## Archivi Categorie: SciPy

### SymPy – 3 – trucchi – 1 Continuo da qui, copio qui.

To begin, we should make something about `SymPy` clear. `SymPy` is nothing more than a Python library, like `NumPy`, `Django`, or even modules in the Python standard library `sys` or `re`. What this means is that `SymPy` does not add anything to the Python language. Limitations that are inherent in the Python language are also inherent in `SymPy`. It also means that `SymPy` tries to use Python idioms whenever possible, making programming with `SymPy` easy for those already familiar with programming with Python. As a simple example, `SymPy` uses Python syntax to build expressions. Implicit multiplication (like `3x` or `3 x`) is not allowed in Python, and thus not allowed in `SymPy`. To multiply `3` and `x`, you must type `3*x` with the `*`.

Simboli
One consequence of this fact is that SymPy can be used in any environment where Python is available. We just import it, like we would any other library: This imports all the functions and classes from SymPy into our interactive Python session. Now, suppose we start to do a computation. Oops! What happened here? We tried to use the variable `x`, but it tells us that `x` is not defined. In Python, variables have no meaning until they are defined. SymPy is no different. Unlike many symbolic manipulation systems you may have used, in SymPy, variables are not defined automatically. To define variables, we must use `symbols`. `symbols` takes a string of variable names separated by spaces or commas, and creates `Symbols` out of them. We can then assign these to variable names. Later, we will investigate some convenient ways we can work around this issue. For now, let us just define the most common variable names, `x`, `y`, and `z`, for use through the rest of this section As a final note, we note that the name of a Symbol and the name of the variable it is assigned to need not have anything to do with one another. Here we have done the very confusing thing of assigning a Symbol with the name `a` to the variable `b`, and a Symbol of the name `b` to the variable `a`. Now the Python variable named a points to the SymPy `Symbol` named `b`, and visa versa. How confusing. We could have also done something like This also shows that `Symbols` can have names longer than one character if we want.

Usually, the best practice is to assign `Symbols` to Python variables of the same name, although there are exceptions: Symbol names can contain characters that are not allowed in Python variable names, or may just want to avoid typing long names by assigning `Symbols` with long names to single letter Python variables.

To avoid confusion, throughout this tutorial, Symbol names and Python variable names will always coincide. Furthermore, the word “Symbol” will refer to a SymPy Symbol and the word “variable” will refer to a Python variable.

Finally, let us be sure we understand the difference between SymPy Symbols and Python variables. Consider the following: What do you think the output of this code will be? If you thought `3`, you’re wrong. Let’s see what really happens Changing `x` to `2` had no effect on `expr`. This is because `x = 2` changes the Python variable `x` to `2`, but has no effect on the SymPy Symbol `x`, which was what we used in creating `expr`. When we created `expr`, the Python variable `x` was a Symbol. After we created, it, we changed the Python variable `x` to `2`. But `expr` remains the same. This behavior is not unique to SymPy. All Python programs work this way: if a variable is changed, expressions that were already created with that variable do not change automatically. For example In this example, if we want to know what `expr` is with the new value of `x`, we need to reevaluate the code that created `expr`, namely, `expr = x + 1`. This can be complicated if several lines created `expr`. One advantage of using a symbolic computation system like SymPy is that we can build a symbolic representation for `expr`, and then substitute `x` with values. The correct way to do this in SymPy is to use `subs`, which will be discussed in more detail later.  ### SciPy – 59 – conclusione Continuo da qui.

Ho così finito la serie iniziata con NumPy, qui. Poi –come le ciliegie– una cosa tira l’altra e sono arrivati centinaia di post che contengono molto più di quello che m’interessa. Per contro per applicarli operativamente ogni singola funzione è da approfondire.
Pochi giorni fa ho assistito un giovane collega über-nerd alle prese con Java, con IDE über e supporto di memos sul telefono; anzi il telefono era un secondo ‘puter. Non riuscivo a seguirlo, sono niubbo, è vero, lo so; ma anche tardigrado. O forse è la specializzazione spinta? (spero).
Con NumPy, Matplotlib, SciPy &co è la stessa cosa: è richiesto un notevole investimento iniziale prima di essere operativi. E, almeno per chi conosco, c’è un rivale affermato: MATLAB. Si deve dire MATLAB (e tutto maiuscolo, à la bimbominkia) anche se in pratica è Octave; tranne casi molto particolari. Mi dicono che tutti usano quello e allora usi quello per evitare di doverti giustificare, spiegare. È cosa già sentita, dai tempi del Fortran IV, poi 77, poi –basta, alla fine si è passato ad altro.
Ma più in generale questo vale anche per Visual Basic, Delphi e altri ancora. Ah! sì Windows, quasi tutti (o sono io che sono sfigato) usano quello.
Poi gli altri, quelli normali, sono alle prese con il Web quindi JavaScript e, poco-poco, Java. C’è ancora chi usa il C++? Chissà…

OK, sono uscito fuori tema con questa tirata, non leggetela.

Io invece, trascinato dalla serie di packages Python penso di iniziare con un ulteriore argomento. Probabilmente senza applicazioni pratiche, ma intrigante. Prossimamente. Forse 😊 ### SciPy – 57 – elaborazione di immagini multidimensionali – 9 Continuo da qui, copio qui.

Misura di oggetti
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_objects` function finds all objects in a labeled array and returns a list of slices that correspond to the smallest regions in the array that contains the object. For instance: The function `find_objects` returns slices for all objects, unless the `max_label parameter` is larger then zero, in which case only the first `max_label` objects are returned. If an index is missing in the label array, `None` is return instead of a slice. For example: 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: Then we can calculate the sum of the elements in the second object: 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: or large arrays and small objects it is more efficient to call the measurement functions after slicing the array: 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: 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 `sum` function calculates the sum of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero label value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `mean` function calculates the mean of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `variance` function calculates the variance of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `standard_deviation` function calculates the standard deviation of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `minimum` function calculates the minimum of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero label value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `maximum` function calculates the maximum of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `minimum_position` function calculates the position of the minimum of the elements of the object with label(s) given by index, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `maximum_position` function calculates the position of the maximum of the elements of the object with label(s) given by index, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `extrema` function calculates the minimum, the maximum, and their positions, of the elements of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation. The result is a tuple giving the minimum, the maximum, the position of the minimum and the position of the maximum. The result is the same as a tuple formed by the results of the functions `minimum`, `maximum`, `minimum_position`, and `maximum_position` that are described above.
• The `center_of_mass` function calculates the center of mass of the of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation.
• The `histogram` function calculates a histogram of the of the object with label(s) given by `index`, using the labels array for the object labels. If `index` is `None`, all elements with a non-zero `label` value are treated as a single object. If `label` is `None`, all elements of input are used in the calculation. Histograms are defined by their minimum (`min`), maximum (`max`) and the number of bins (`bins`). They are returned as one-dimensional arrays of type `Int32`.

A questo punto c’è Extending `scipy.ndimage` in C ma salto l’argomento e passo al capitolo successivo.

Avevo già provato in passato senza riuscirci. Adesso ho approfondito, ho consultato la documentazione ufficiale di Python, scoprendo il comando `python3 setup.py` build, dove `setup.py` è il file che devi fornire (come indicato nalla guida che sto seguendo). Creo regolarmente la libreria (da spostare e rinominare) ma alla fine non trova la funzione di callback. Considerando che la creazione di funzioni in C (e, credo, in C++) sarebbe cosa estremamente avanzata e fuori dai miei interessi non insisto. ### SciPy – 56 – elaborazione di immagini multidimensionali – 8 Continuo da qui, copio qui.

Segmentazione ed etichettatura
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: 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 `label` function generates an array where the objects in the input are labeled with an integer index. It returns a tuple consisting of the array of object labels and the number of objects found, unless the output parameter is given, in which case only the number of objects is returned. The connectivity of the objects is defined by a structuring element. For instance, in two dimensions using a four-connected structuring element gives: 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: If no structuring element is provided, one is generated by calling `generate_binary_structure` (see Binary morphology [qui]) 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: Note: The structuring element used by label is assumed to be symmetric.

There is a large number of other approaches for segmentation, for instance from an estimation of the borders of the objects that can be obtained for instance by derivative filters. One such an approach is watershed segmentation. The function `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_ift` function 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: 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: The result is that the object (`marker = 2`) is smaller because the second marker was processed earlier. This may not be the desired effect if the first marker was supposed to designate a background object. Therefore `watershed_ift` treats markers with a negative value explicitly as background markers and processes them after the normal markers. For instance, replacing the first marker by a negative marker gives a result similar to the first example: 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` (see Binary morphology [stesso link]) 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: Note: The implementation of `watershed_ift` limits the data types of the input to `UInt8` and `UInt16`. ### SciPy – 55 – elaborazione di immagini multidimensionali – 7 Continuo da qui, copio qui.

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

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

The distances and indices arguments can be used to give optional output arrays that must be of the correct size and type (both `Int32`). 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_edt` calculates the exact euclidean distance transform of the input, by replacing each object element (defined by values larger than zero) with the shortest euclidean distance to the background (all non-object elements).

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

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

The `distances` and `indices` arguments can be used to give optional output arrays that must be of the correct size and type (`Float64` and `Int32`).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_bf` uses a brute-force algorithm to calculate the distance transform of the input, by replacing each object element (defined by values larger than zero) with the shortest distance to the background (all non-object elements). The metric must be one of `'euclidean'`, `'cityblock'`, or `'chessboard'`.

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

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

The `distances` and `indices` arguments can be used to give optional output arrays that must be of the correct size and type (`Float64` and `Int32`).

Note: This function uses a slow brute-force algorithm, the function `distance_transform_cdt` can be used to more efficiently calculate `cityblock` and `chessboard` distance transforms. The function `distance_transform_edt` can be used to more efficiently calculate the exact `euclidean` distance transform. ### SciPy – 54 – elaborazione di immagini multidimensionali – 6 Continuo da qui, copio qui.

Morfologia
morfologia binaria
The `generate_binary_structure` functions generates a binary structuring element for use in binary morphology operations. The rank of the structure must be provided. The size of the structure that is returned is equal to three in each direction. The value of each element is equal to one if the square of the Euclidean distance from the element to the center is less or equal to connectivity. For instance, two dimensional 4-connected and 8-connected structures are generated as follows: Most binary morphology functions can be expressed in terms of the basic operations erosion and dilation.

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

Here is an example of using `binary_dilation` to find all elements that touch the border, by repeatedly dilating an empty array from the border using the data array as the mask: 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_structure` function returns a structure by dilation of the input structure `iteration-1` times with itself.

For instance: If the origin of the original structure is equal to 0, then it is also equal to 0 for the iterated structure. If not, the origin must also be adapted if the equivalent of the `iterations` erosions or dilations must be achieved with the iterated structure. The adapted origin is simply obtained by multiplying with the number of iterations. For convenience the `iterate_structure` also returns the adapted origin if the `origin` parameter is not `None`: Other morphology operations can be defined in terms of erosion and d dilation. The following functions provide a few of these operations for convenience:

• The `binary_opening` function implements binary opening of arrays of arbitrary rank with the given structuring element. Binary opening is equivalent to a binary erosion followed by a binary dilation with the same structuring element. The `origin` parameter controls the placement of the structuring element as described in Filter functions [stesso link]. If no structuring element is provided, an element with connectivity equal to one is generated using `generate_binary_structure`. The `iterations` parameter gives the number of erosions that is performed followed by the same number of dilations.
• The `binary_closing` function implements binary closing of arrays of arbitrary rank with the given structuring element. Binary closing is equivalent to a binary dilation followed by a binary erosion with the same structuring element. The `origin` parameter controls the placement of the structuring element as described in Filter functions [stesso link]. If no structuring element is provided, an element with connectivity equal to one is generated using `generate_binary_structure`. The `iterations` parameter gives the number of dilations that is performed followed by the same number of erosions.
• The `binary_fill_holes` function is used to close holes in objects in a binary image, where the structure defines the connectivity of the holes. The `origin` parameter controls the placement of the structuring element as described in Filter functions [stesso link]. If no structuring element is provided, an element with connectivity equal to one is generated using `generate_binary_structure`.
• The `binary_hit_or_miss` function implements a binary hit-or-miss transform of arrays of arbitrary rank with the given structuring elements. The hit-or-miss transform is calculated by erosion of the input with the first structure, erosion of the logical not of the input with the second structure, followed by the logical and of these two erosions. The `origin` parameters control the placement of the structuring elements as described in Filter functions [stesso link]. If `origin2` equals None it is set equal to the `origin1` parameter. If the first structuring element is not provided, a structuring element with connectivity equal to one is generated using `generate_binary_structure`, if `structure2` is not provided, it is set equal to the logical not of `structure1`.

morfologia grey-scale
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 [stesso link], 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_erosion` function calculates a multidimensional grey- scale erosion.
• The `grey_dilation` function calculates a multidimensional grey-scale dilation.

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

• The `grey_opening` function implements grey-scale opening of arrays of arbitrary rank. Grey-scale opening is equivalent to a grey-scale erosion followed by a grey-scale dilation.
• The `grey_closing` function implements grey-scale closing of arrays of arbitrary rank. Grey-scale opening is equivalent to a grey-scale dilation followed by a grey-scale erosion.
• The `morphological_gradient` function implements a grey-scale morphological gradient of arrays of arbitrary rank. The grey-scale morphological gradient is equal to the difference of a grey-scale dilation and a grey-scale erosion.
• The `morphological_laplace` function implements a grey-scale morphological laplace of arrays of arbitrary rank. The grey-scale morphological laplace is equal to the sum of a grey-scale dilation and a grey-scale erosion minus twice the input.
• The `white_tophat` function implements a white top-hat filter of arrays of arbitrary rank. The white top-hat is equal to the difference of the input and a grey-scale opening.
• The `black_tophat` function implements a black top-hat filter of arrays of arbitrary rank. The black top-hat is equal to the difference of a grey-scale closing and the input. ### SciPy – 53 – elaborazione di immagini multidimensionali – 5 Continuo da qui, copio qui.

Filtri del dominio di Fourier
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_shift` function multiplies the input array with the multidimensional Fourier transform of a shift operation for the given `shift`. The `shift` parameter is a sequences of shifts for each dimension, or a single value for all dimensions.
• The `fourier_gaussian` function multiplies the input array with the multidimensional Fourier transform of a Gaussian filter with given standard-deviations `sigma`. The `sigma` parameter is a sequences of values for each dimension, or a single value for all dimensions.
• The fourier_uniform function multiplies the input array with the multidimensional Fourier transform of a uniform filter with given sizes `size`. The `size` parameter is a sequences of values for each dimension, or a single value for all dimensions.
• The `fourier_ellipsoid` function multiplies the input array with the multidimensional Fourier transform of a elliptically shaped filter with given sizes `size`. The `size` parameter is a sequences of values for each dimension, or a single value for all dimensions. This function is only implemented for dimensions 1, 2, and 3.

No, niente esempi.

Funzioni di interpolazione
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.

pre-filtri per splines
Interpolation using splines of an order larger than 1 requires a pre-filtering step. The interpolation functions [here] described […] 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_filter1d` function calculates a one-dimensional spline filter along the given axis. An output array can optionally be provided. The order of the spline must be larger then 1 and less than 6.
• The `spline_filter` function calculates a multidimensional spline filter.

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

funzioni di interpolazione
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 [qui] 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_transform` function applies an arbitrary geometric transform to the input. The given mapping function is called at each point in the output to find the corresponding coordinates in the input. mapping must be a callable object that accepts a tuple of length equal to the output array rank and returns the corresponding input coordinates as a tuple of length equal to the input array rank. The output shape and output type can optionally be provided. If not given they are equal to the input shape and type.

For example: Optionally extra arguments can be defined and passed to the filter function. The `extra_arguments` and `extra_keywords` arguments can be used to pass a tuple of extra arguments and/or a dictionary of named arguments that are passed to derivative at each call. For example, we can pass the shifts in our example as arguments or Note: The mapping function can also be written in C and passed using a `scipy.LowLevelCallable`. See Extending scipy.ndimage in C [prossimamente] for more information.

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

Here is an example that interpolates a 2D array at `(0.5, 0.5)` and `(1, 2)`: The `affine_transform` function applies an affine transformation to the input array. The given transformation matrix and offset are used to find for each point in the output the corresponding coordinates in the input. The value of the input at the calculated coordinates is determined by spline interpolation of the requested order. The transformation matrix must be two-dimensional or can also be given as a one-dimensional sequence or array. In the latter case, it is assumed that the matrix is diagonal. A more efficient interpolation algorithm is then applied that exploits the separability of the problem. The output shape and output type can optionally be provided. If not given they are equal to the input shape and type.

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

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

The `rotate` function returns the input array rotated in the plane defined by the two axes given by the parameter axes, using spline interpolation of the requested order. The angle must be given in degrees. If reshape is true, then the size of the output array is adapted to contain the rotated input. ### SciPy – 52 – elaborazione di immagini multidimensionali – 4 Continuo da qui, copio qui.

Funzioni filtro generiche
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 `PyCapsule` (see Extending `scipy.ndimage` in C [prossimamente] for more information).

The `generic_filter1d` function implements a generic one-dimensional filter function, where the actual filtering operation must be supplied as a Python function (or other callable object). The `generic_filter1d` function iterates over the lines of an array and calls function at each line. The arguments that are passed to function are one-dimensional arrays of the `tFloat64` type. The first contains the values of the current line. It is extended at the beginning end the end, according to the `filter_size` and `origin` arguments. The second array should be modified in-place to provide the output values of the line. For example consider a correlation along one dimension: The same operation can be implemented using generic_filter1d as follows: Here the origin of the kernel was (by default) assumed to be in the middle of the filter of length 3. Therefore, each input line was extended by one value at the beginning and at the end, before the function was called.

Optionally extra arguments can be defined and passed to the filter function. The `extra_arguments` and `extra_keywords` arguments can be used to pass a tuple of extra arguments and/or a dictionary of named arguments that are passed to derivative at each call. For example, we can pass the parameters of our filter as an argument or The `generic_filter` function implements a generic filter function, where the actual filtering operation must be supplied as a Python function (or other callable object). The `generic_filter` function iterates over the array and calls function at each element. The argument of function is a one-dimensional array of the `tFloat64` type, that contains the values around the current element that are within the footprint of the filter. The function should return a single value that can be converted to a double precision number. For example consider a correlation: The same operation can be implemented using generic_filter as follows: 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. The `size` parameter, if provided, must be a sequence of sizes or a single number in which case the size of the filter is assumed to be equal along each axis. The `footprint`, if provided, must be an array that defines the shape of the kernel by its non-zero elements.

Optionally extra arguments can be defined and passed to the filter function. The `extra_arguments` and `extra_keywords` arguments can be used to pass a tuple of extra arguments and/or a dictionary of named arguments that are passed to derivative at each call. For example, we can pass the parameters of our filter as an argument or 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. Non sono riuscito a riprodurlo, mi da errori per `range` 😡 ### SciPy – 51 – elaborazione di immagini multidimensionali – 3 Continuo da qui, copio qui.

Derivate
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 `prewitt` function calculates a derivative along the given axis.
• The `sobel` function calculates a derivative along the given axis.

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

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

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

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

For example: To demonstrate the use of the extra_arguments argument we could do or The following two functions are implemented using `generic_laplace` by providing appropriate functions for the second derivative function:

• The function `laplace` calculates the Laplace using discrete differentiation for the second derivative (i.e. convolution with `[1, -2, 1]`).
• The function `gaussian_laplace` calculates the Laplace filter using `gaussian_filter` to calculate the second derivatives. The standard-deviations of the Gaussian filter along each axis are passed through the parameter `sigma` as a sequence or numbers. If `sigma` is not a sequence but a single number, the standard deviation of the filter is equal along all directions.

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

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

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

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

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

For example, the `sobel` function fits the required signature See the documentation of `generic_laplace` for examples of using the extra_arguments and extra_keywords arguments.

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

The function `gaussian_gradient_magnitude` calculates the gradient magnitude using `gaussian_filter` to calculate the first derivatives. The standard-deviations of the Gaussian filter along each axis are passed through the parameter `sigma` as a sequence or numbers. If `sigma` is not a sequence but a single number, the standard deviation of the filter is equal along all directions. ### SciPy – 50 – elaborazione di immagini multidimensionali – 2 Continuo da qui, copio qui.

Correlazione e convoluzione
The `correlate1d` function calculates a one-dimensional correlation along the given axis. The lines of the array along the given axis are correlated with the given weights. The `weights` parameter must be a one-dimensional sequences of numbers.

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

The `convolve1d` function calculates a one-dimensional convolution along the given axis. The lines of the array along the given axis are convoluted with the given weights. The `weights` parameter must be a one-dimensional sequences of numbers.

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

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

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

Filtri di appianamento (smoothing)
The `gaussian_filter1d` function implements a one-dimensional Gaussian filter. The standard-deviation of the Gaussian filter is passed through the parameter `sigma`. Setting `order = 0` corresponds to convolution with a Gaussian kernel. An order of 1, 2, or 3 corresponds to convolution with the first, second or third derivatives of a Gaussian. Higher order derivatives are not implemented.

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

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

The `uniform_filter1d` function calculates a one-dimensional uniform filter of the given size along the given axis.

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

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

Filtri basati sulle statistiche degli ordini
The `minimum_filter1d` function calculates a one-dimensional minimum filter of given size along the given axis.

The `maximum_filter1d` function calculates a one-dimensional maximum filter of given size along the given axis.

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

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

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

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

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