Diffusion maps

The feature matrix, F, found by Diffusion Maps contains the diffusion distances between data points after t time steps. The diffusion distance can be found by computing a Markov random walk on a weighted graph G = V, E), with the vertex set 

Using the forward probabilities from the random walk at timestep t, the diffusion distance between x, and xj can be defined by 

As shown in [11], the low-dimensional representation is found by the decomposition of a modified form of Eq. (2.1) such that 

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A measure of the echo attenuation within each pixel of an image created using the pulse sequence of figure Bl.14,9 perhaps by repeating the experiment with different values of and/or 8, gives data from which a true diffusion map can be constructed [37, 38],... 

Fig. 20. Spin density, and water diffusion images for a 2.2-inm-diameter, spherical silica catalyst support pellet. In-plane pixel resolution was 45 pm x 45 pm image slice thickness was 0.3 mm. (a) Spin-density map lighter shades indicate higher liquid content, (b) map (150 00 ms) lighter shades indicate longer values of Ti. (c) Diffusivity map ((0-1.5) x 10 m s ) lighter shades indicate higher values of water diffusivity within the pellet.

Fi(i. 7 2,15 [Gie 11 Functional diffusion maps according to (7.2.16) of an intcrverlebal disk from a rabbit acquired in viim. (a) Without compression, (b) With compression. The diffusion gradients have been applied in the direction of applied pressure. 

O. Fudym, H.R.B. Orlande, M. Bamford, and J.C. Batsale, Bayesian Approach for Thermal Diffusivity Mapping from Infrared Images Processing with Spatially Random Heat Pulse Heating, Journal of Physics. Conference Series (Online),... 

The concentration of a toxic substance is critical however, it is necessary to compare it with a reference "background pollution." The background pollution is specific to a zone free of toxic substances. It is then possible to make a diffusion map of toxic substances. 

Figure 9.10 (a) Normalised velocity profiles for different concentration solutions of polyfethylene oxide) in water obtained using dynamic NMR microscopy. The concentrations increase in equal steps from 0.5% (w/v) ( ) to 4.5% (w/v) ( ). (b) The polymer self-diffusion profile for the highest concentration solution in units of 10 m s" Note that this was obtained in a separate experiment so that the capillary wall does not fall at precisely the same pixel as in (a), (c) Water solvent velocity and (d) diffusion maps for the 4.5% (w/v) poly(ethylene oxide) solution. (From Y. Xia and P.T. Callaghan [18] and reproduced by permission of the American Chemical Society.)... 

Reith W, Hasegawa Y, Latour LL, Dardzinski BJ, Sotak CH, Fisher M (1995) Multislice diffusion mapping for 3-D evolution of cerebral ischemia in a rat stroke model. Neurology 45 172-177... 

Figure 6 (A) Pulsed gradient spin echo sequence used to encode spin magnetization phase for molecular translational motion. (B) Velocity and diffusion maps for a water molecule flowing through a 2 mm diameter capillary. The images are shown as stackplots. The velocity profile is Poiseuille while the diffusion map is uniform. Courtesy of RW Mair, MM Britton and the author.

Wheat grains Water proton density, velocity and diffusion maps Relationship between flow and grain development... 

Fig. 2.6 Example 2-dimensional embeddings of the S-Curve dataset found by Diffusion Maps with t = l,a =0.1...

Lafon, S., Lee, A.B. Diffusion maps and coarse-graining A unified framework for dimensionality reduction, graph partitioning, and data set parameterization. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(9), 1393-1403 (2006)... 

Unlike other global methods for spectral dimensionality reduction. Diffusion Maps does not compute a -nearest neighbour graph. Rather, the affinities are computed for each pairing of data points, that is, the affinities (Eq.2.6) are calculated for all points such that the affinity matrix W is dense. Since the affinities are calculated for all n points, the cost of computing the affinity matrix is 0 rp-). This computational... 

Although methods may be employed to reduce the cost of constructing the affinity matrix, the overall complexity of Diffusion Maps is still hampered by the eigende-composition of the forward transition probability matrix F. As such, the overall computational complexity of Diffusion Maps is 0(n ) as this is the cost of performing eigendecomposition on the n x n matrix F. 

As with other spectral dimensionality reduction techniques, Diffusion Maps has a large computational overhead due to the eigendecomposition of a large square kernel matrix. One way to overcome this computational cost is to approximate the final embedding based on a known set of learnt points. Such a method lies at the heart of /u.-isometric Difussion Maps [27] (/u,-DM). The term /u-isometric... 

Mishne, G., Cohen, 1. Multiscale anomaly detectiong using diffusion maps. IEEE Journal of Selected Topics in Signal Processing 7(1), 111-123 (2013)... 

Another similar method used Diffusion Maps for edge aware image editing [18]. All operations are performed using the diffusion distance as opposed to the Euclidean distance and so diffusion maps are used to estimate the diffusion distances in the lowdimensional space. What is of real interest is that this is a large scale problem and so the Nystrom extension [19] (as discussed in Chap. 6) is used to alleviate the computational bottlenecks. The results obtained using the diffusion distance show that manifold distances can better account for the global distribution of features in the feature space [18]. 

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