Spatial smoothness

Shan et al., 1985] Shan, T.-J., Wax, M., and Kailath, T. (1985). On spatial smoothing for direction-of-arrival estimation of coherent signals. IEEE Trans. Acoust. Speech andSig. Proc., ASSP-33 806-811. 

Fig 5.3. The latter result from differences in sea-level air pressure between the stations Gibraltar (Iberian Peninsula), Reykjavik (Iceland), and Oulu (Finland) for the period 1881-2002. Due to spatial smoothing and monthly averaging, this graph does not resolve the temporal details of the ice saint loop in April/May. Although the summer behavior... 

The exponential asymptotic decay also holds for temporally random (i.e. with finite correlation time) spatially smooth flows. In this case the corresponding eigenmode is a stationary random function (Fig. 2.21) and the decay process can be characterized by a set of exponents 7n, associated to the moments of the concentration field, defined by... 

When the Schmidt number (or Prandtl number in case of temperature) is larger than unity the Obukhov-Corrsin scaling (2.120) remains valid for scales above the Kolmogorov scale. However, the scalar fluctuations are not yet dissipated at the viscous cut-off, rj, and are transferred further to smaller scales by chaotic advection in a spatially smooth unsteady flow. This is the so called Batchelor regime, where the advection is dominated by vortices of a single characteristic size l rj, with characteristic timescale tv e l/2v1/2. 

Note that the inertial range and high Re are not necessary for the existence of this type of Batchelor scaling and it can be also produced by chaotic advection in spatially smooth unsteady large scale flows with relatively small or moderate Reynolds number, or in two-dimensional flows (Jullien et ah, 2000 Pierrehumbert, 2000). 

It is interesting to observe that the only difference between the meaningful, if approximate, predictions of the equation above, and the prediction of nonsense comes in the introduction of Amax) that is, in the introduction of spatial smoothing. Since... 

In the analysis of diffusion and reaction in the micropores, we are confronted with the fact that catalysts are not uniformly distributed on the surface of the solid phase thus the so-called catalytic surface is highly non-uniform and spatial smoothing is required in order to achieve a complete analysis of the process. This leads to yet another averaging volume illustrated as level IV in Figure 1.5. The analysis at this level should make use of the method of area averaging (Ochoa-Tapia etal., 1993 Wood etal., 2000) in order to obtain a spatially smoothed jump condition associated with the non-uniform catalytic surface. It would appear that this aspect of the diffusion and reaction process has received little attention and the required information associated with level IV is always obtained by experiment based on the assumption that the experimental information can be used directly at level III. 

Tibshiratti, R., and Wang, P. (2008). Spatial smoothing and hot spot detection for CGH data using the fused lasso. Biostatistics, 9 18—29. 

If the seismic source is spread over an area, then it may not be appropriate to model it as a linear source. To overcome this limitation areal sources, sources which are spread over an area, were also considered in this study. The smoothed historic seismicity approach suggested by Frankel (1995) was adopted for smoothing the areal seismic sources. For the evaluation of seismic hazard, spatially smoothed areal sources, identified based on earthquakes of magnitude 4 and above were used. 

If speed and direction of object movement vary smoothly along the space, the smoothness of the vector field can be set as the second constraint This method is called the spatial global optimization (SGO). The spatial smoothness of the optical flow can be assumed as follows [27] ... 

By the least squares method, v u,v) can be determined by two equations 3 /3m=0 and 3 /3v=0. The constraint Eq. (18) does not assume the spatial smoothness of the vectors. For this reason, TLO method does not depend on the constant object shape unlike SGO and SLO methods. In addition, velocity vectors at each pixel can be independently determined that the resulting optical flow estimation has a higher spatial resolution. TLO method was applied to the analysis of cell movement of Dictyostelium discoideum to study the cell movement pattern within multicellular structure and revealed the geometry of cell-to-cell signaling waves that coordinate the movement of tens of thousands of cells [45,46]. 

The SLO approach has been extended to the structure tensor model approach. This approach considers the temporal domain in addition to the spatial smoothness by treating the tensor field of the matrix l(x,y,t) [40]. The trajectory of moving objects appears as a structure that penetrates the I(x,/,t) matrix that is inclined against the t axis. The structural tensor of this trajectory is subjected to the eigenvalue analysis for obtaining the vector of the brightest part of the trajectory. This approach updated the minimization procedure from... 

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