The steps of Isomap algorithm used for mapping data into lower dimension d: (1) find K nearest neighbors for each data point X i ; (2) calculate pair-wise geodesic distance matrix and reconstruct neighborhood graph using Dijkstra search algorithm; (3) apply Multidimensional scaling (MDS) on the reconstructed neighborhood graph (matrix D) to compute the low-dimensional embedding vectors.
Steps of Locally Linear Embedding (LLE) algorithm used for mapping data into lower dimension d (1) search for K nearest neighbors for each data point X i ; (2) solve the constrained least-squares problem in Eq. (7) to obtain weights W ij that best linearly reconstruct data point X i from its K neighbors, (3) Compute the low-dimensional embedding vectors Y i best reconstructed by W ij .
(a) Proposed schema for multidimensional image data integration using the Non Linear Dimensional Reduction (NLDR) methods to construct the embedded image and soft tissue segmentation; (b) Proposed schema for extraction of tumor hard boundaries for direct comparison with the postcontrast image as the current standard technique. The same procedure is used for the other breast tissue (fatty and glandular).
Signal (a) and image (b) decomposition using Fast Wavelet Transform (FWT); Signal (c) and Image (d) reconstruction using Inverse FWT (IFWT). a 0 is original signal (image). a j and d j , respectively, are the approximation (coarse or low pass) and detail (high pass) components corresponding to vertical, horizontal, and diagonal at decomposition level j.
Dimension reduction of the Swiss Roll (a) from 3 to 2 dimensions using MDS, PCA, LLE, Isomap and diffusion maps (DfM). Neighborhood size for LLE and Isomap, respectively, were 5 and 10. Sigma for DfM was 0.2. The linear methods (MDS and PCA) both failed to unfold the Swiss Roll in the reduced dimension, while all nonlinear methods (LLE, Isomap, and DfM) were able to unfold the data and preserve structure. The best result was obtained by Isomap.
Demonstration of dimension reduction of 3D clusters (a) from 3 to 2 dimensions using Multidimensional Scaling (MDS), Principal Component Analysis (PCA), Locally Linear Embedding (LLE), Isomap and diffusion maps (DfM). Sigma for DfM was 0.2. Neighborhood size for LLE and Isomap, respectively, were 5 and 10. In this example, all methods except LLE were able to preserve clusters in the reduced dimension. Isomap was not able to fully preserve structure of for the cyan color cluster in the embedding space. LLE was not able to preserve structure of all three clusters and converted the clusters to points in the embedding space.
Embedding of sparse data (a) from 3 to 2 dimensions using MDS, PCA, LLE, Isomap and diffusion maps (DfM). Neighborhood size for LLE and Isomap, respectively, were 5 and 10. Sigma for DfM was 0.2. Both linear methods (MDS and PCA) failed to preserve the sparse data structure in the reduced dimension. DfM was able to fully preserve the sparse data pattern in the embedding space. LLE successfully mapped the sparse data structure to the embedded space. Isomap also was able to preserve most of the data structure but was unable to correctly map the all the blue color to the embedding space.
Correction of B1 inhomogeneity in the MRI data with the local entropy minimization with a bicubic spline model (LEMS) method: Shown are original and corrected images, respectively, for T1-weighted images. After correction, better visualization of breast tissues is noted and they isointense across the image, compared with the images on the left.
Typical diffusion-weighted image (b = 500) in the original size 256 × 256, after compression (64 × 64) and decompression (reconstructed image: 256 × 256). For compression, 2D biorthogonal spline wavelets were used. , respectively, are detail components corresponding to vertical, horizontal, and diagonal. a j , is the approximation (coarse) component at decomposition level j.
Demonstration of the coregistration of the MRI data using a locally affine model: (a) the T2WI source image; (b) the T1WI reference image; (c) warping map; (d) the final coregistered image and (e) the difference image obtained from subtracting the reference image from the registered image.
(a) Typical multiparametric MRI data and (b) resulting embedded images and scattergrams for a malignant breast case from three NLDR algorithms Diffusion Maps (DfM), Isometric feature mapping (ISOMAP), Locally Linear Embedding (LLE). The lower scattergram shown is derived from Isomap. Clear demarcation of the lesion and surrounding breast are shown.
(a) Typical axial multiparametric MRI data from a patient with no breast lesion. (b) resulting embedded images and scattergrams demonstrating the separation of fatty and glandular tissue in the embedded image, the scattergram shown is derived from Isometric feature mapping (ISOMAP).
Demonstration of the sensitivity of NLDR methods to control parameters: (a) effects of Sigma value on the dice similarity index between Diffusion Maps (DfM) based embedded image and the postcontrast; (b) and (c) effects of neighborhood size (K) on the dice similarity index between the embedded image and the postcontrast, respectively, for Isomap and LLE. Example input MRI data for the NLDR methods are shown in Fig. 11.
Dice similarity metric.
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