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# Convolution and Filtering (`astropy.convolution`)

## Introduction

`astropy.convolution` provides convolution functions and kernels that offer improvements compared to the SciPy `scipy.ndimage` convolution routines, including:

- Proper treatment of NaN values (ignoring them during convolution and replacing NaN pixels with interpolated values)

- Both direct and Fast Fourier Transform (FFT) versions

- Built-in kernels that are commonly used in Astronomy

The following thumbnails show the difference between SciPy and Astropy’s convolve functions on an astronomical image that contains NaN values. Scipy’s function returns NaN for all pixels that are within a kernel-sized region of any NaN value, which is often not the desired result.

(`Source code`)

## Getting Started

Two convolution functions are provided. They are imported as:

    from astropy.convolution import convolve, convolve_fft

and are both used as:

    result = convolve(image, kernel)
    result = convolve_fft(image, kernel)

`convolve()` is implemented as a direct convolution algorithm, while `convolve_fft()` uses a Fast Fourier Transform (FFT). Thus, the former is better for small kernels, while the latter is much more efficient for larger kernels.

### Example

To convolve a 1D dataset with a user-specified kernel, you can do:

    >>> from astropy.convolution import convolve
    >>> convolve([1, 4, 5, 6, 5, 7, 8], [0.2, 0.6, 0.2])
    array([1.4, 3.6, 5. , 5.6, 5.6, 6.8, 6.2])

The `boundary` keyword determines how the input array is extended beyond its boundaries. The default value is `'fill'`, meaning values outside of the array boundary are set to the input `fill_value` (default is 0.0). Setting `boundary='extend'` causes values near the edges to be extended using a constant extrapolation beyond the boundary. The values at the end are computed assuming that any value below the first point is `1`, and any value above the last point is `8`:

    >>> from astropy.convolution import convolve
    >>> convolve([1, 4, 5, 6, 5, 7, 8], [0.2, 0.6, 0.2], boundary='extend')
    array([1.6, 3.6, 5. , 5.6, 5.6, 6.8, 7.8])

For a more detailed discussion of boundary treatment, see Using the Convolution Functions.

### Example

The convolution module also includes built-in kernels that can be imported as, for example:

    >>> from astropy.convolution import Gaussian1DKernel

To use a kernel, first create a specific instance of the kernel:

    >>> gauss = Gaussian1DKernel(stddev=2)

`gauss` is not an array, but a kernel object. The underlying array can be retrieved with:

    >>> gauss.array
    array([6.69162896e-05, 4.36349021e-04, 2.21596317e-03, 8.76430436e-03,
           2.69959580e-02, 6.47599366e-02, 1.20987490e-01, 1.76035759e-01,
           1.99474648e-01, 1.76035759e-01, 1.20987490e-01, 6.47599366e-02,
           2.69959580e-02, 8.76430436e-03, 2.21596317e-03, 4.36349021e-04,
           6.69162896e-05])

The kernel can then be used directly when calling `convolve()`:

(`Source code`, `png`, `svg`, `pdf`)

### Using `astropy`’s Convolution to Replace Bad Data

`astropy`’s convolution methods can be used to replace bad data with values interpolated from their neighbors. Kernel-based interpolation is useful for handling images with a few bad pixels or for interpolating sparsely sampled images.

The interpolation tool is implemented and used as:

    from astropy.convolution import interpolate_replace_nans
    result = interpolate_replace_nans(image, kernel)

Some contexts in which you might want to use kernel-based interpolation include:

> - Images with saturated pixels. Generally, these are the highest-intensity regions in the imaged area, and the interpolated values are not reliable, but this can be useful for display purposes.

> - Images with flagged pixels (e.g., a few small regions affected by cosmic rays or other spurious signals that require those pixels to be flagged out). If the affected region is small enough, the resulting interpolation will have a small effect on source statistics and may allow for robust source-finding algorithms to be run on the resulting data.

> - Sparsely sampled images such as those constructed with single-pixel detectors. Such images will only have a few discrete points sampled across the imaged area, but an approximation of the extended sky emission can still be constructed.

Note

Care must be taken to ensure that the kernel is large enough to completely cover potential contiguous regions of NaN values. An `AstropyUserWarning` is raised if NaN values are detected post- convolution, in which case the kernel size should be increased.

#### Example

The script below shows an example of kernel interpolation to fill in flagged-out pixels:

(`Source code`)

#### Example

This script shows the power of this technique for reconstructing images from sparse sampling. Note that the image is not perfect: the pointlike sources are sometimes missed, but the extended structure is very well recovered by eye.

(`Source code`)

## Using `astropy.convolution`

- Using the Convolution Functions
  - Overview
  - Kernels
- Convolution Kernels
  - Introduction and Concept
  - Examples
  - Available Kernels
  - Kernel Arithmetic
  - Discretization
  - Normalization
- Convolving with Unnormalized Kernels
  - Examples

## Performance Tips

The `convolve()` function is best suited to small kernels, and can become very slow for larger kernels. In this case, consider using `convolve_fft()` (though note that this function uses more memory, and consider the different padding options).

## Reference/API

- Reference/API
  - astropy.convolution Package