Fig 1.
a) Grouping of similar sub-blocks obtained from an image allows for enhanced sparsity. A reference image block x is denoised by stacking it along with similar image blocks xi into a multi-dimensional array and performing collaborative filtering in a transform domain. An upper cut-off criterion for the maximum distance between reference block x and adjacent blocks determines how many image blocks are used for the transform; all image blocks contribute equally (linear regime). b) Nonlinear methods can be used for transforms where image blocks contribute depending on a nonlinear function, e.g. a Gaussian function.
Fig 2.
a) Kernel PCA deduces an implicit transformation Φ from input space (green circles) into a high-dimensional feature space where linear algorithms can be employed to separate image data from artifacts (red circles). b) Denoising is performed by projecting the test vector x onto the first q principal components by Pq. Backmapping of the projected data is done by finding a so-called pre-image z in image space which minimizes the Euclidean distance between Φ(z) and PqΦ(x).
Fig 3.
Workflow for image reconstruction with nonlinear kernel PCA.
Each iteration consists of two steps: (1) a gradient update ensures consistency with the acquired k-space data, and (2) kernel PCA denoising of image blocks.
Fig 4.
Systolic and diastolic reference data and reconstruction results for 2D dynamic data with a reduction factor of 5.
Temporal profile plots are taken along the indicated line. RMSE values for the shown 3D subvolume covering the heart over time are quoted. The figures on the right show the full field of view of the data set as well as the sampling pattern.
Fig 5.
Reference and reconstruction results for 2D dynamic data with reduction factors of 6.5 and 8.
RMSEs for the shown 3D subvolume covering the heart over time are indicated.
Fig 6.
Signal-intensity profiles through the left ventricle for reduction factors 5, 6.5 and 8.
RMSEs for a 3D subvolume covering the heart over time are indicated.
Fig 7.
Mean and standard deviation of the RMSE relative to the fully sampled reference for all 2D cine data sets (# of volunteers = 6) in a 3D ROI around the heart as indicated in the upper right corner.