HyperSpy signal.

### Phase unwrapping¶

With the `unwrapped_phase()`

method the
complex phase of a signal can be unwrapped and returned as a new signal. The
underlying method is `unwrap()`

, which uses the
algorithm described in [Herraez].

### Add a linear phase ramp¶

For 2-dimensional complex images, a linear phase ramp can be added to the
signal via the
`add_phase_ramp()`

method.
The parameters ramp_x and ramp_y dictate the slope of the ramp in x-
and y direction, while the offset is determined by the offset parameter.
The fulcrum of the linear ramp is at the origin and the slopes are given in
units of the axis with the according scale taken into account. Both are
available via the `AxesManager`

of the signal.