The vectorial basis of patches (left) learnt from a high quality tomographic reconstruction of the phase image of a human breast(right).

<p>In each atom of the dictionary the upper part is the component while the lower part is the component.</p

The quality improvement factor versus the regularisation parameter (squares) for our overlapping patches method, and versus (dots) for the total variation method.

<p>The used phantom is the image of Lena, the reconstruction is performed using only noised projections. For our method we have used a value fixed to , the patches basis shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0114325#pone-0114325-g001" target="_blank">figure 1</a> and a step size of .</p

Reconstruction of a computed tomographic slice of the breast.

<p>The images on the first and second row are the X and Y phase gradients, respectively. In the left column the results of the reconstruction obtained with the FBP method using the full set of data are reported. In the right column the results of our method using one projection over five are shown. For these reconstructions we set and .</p

Results from the numerical experiment on the Lena image .

<p>The sinogram is obtained by projecting the image at 80 angles between 0 and 180 degree. A comparison of the images reconstructed with the FBP (a), the EST (b), the algorithm (c), and our DL method (d) is shown. The final image (e) is the same cropped zone in the original image for the sake of comparison (e).</p

The objective functions and (floating solution form) versus iteration number for FISTA optimisation.

<p>The objective functions and (floating solution form) versus iteration number for FISTA optimisation.</p

Results from the numerical experiment on the Lena image with additive Poisson noise.

<p>The sinogram is obtained by projecting the image at 80 angles between 0 and 180 degree, and adding a Poisson noise with a standard deviation equal to 0.3% of the maximal sinogram value. A comparison of the images reconstructed with the FBP (a), the EST (b), the algorithm (c), and our DL method (d) is shown. The final image (e) is the same cropped zone in the original image for the sake of comparison (e).</p

The training image used in our work (a) and the dictionary elements obtained by using the K-SVD algorithm (b).

<p>This basis is then used for the numerical experiment.</p

Comparison of the tomographic image reconstructions of the breast obtained with the FBP (b and g), the TV minimization (c and h), the EST (d and i), and our method (e and j) using 200 projections over the 1000 available.

<p>The right images (f and k) are the images computed with the FBP using the entire set of projections. The top image (a) is the result obtained by the FBP using the entire set of projections. It is reported for showing the location of the insets. The SSIM values are reported based on the FBP full dose image.</p

An illustration of the SIM feature extraction process.

<p>Examples of a normal and osteoarthritic VOIs (left), and their corresponding SIM transformations for radius <i>r</i> = 1 (middle). In the SIM transformations, dark regions correspond to lower magnitudes of <i>α</i> while brighter regions reflect higher magnitudes of <i>α</i>. The distribution of <i>α</i>-values from each SIM transformation are represented by histograms (top right) and by 9 percentiles (10^<i>th</i>–90^<i>th</i>) (bottom right).</p

An exploration of the SIM feature space.

<p>In all plots, representations of feature vectors extracted from healthy VOIs are colored gray, those from osteoarthritic VOIs are colored black. Cluster separation is quantified with Dunn’s separation index (SI), and specified in each plot. (Top Left) SIM feature vectors extracted from normal diseased ROIs area. The distribution of curves corresponding to each class is enclosed by the 25^<i>th</i> and 75^<i>th</i> percentile curves, the solid line represents the median curve. (Top Right) Plotting the 2-D reduced feature representation of the SIM feature set as obtained through evaluation of mutual information criteria. (Middle) Plots of 2-D projections of the SIM feature set obtained with PCA (left) and Sammon’s mapping (right). (Bottom) Plots of 2-D projections of the SIM feature set obtained with XOM (left) and t-SNE (right). As seen, here all feature reduction techniques yield discernible clusters of healthy and osteoarthritic VOIs, but with varying degrees of overlap.</p