# DWT and IDWT

## Discrete Wavelet Transform

Let’s do a `Discrete Wavelet Transform` of a sample data `x` using the `db2` wavelet. It’s simple..

    >>> import pywt
    >>> x = [3, 7, 1, 1, -2, 5, 4, 6]
    >>> cA, cD = pywt.dwt(x, 'db2')

And the approximation and details coefficients are in `cA` and `cD` respectively:

    >>> print(cA)
    [ 5.65685425  7.39923721  0.22414387  3.33677403  7.77817459]
    >>> print(cD)
    [-2.44948974 -1.60368225 -4.44140056 -0.41361256  1.22474487]

## Inverse Discrete Wavelet Transform

Now let’s do an opposite operation - `Inverse Discrete Wavelet Transform`:

    >>> print(pywt.idwt(cA, cD, 'db2'))
    [ 3.  7.  1.  1. -2.  5.  4.  6.]

Voilà! That’s it!

## More Examples

Now let’s experiment with the `dwt()` some more. For example let’s pass a `Wavelet` object instead of the wavelet name and specify signal extension mode (the default is symmetric) for the border effect handling:

    >>> w = pywt.Wavelet('sym3')
    >>> cA, cD = pywt.dwt(x, wavelet=w, mode='constant')
    >>> print(cA)
    [ 4.38354585  3.80302657  7.31813271 -0.58565539  4.09727044  7.81994027]
    >>> print(cD)
    [-1.33068221 -2.78795192 -3.16825651 -0.67715519 -0.09722957 -0.07045258]

Note that the output coefficients arrays length depends not only on the input data length but also on the :class:Wavelet type (particularly on its `filters length` that are used in the transformation).

To find out what will be the output data size use the `dwt_coeff_len()` function:

    >>> # int() is for normalizing Python integers and long integers for documentation tests
    >>> int(pywt.dwt_coeff_len(data_len=len(x), filter_len=w.dec_len, mode='symmetric'))
    6
    >>> int(pywt.dwt_coeff_len(len(x), w, 'symmetric'))
    6
    >>> len(cA)
    6

Looks fine. (And if you expected that the output length would be a half of the input data length, well, that’s the trade-off that allows for the perfect reconstruction…).

The third argument of the `dwt_coeff_len()` is the already mentioned signal extension mode (please refer to the PyWavelets’ documentation for the modes description). Currently there are six extension modes available:

    >>> pywt.Modes.modes
    ['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect', 'antisymmetric', 'antireflect']

As you see in the above example, the periodization (periodization) mode is slightly different from the others. It’s aim when doing the `DWT` transform is to output coefficients arrays that are half of the length of the input data.

Knowing that, you should never mix the periodization mode with other modes when doing `DWT` and `IDWT`. Otherwise, it will produce invalid results:

    >>> x
    [3, 7, 1, 1, -2, 5, 4, 6]
    >>> cA, cD = pywt.dwt(x, wavelet=w, mode='periodization')
    >>> print(pywt.idwt(cA, cD, 'sym3', 'symmetric')) # invalid mode
    [ 1.  1. -2.  5.]
    >>> print(pywt.idwt(cA, cD, 'sym3', 'periodization'))
    [ 3.  7.  1.  1. -2.  5.  4.  6.]

## Tips & tricks

### Passing `None` instead of coefficients data to `idwt()`

Now some tips & tricks. Passing `None` as one of the coefficient arrays parameters is similar to passing a zero-filled array. The results are simply the same:

    >>> print(pywt.idwt([1,2,0,1], None, 'db2', 'symmetric'))
    [ 1.19006969  1.54362308  0.44828774 -0.25881905  0.48296291  0.8365163 ]

    >>> print(pywt.idwt([1, 2, 0, 1], [0, 0, 0, 0], 'db2', 'symmetric'))
    [ 1.19006969  1.54362308  0.44828774 -0.25881905  0.48296291  0.8365163 ]

    >>> print(pywt.idwt(None, [1, 2, 0, 1], 'db2', 'symmetric'))
    [ 0.57769726 -0.93125065  1.67303261 -0.96592583 -0.12940952 -0.22414387]

    >>> print(pywt.idwt([0, 0, 0, 0], [1, 2, 0, 1], 'db2', 'symmetric'))
    [ 0.57769726 -0.93125065  1.67303261 -0.96592583 -0.12940952 -0.22414387]

Remember that only one argument at a time can be `None`:

    >>> print(pywt.idwt(None, None, 'db2', 'symmetric'))
    Traceback (most recent call last):
    ...
    ValueError: At least one coefficient parameter must be specified.

### Coefficients data size in `idwt`

When doing the `IDWT` transform, usually the coefficient arrays must have the same size.

    >>> print(pywt.idwt([1, 2, 3, 4, 5], [1, 2, 3, 4], 'db2', 'symmetric'))
    Traceback (most recent call last):
    ...
    ValueError: Coefficients arrays must have the same size.

Not every coefficient array can be used in `IDWT`. In the following example the `idwt()` will fail because the input arrays are invalid - they couldn’t be created as a result of `DWT`, because the minimal output length for dwt using `db4` wavelet and the symmetric mode is `4`, not `3`:

    >>> pywt.idwt([1,2,4], [4,1,3], 'db4', 'symmetric')
    Traceback (most recent call last):
    ...
    ValueError: Invalid coefficient arrays length for specified wavelet. Wavelet and mode must be the same as used for decomposition.

    >>> int(pywt.dwt_coeff_len(1, pywt.Wavelet('db4').dec_len, 'symmetric'))
    4