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.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/applications/plot_tomography_l1_reconstruction.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_applications_plot_tomography_l1_reconstruction.py>`
        to download the full example code or to run this example in your browser via Binder

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_applications_plot_tomography_l1_reconstruction.py:


======================================================================
Compressive sensing: tomography reconstruction with L1 prior (Lasso)
======================================================================

This example shows the reconstruction of an image from a set of parallel
projections, acquired along different angles. Such a dataset is acquired in
**computed tomography** (CT).

Without any prior information on the sample, the number of projections
required to reconstruct the image is of the order of the linear size
``l`` of the image (in pixels). For simplicity we consider here a sparse
image, where only pixels on the boundary of objects have a non-zero
value. Such data could correspond for example to a cellular material.
Note however that most images are sparse in a different basis, such as
the Haar wavelets. Only ``l/7`` projections are acquired, therefore it is
necessary to use prior information available on the sample (its
sparsity): this is an example of **compressive sensing**.

The tomography projection operation is a linear transformation. In
addition to the data-fidelity term corresponding to a linear regression,
we penalize the L1 norm of the image to account for its sparsity. The
resulting optimization problem is called the :ref:`lasso`. We use the
class :class:`~sklearn.linear_model.Lasso`, that uses the coordinate descent
algorithm. Importantly, this implementation is more computationally efficient
on a sparse matrix, than the projection operator used here.

The reconstruction with L1 penalization gives a result with zero error
(all pixels are successfully labeled with 0 or 1), even if noise was
added to the projections. In comparison, an L2 penalization
(:class:`~sklearn.linear_model.Ridge`) produces a large number of labeling
errors for the pixels. Important artifacts are observed on the
reconstructed image, contrary to the L1 penalization. Note in particular
the circular artifact separating the pixels in the corners, that have
contributed to fewer projections than the central disk.

.. GENERATED FROM PYTHON SOURCE LINES 38-146



.. image-sg:: /auto_examples/applications/images/sphx_glr_plot_tomography_l1_reconstruction_001.png
   :alt: original image, L2 penalization, L1 penalization
   :srcset: /auto_examples/applications/images/sphx_glr_plot_tomography_l1_reconstruction_001.png
   :class: sphx-glr-single-img





.. code-block:: default


    # Author: Emmanuelle Gouillart <emmanuelle.gouillart@nsup.org>
    # License: BSD 3 clause

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import ndimage, sparse

    from sklearn.linear_model import Lasso, Ridge


    def _weights(x, dx=1, orig=0):
        x = np.ravel(x)
        floor_x = np.floor((x - orig) / dx).astype(np.int64)
        alpha = (x - orig - floor_x * dx) / dx
        return np.hstack((floor_x, floor_x + 1)), np.hstack((1 - alpha, alpha))


    def _generate_center_coordinates(l_x):
        X, Y = np.mgrid[:l_x, :l_x].astype(np.float64)
        center = l_x / 2.0
        X += 0.5 - center
        Y += 0.5 - center
        return X, Y


    def build_projection_operator(l_x, n_dir):
        """Compute the tomography design matrix.

        Parameters
        ----------

        l_x : int
            linear size of image array

        n_dir : int
            number of angles at which projections are acquired.

        Returns
        -------
        p : sparse matrix of shape (n_dir l_x, l_x**2)
        """
        X, Y = _generate_center_coordinates(l_x)
        angles = np.linspace(0, np.pi, n_dir, endpoint=False)
        data_inds, weights, camera_inds = [], [], []
        data_unravel_indices = np.arange(l_x**2)
        data_unravel_indices = np.hstack((data_unravel_indices, data_unravel_indices))
        for i, angle in enumerate(angles):
            Xrot = np.cos(angle) * X - np.sin(angle) * Y
            inds, w = _weights(Xrot, dx=1, orig=X.min())
            mask = np.logical_and(inds >= 0, inds < l_x)
            weights += list(w[mask])
            camera_inds += list(inds[mask] + i * l_x)
            data_inds += list(data_unravel_indices[mask])
        proj_operator = sparse.coo_matrix((weights, (camera_inds, data_inds)))
        return proj_operator


    def generate_synthetic_data():
        """Synthetic binary data"""
        rs = np.random.RandomState(0)
        n_pts = 36
        x, y = np.ogrid[0:l, 0:l]
        mask_outer = (x - l / 2.0) ** 2 + (y - l / 2.0) ** 2 < (l / 2.0) ** 2
        mask = np.zeros((l, l))
        points = l * rs.rand(2, n_pts)
        mask[(points[0]).astype(int), (points[1]).astype(int)] = 1
        mask = ndimage.gaussian_filter(mask, sigma=l / n_pts)
        res = np.logical_and(mask > mask.mean(), mask_outer)
        return np.logical_xor(res, ndimage.binary_erosion(res))


    # Generate synthetic images, and projections
    l = 128
    proj_operator = build_projection_operator(l, l // 7)
    data = generate_synthetic_data()
    proj = proj_operator @ data.ravel()[:, np.newaxis]
    proj += 0.15 * np.random.randn(*proj.shape)

    # Reconstruction with L2 (Ridge) penalization
    rgr_ridge = Ridge(alpha=0.2)
    rgr_ridge.fit(proj_operator, proj.ravel())
    rec_l2 = rgr_ridge.coef_.reshape(l, l)

    # Reconstruction with L1 (Lasso) penalization
    # the best value of alpha was determined using cross validation
    # with LassoCV
    rgr_lasso = Lasso(alpha=0.001)
    rgr_lasso.fit(proj_operator, proj.ravel())
    rec_l1 = rgr_lasso.coef_.reshape(l, l)

    plt.figure(figsize=(8, 3.3))
    plt.subplot(131)
    plt.imshow(data, cmap=plt.cm.gray, interpolation="nearest")
    plt.axis("off")
    plt.title("original image")
    plt.subplot(132)
    plt.imshow(rec_l2, cmap=plt.cm.gray, interpolation="nearest")
    plt.title("L2 penalization")
    plt.axis("off")
    plt.subplot(133)
    plt.imshow(rec_l1, cmap=plt.cm.gray, interpolation="nearest")
    plt.title("L1 penalization")
    plt.axis("off")

    plt.subplots_adjust(hspace=0.01, wspace=0.01, top=1, bottom=0, left=0, right=1)

    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 5.923 seconds)


.. _sphx_glr_download_auto_examples_applications_plot_tomography_l1_reconstruction.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
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        :alt: Launch binder
        :width: 150 px



    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_tomography_l1_reconstruction.py <plot_tomography_l1_reconstruction.py>`

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      :download:`Download Jupyter notebook: plot_tomography_l1_reconstruction.ipynb <plot_tomography_l1_reconstruction.ipynb>`


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