Shape Analysis and Spatio-Temporal Tracking of Mesoscale Eddies in ...
Shape Analysis and Spatio-Temporal Tracking of Mesoscale Eddies in Miami Isopycnic Coordinate Ocean Model Veena Moolani, Ramprasad Balasubramanian, Li Shen Amit Tandon Department of Computer and Information Science Department of Physics University of Massachusetts Dartmouth, Dartmouth, MA, USA {g_vmoolani, rbalasubrama, lshen, atandon}@umassd.edu Abstract Detection and analysis of ocean surface phenomena have so far relied on manual analysis of long sequences of satellite images or images produced from the mathematical models. In this paper a technique for the three-dimensional shape analysis and spato-temporall tracking of mesoscale eddies from MICOM dataset is presented. Mesoscale eddies play a strong role in carrying heat poleward. The shape analysis of eddies provides a volumetric estimation which can be further used to determine the heat trapped in eddies. Eddies are detected and the region of swirling currents around the detected center is segmented. Segmented eddies are stacked to generate 3D visualization. A 3D skeleton is obtained with the centroids of all the layers in the segmented eddy and represented parametrically. Coefficients of the parametric representation are used for motion tracking of eddies.
1. Introduction The ocean is a huge body of water that is constantly in motion. Eddies transport heat and momentum in the ocean. They also trace chemicals, biological communities, and the oxygen and essential nutrients concerning to life in the sea. Although sporadic in time and space, these eddies are energetic, vigorous, swirling, time-dependent circulations about 100km in width, and can exist for months and move hundreds to thousands of kilometers in their lifetime [13]. These are called "mesoscale eddies," meaning they are of intermediate size and lifespan. These eddies contain huge kinetic energy, comparable with that of the time averaged ocean circulation [14]. Miami Isopycnic Ocean Circulation Model (MICOM) is a three dimensional, time-dependent model, which uses density as a vertical coordinate for the ocean. In this model, the
data is saved at different spatial and temporal locations in the ocean. The MICOM group is based in the RSMAS Division of Meteorology and Physical Oceanography (MPO) [3, 7]. Typically the data is collected daily (or every three days) for 6 or 12 months at a time. In this research attention is confined to the Atlantic Ocean (from latitude 0º- 60º, longitude -95º-16º). The high resolution data is available every 0.6 degree along the longitude and approximately about 0.8 degree in the latitude [15] In MICOM, measurements are available for 16 layers, for each layer, four variables have been investigated in this study, namely depth of the layer (also known as layer thickness), temperature, eastward velocity component and northward velocity component, for each point on the grid. Since depth of the layer is an independent variable, the third velocity component is represented by the change in layer thickness. After converting the MICOM numerical data from its native format to a matrix of longitude, latitude, depth, temperature and U-V velocity components the Poleward and Eastward heat flux are computed. The fluxes provide a measure for heat transport for each Latitude and Longitude and are important metrics. Using these metrics, mesoscale eddies are identified and their three dimensional shape is determined using visualization techniques and they are tracked spatially.
2. Previous work While tremendous progress has been made in the general area of Scientific Visualization and 3D visualization in specific, we confine our attention to the visualization issues of eddies in this section. Eddy motion over the spatial and temporal domains is very important for the research of an unsteady flow field, because it can tell us how many moving eddies there exist and which eddy will appear or disappear thus leading to a net heat transport. The sum over annual period provides the total amount of heat transferred poleward in a year. A lot work has been done by
using the satellite images, but a very little have been accomplished using visualizations of the mathematical numeric oceanographic models. There are many 2D techniques available for oceanic visualizations. Two-dimensional visualizations include basic plots like time-parameter and parameter-parameter. Maps are included in twodimensional. Tools like Live Access Server, Java Ocean Atlas, and Ocean Data View provide excellent plots and maps [8]. A number of researchers have developed 3D visualization tools, and 3D volumetric approach which models the ocean as a solid with variations in density and temperature. 3D volume rendering visualization of oceanographic include template matching and time tracking applied to NOAA/Levitus ocean data [16], color mapping of volumetric data [5], POP and MICOM data [9], VRML for volume visualization [2], OVIRT [12], motion estimation in volumetric data [10], and ParVox [11]. However most of them require servers, mainframes, supercomputers, special-purpose workstations, immersive virtual reality environments, or custom application software [6]. Detection and temporal segmentation of eddies have been done earlier [1, 20]. In these studies, Eddy is detected by pattern recognition techniques. This pattern matching technique is based on minimizing false negatives in the results without allowing the found patterns to be highly deviant from the structuring element [1, 20]. Once the eddy centers have been detected, the second step towards the tracking is to segment the eddy structures to mark their boundaries. The two approaches that were considered before were to draw an ellipse or a circle around an eddy [1, 20]. This approach segments eddies in either a circle shape or an elliptical shape. Figures and data from MICOM show that the oceanic eddies while being coherent structure, are not regular in shape. Therefore these shapes will not capture the volume and heat transfer. Hence, an exact method to segment eddies accurately is important for accurate volume and heat transfer estimation. Eddies move slowly in the ocean, and their path is of interest to oceanographers. Previous work has been done in tracking eddies non-spatially; i.e. considering movement of eddies in a particular layer. This gives an idea of motion of eddies only in that particular layer, whereas the motion for the same eddy could be different in other layers. To find out the total heat transported, eddies need to be tracked through their life cycle. In this period, new eddies are created, travel and change in size, and finally dissipate. 3D shape analysis and tracking of eddies spatially will reveal important information which can be of great interest to many climate researchers and oceanographers.
3. Motivation Mesoscale eddies and jets in the ocean are important contributors to poleward heat transport: the amount of heat the ocean moves from the equatorial regions, to the north and south poles. In addition, biological communities are also strongly affected by ocean circulation and mesoscale eddies. Nutrients exist abundantly deep in the ocean. These nutrients need to be drawn up to the bright surface in order to produce chlorophyll rich phytoplankton. However, constant density layering in the ocean provides a powerful barrier to vertical movement of water. The processes that lift deep water nearer to the surface, are likely to promote life and cold cyclonic eddies do the same [13]. Nutrient rich cold waters promote growth in the entire food chain. Eddy edges act as ocean fronts where biological communities aggregate. Eddies act to exchange water between shallow continental shelf and deeper parts of the ocean. Visualization of ocean circulation also helps cable laying vessels and offshore oil operations, avoiding and minimizing the impacts of strong currents. In addition, volume rendering of ocean assists offshore engineering projects worldwide [4]. The overall effect of mesoscale structures is important for climate. The understanding of ocean circulation is very important for diagnosing and predicting climate changes and their effects. The earth's weather is controlled by the ocean, as they heat and cool, humidify and dry the air and control the direction as well as speed of the wind. Visualizing and modeling changes in the distribution of heat of the ocean, enable researchers and oceanographers, to have ability of projecting future climate and their effect [14]. Representation and analysis of shape is considered a difficult and challenging problem in computer vision and image analysis. Main motivations for shape characterization in vision are extraction of characteristic features for object recognition and retrieval in image databases, building shape models for model-based segmentation, and object tracking for navigation and surveillance. The 3-D shape analysis and tracking of eddies provides a volumetric estimation which can be further used to determine the heat trapped in eddies. This is important for climate studies relating to poleward heat transport, and to oceanographers studying the biological community of the ocean. The spatial tracking method considers eddies in all the available layers of the ocean simultaneously, and provides a three-dimensional view for tracking eddies. Thus, the complete life span of eddies can be tracked spatially. This can provide answers to many important
questions such as: how much heat is trapped in an eddy? How much heat was transferred by the eddy in its complete life-span, etc? With a technique available to track eddies spatially, researchers and oceanographers would be able to quantify the contribution of eddies to poleward heat transport, and the influence on the biological community. Volume rendering techniques can be used to determine ocean properties that may explain fish and mammal behavior [4].
4. Methodology This section describes a methodology used to obtain the 3D shape analysis and modeling of mesoscale eddies. For shape modeling of eddies; specific eddies are considered. For this paper a fast moving eddy-rich region is used (20 N to 8 S and 74 E to -65 W). The MICOM data is available for 6 years (15-20) and for 360 days (000-360). This research uses year 15 for 10 (between day 036 and day 063) days.
4.1 Eddy Segmentation Visualizations with 2D plots of velocity, temperature, depth etc were considered to find the general trends in this oceanic feature. A quiver plot is a vector flow field that displays velocity vectors as arrows with components (u,v) at the points (x,y). Quiver plots are very useful in representing data displaying vector quantities with arrows indicating both direction as well as the magnitude [8]. Segmentation is an important step for accurate shape modeling of the mesoscale eddies.
of considering an eddy in circle or ellipse shape. The eddy detection algorithm, as presented in [1, 17] was used to locate eddies. Pre-developed tracing MATLAB algorithms were used to segment the boundaries of eddies accurately [19]. Thus, a trace can be drawn on the pseudo color map (with the color representing the intensity of the parameter) to segment the region of interest. The boundary points are carefully chosen based on the direction and magnitude of the quiver plots. The segment inside the boundary points is considered as region of interest and the boundary points are saved. This provides with the segmentation of the desired eddy for a particular layer on a specific day. Figure 1 shows a snap shot of the trace program developed to segment eddies manually.
4.2 Voxel Visualization Segmented samples are collected for the desired eddy for all 16 available layers, one per layer. The segmented samples are converted into binary images using polygonal region of interest. A MATLAB function called ROIPOLY was used for this purpose. ROIPOLY returns a binary image that can be used as a mask for masked filtering. Thus for each segmented sample a binary region is obtained. All the binary images are stacked on each other layer-wise to provide a spatial shape of the considered eddy. Layer thickness is an important contribution in the spatial shape analysis. In the ocean, each layer has some associated thickness and in order to represent that layer thickness accurately, several sheets of layers are interpolated between two consecutive layers of the ocean. The interpolation is done in proportion to the density layer thickness. Interpolation also helps in smoothening of the three-dimensional shape of the eddy. All the binary images are stacked to provide a three-dimensional view.
Figure 1 Manual marking of an eddy for segmentation This work uses manual segmentation approach in order to get the shape of the oceanic features, instead
Figure 2 Voxel Representation- Day 36 (Eddy Latitude:-68.25; Longitude: 13.53)
Figure 2 shows the voxel image of an eddy. It gives a good representation of the eddy about its total surface area and volume. Pre-developed algorithms to visualize a binary object are used to provide the voxel representation of the three-dimensional data [18]. A voxel represents a quantity of 3D data just as a pixel represents a point or cluster of points in 2D data. In our case considering the three dimensional data, the x and y coordinates are the latitude and longitude of the eddy in the ocean and the z coordinate is the depth (layer thickness). The input to the algorithm [18] is the 3D binary data and output is the voxel visualization of the binary data.
a
b
4.3 Three-Dimensional Surface Visualization A smoothed three-dimensional visualization of the considered eddy is shown in Figure 3.
c
d
Figure 4: Wireframe Visualizations of Eddy2 (Lat: -65.3, Long: 15.94). a) All Layers, actual depth, b) 16 layers, actual depth, c) 16 layers equidistant, d) all layers
Figure 3 Three-dimensional Surface Visualization- Day 36 (Eddy Lat: -70.85; Long: 13.8) Here the x and y coordinates represent longitude and latitude respectively and the z coordinate is the depth. MATLAB built-in functions smoothes the input data and returns the smoothed data in an array. The original unsmoothed data can be used to view details of the interior. The wire-frame model shows the colored parametric surfaces in Figure 4. Color of the mesh plot is proportional to surface height. Figure 4 shows a wire-frame model of the eddy with interpolated layers along with actual thickness of the layer at that point. Here the z coordinate represents the actual depth value of the ocean layer. Hence this figure shows the actual shape of the eddy via wire frame visualization. Interpolations of layers are in proportion to the actual thickness values. Figure 4 shows various wireframe visualizations of an eddy at latitude -65.36 and longitude 15.9499 on day 63.
Surface plot shows the colored parametric surfaces. Color of the surface plot is proportional to surface height. Surface color can be specified in two different ways: at the vertices or at the centers of each patch. In this general setting, the surface need not be a single-valued function of latitude (x) and longitude (y). Moreover, the four-sided surface patches need not be planar. Surface visualization is primarily a surface plot on the wire frame plot. Figure 5 shows all the surface plot visualizations of another eddy.
a
b
c
d
Figure 5: Surface Visualizations of Eddy3 (Lat: -2.56, Long: 16.18). a) All Layers, actual depth, b) 16 layers, actual depth, c) 16 layers equidistant, d) all layers
4.5 Volume Estimation
The voxel representation is a three-dimensional representation of the binary segmented eddies. The total volume of the eddy is the total number of voxels in all the layers. The Volume estimation of the above considered eddy is the number of 1’s in the threedimensional binary data set. The spatial resolution along the latitude (1/12th of a degree) and longitude (1/8th of a degree) provide a limit on how accurately the volume estimation can be made. Table 1 shows the volume estimation for an eddy, tracked temporally. Here, 1 Voxel = δ-lat * δlong * depth-of-layer = (1/12) * (1/8) * (total depth/73); where: lat is the latitude and long is the longitude for the considered eddy, δ signifies the change in latitude and longitude and 73 represents the total number of layers including the interpolated layers. The variable total depth in the formula is the total sum of individual layer thickness values. Table 1 Volume Estimation Results (Eddy1 Lat:-68 to -66; Long: 15 to 13)
36
Volume (in voxels) 1515082
39
1699010
42
1759606
45
1723301
48
1646723
51
1699181
54
1773376
57
1746581
60
1699601
63
1680259
Day
Change (in voxels)
Displacement X(in degrees)
model is needed. The temporal and spatial tracking of eddies is done with the parametric representation of this three-dimensional curve and independent of the volume. A curve (or skeleton, medial axis) representation of eddies was deemed best as tracking of the curve is easier and more convenient than tracking of the complete shape. Curve is easy to represent and track as it constitutes less number of points and hence makes motion tracking relatively easy. A 3D curve representation of an eddy with a line joining the centroids of all the 16 layers in the segmented eddy was determined the best (justification later). In order to represent a spatial model in 3D curve, first of all centroids are calculated for each layer in the model. The centroid in a particular layer will be a three dimensional value where x and y coordinate will be the longitude and latitude respectively and the z coordinate will be the thickness value of that layer. The centroids can be joined by a 3D line. Thus, a 3D curve representation was generated for the considered eddy. Figure 6 shows the three-dimensional curve representation of the eddy considered.
Y(in degrees) Z(in km)
183928
0.2739
0.2164
0.0256
60596
0.2150
-0.0295
-0.3400
-36305
0.2809
-0.1199
-0.2051
-76578
0.1661
-0.0960
0.3563
52458
0.1418
-0.0329
-0.2774
74195
0.3200
-0.1141
0.6414
-26795
0.2555
0.2352
0.8685
-46980
0.3702
-0.0861
-0.3217
-19342
0.2074
-0.0383
0.1276
The change in the volume of an eddy can be negative or positive. In this method the volume estimated is depended only on the shape of the eddy, so if the eddy breaks into two eddies the volume is reduced significantly. When the eddy breaks, volume of the eddy with centers closer to the original eddy is considered for correspondence. Similarly, as soon as any other eddy unites with the original eddy, the total volume increases. The volume can change even without the break-up or merging of eddies
4.6 3D Line or Skelton representations In order to track the motion of eddies spatially a three-dimensional curve representation of the spatial
Figure 6: Surface Visualization of eddy and respective three-dimensional curve. (Eddy1 Lat: -66.48, Long: 13.5518) 4.7 Why Centroids and not the detected centers? For three-dimensional curve representation, centroid for each eddy was chosen as the connecting point to generate the three dimensional curve for several reasons. Algorithms were also implemented to consider the center of an eddy at each layer to generate the curve, as centers are detected automatically for all eddies in the considered region. Centers of eddies are automatically generated by the center detection algorithm [1, 20]. Since the centers are automatically detected for each layer prior to the
segmentation of eddies; center is independent of the shape of eddies. Table 2: Volume vs. Displacement (Lat:-68 to -66; Long: 15 to 13) Day
Volume (in voxels)
36
1515082
39
1699010
183928
0.2429
0.1435
42
1759606
60596
0.2739
0.2764
0.2118
45
1723301
-36305
0.2809
0.1205
-0.2051
48
1646723
-76578
0.1661
-0.0960
0.3563
51
1699181
52458
0.1418
-0.0329
-0.2774
54
1773376
74195
0.3200
-0.1141
0.6414
57
1746581
-26795
0.2555
0.2352
0.8685
60
1699601
-46980
0.3702
-0.0861
-0.3217
63
1680259 -19342
0.2074
-0.0383
0.1276
Change
Displacement X(in Y(in degrees) degrees) Z(in km)
4.9 Motion Analysis Motion tracking deals with tracing the motion of the considered eddy in a given time frame. Relative difference in the positioning of the eddy between any two days provides the tracking temporally. Motion tracking gives a three-dimensional displacement vector. Figure 7 shows the displacement of an eddy for 10 days. The black curve shows the eddy at day 36 and the blue curve shows the eddy at day 63.
0.2021
For three-dimensional curve representation a parameter which should be dependent on the shape of eddies is needed, as the shape can vary even between layers. Thus, in order to represent the threedimensional curve for an eddy in accordance to its three-dimensional shape, centroid was used. The other reason to consider a centroid was its ease of computation and its attribute to be represented as the "centre of gravity" or the "center of mass” for any given polygon.
Figure 7: Motion Visualization (Lat:-68 to -66; Long: 15 to13) For motion tracking, each eddy is represented by a parametric equation. The coefficients of parametric equations of curves are compared to provide the x, y and z displacements. The temporal displacement computed using the following equation -
4.8 Parametric Representation Next step towards motion tracking was the calculation of a parametric representation for the generated three-dimensional curve. In order to track the motion of an eddy between two days, the coefficients of their mathematical equations can be compared which can further provide information about the displacement of the eddy from one day to the next. Algorithms were developed to generate parametric equations using 3 degree, 4 degree, 5 degree, 6 degree and 7 degree polynomials. Various curve fitting algorithms as Bezier curves were also considered. All the centroids, including the ones for the interpolated layers were considered for the curvefitting, to avoid any piece-wise fitting. For each polynomial degree a system of equations was set up and minimized over the error-of-fit. The study showed that a 5th degree polynomial (with 18 coefficients) was sufficient. Coefficients of the parametric representation are then used for motion tracking of eddies.
⎛ ∫ ⎜⎝ ⎡⎣( X
1
2 2 2 − X 2 ) + (Y1 − Y2 ) + ( Z1 − Z 2 ) ⎤ ⎟⎞ dt ⎦⎠
(1)
Here X, Y and Z are the x, y and z coordinates of an eddy at a particular time t for day m and day n. Here t varies from t = 0 to t = 1. The relative difference in the positioning of the eddy in any two consecutive days provides the shift of eddy from one day to another. The algorithm uses the difference in coefficients of the two parametric equations to estimate the displacement. Thus, the displacement of the eddy from a day (day m) to the day (day n) in the frame is determined. The result is the distance value representing the displacement of the eddy from one day to another.
5. Results Table 2 shows the change in volume when the considered eddy moves from a particular day to
another. It also shows the displacement of the eddy in its X, Y and Z coordinates. The displacement calculations were done by two methods. First was computed through equation (1), mentioned in the previous section.
segmentation, which will directly feed into this system for shape modeling and volume estimation.
Table 3: Temporal Distance covered by three eddies in 10 days Day 36-39
Eddy1 (in degrees) 0.3097
Eddy2(in degrees) 0.1550
Eddy3 (in degrees) 0.3332
39-42
0.3669
0.1762
0.2087
42-45
0.3132
0.9627
0.3460
45-48
0.1928
1.1814
0.0710
48-51
0.1738
0.1305
0.0791
51-54
0.3468
0.1743
0.2081
54-57
0.3611
0.0880
0.2255
57-60
0.3482
0.2109
0.3255
60-63 Total
0.2180 2.6305
0.0989 2.1779
0.3047 2.1018
In the second method each point on the 3D curve was considered. Difference of respective points for each coordinated in both the curves was computed. The average value gave us an idea about the average displacement in X, Y and Z coordinate. The results from the two methods were about the same and they were further used in the distance calculation of eddies. The displacement shows the movement of the eddy from one day to another in X, Y and Z coordinates. Temporal distance indicates the distance covered by the eddy from one day to another in X and Y coordinates and can be expressed in degrees. Table 3 shows the temporal distance covered by three eddy marked as eddy1, eddy2 and eddy3 over 10 days. Figure 8 shows a pseudo color plot for temporal displacement of three eddies from day 36 to day 63.
6. Conclusion and future work Techniques presented here provide a successful way to analyze three-dimensional shape of mesoscale eddies. Various visualization techniques are discussed and results are shown, which provides very detailed understanding of an eddy structure. Methods for Volume estimation and displacement calculation are also developed. In this research, a spatial eddy tracking method was presented. The future will aim to make the three–dimensional tracking method automated. We have automated methods for accurate
Figure: 8 Temporal Displacements of Eddy1, Eddy2 and Eddy3
7. References [1] Ramprasad Balasubramanian, Amit Tandon, Bin John and Vishal Sood. “Detecting and Tracking of Mesoscale Oceanic Features in the Miami Isopycnic Circulation Ocean Model”. Proceedings of The IASTED International Conference on Visualization, Imaging, and Image Processing (VIIP) 2003, Pg. 169-174. Sept 8-10, Benalmedena, Spain. [2] J. Behr and M. Alexa, “Volume visualization in VRML”. Proceedings of the sixth international conference on 3D Web technology. ACM Press, 2001, pp. 23–27. [3] Chassignet, E. P., and Z. D. Garraffo, 2001: Viscosity parameterization and the Gulf Stream separation. In "From Stirring to Mixing in a Stratified Ocean". Proceedings 'Aha Huliko'a Hawaiian Winter Workshop. U. of Hawaii. January 15-19, 2001. P. Muller and D. Henderson, Eds., 37-41. [4] Alexandre Coelho,Marcio Nascimento,Cristiana Bentes, Maria Clicia Stelling de Castro,and Ricardo
Farias. “Parallel Volume Rendering for Ocean Visualization in a Cluster of PC’s”. In: Brazilian Symposium on GeoInformatics - GeoInfo 2004, 2004, Campos do Jordão. Proceedings of the VI Brazilian Symposium on GeoInformatics - GeoInfo 2004, 2004. v. 1. p. 291-304. [5] P. Crossno, E. Angel, and D. Munich, “Case study: visualizing ocean currents with color and dithering”. Proceedings of the IEEE 2001 symposium on parallel and large-data visualization and graphics. IEEE Press, 2001, pp. 37–40. [6] D. Ehrens, A. Tandon, R. Balasubramanian, J.P. Schulze. “Volume Rendering with Animation of Gulf Stream Currents”. Technical Report CS-05-03, Brown University, Department of Computer Science, MA, USA. March 2005 [7] Garraffo, Z. D., A. J. Mariano, A. Griffa, C. Veneziani, and E. P. Chassignet, 2001: Lagrangian data in a high resolution numerical simulation of the North Atlantic. I: Comparison with in-situ drifter data. J. Mar. Sys., 29, 157-176. [8] John Graybeal. “Oceanographic Visualizations”. Monterey Bay Aquarium Research Institute, CA, USA. [9] P. Gruzinskas, A. Haas, L. Goon, and M. Adams, “Data mining ocean model output at the naval oceanographic office major shared resource center”. MTS/IEEE Oceans 2002 Conference. IEEE, 2002. [10] W. de Leeuw and R. van Liere. “Bm3d: motion estimation in time dependent volume data”. Proceedings of the conference on Visualization ’02. IEEE Computer Society, 2002, pp. 427–434. [11] P. P. Li, “Supercomputing visualization for earth science datasets”. Earth Science Technology Conference. Earth Science Technology Office (NASA), 2002. [12] Z. Zhu and R. J. Moorhead, "Extracting and Visualizing Ocean Eddies in Time-Varying Flow Fields," 7th International Symposium on Flow Visualization, Seattle, WA, Sept. 11-15, 1995, pp. 206-211 [13] Peter B Rhines. University of Washington, Seattle, Washington, USA. http://www.ocean.washington.edu/research/gfd/apeddies-gall.pdf [14] Peter B Rhines. University of Washington, Seattle, Washington, USA. “Ocean Circulation and its Role in Global Climate”. http://fluid.ippt.gov.pl/ictam04/text/sessions/docs/SL 14/12781/SL14_12781.pdf [15] http://gcmd.nasa.gov/records/RSMAS_MICOM.html [16] D. Silver and X. Wang, “Tracking scalar features in unstructured datasets,”. In proceedings of
the conference on Visualization ’98. IEEE Computer Society Press, 1998, pp. 79–86. [17] D. Silver and Xin Wang. “Volume Tracking”. Proceedings of IEEE Visualization, San Francisco, CA, 1996, 157-164 [18] Li Shen, James Ford, Fillia Makedon, Andrew Saykin. “A Surface-based Approach for Classification of 3D Neuroanatomic Structures”. Intelligent Data Analysis, An International Journal, Volume 8(6), pp. 519-542, 2004. [19] Li Shen, Ling Gao, Zhenwu Zhuang, Ebo DeMuinck, Heng Huang, Fillia Makedon, Justin Pearlman. “An interactive 3D visualization and manipulation tool for effective assessment of angiogenesis and arteriogenesis using computed tomographic angiography”. SPIE Medical Imaging 2005, Proc. of SPIE Vol 5744, pages 848-858, San Diego, California, 12-17 February 2005. [20] Vishal Sood, Bin John, Ramprasad Balasubramanian and Amit Tandon. “Segmentation and Tracking of Mesoscale Eddies in Numeric Ocean Models”. Proceedings of the IEEE International Conference on Image Processing (ICIP 2005), Volume 3, 11-14 Sept. 2005 Page(s):469 – 472, Genova, Italy, September, 2005.
1. Introduction The ocean is a huge body of water that is constantly in motion. Eddies transport heat and momentum in the ocean. They also trace chemicals, biological communities, and the oxygen and essential nutrients concerning to life in the sea. Although sporadic in time and space, these eddies are energetic, vigorous, swirling, time-dependent circulations about 100km in width, and can exist for months and move hundreds to thousands of kilometers in their lifetime [13]. These are called "mesoscale eddies," meaning they are of intermediate size and lifespan. These eddies contain huge kinetic energy, comparable with that of the time averaged ocean circulation [14]. Miami Isopycnic Ocean Circulation Model (MICOM) is a three dimensional, time-dependent model, which uses density as a vertical coordinate for the ocean. In this model, the
data is saved at different spatial and temporal locations in the ocean. The MICOM group is based in the RSMAS Division of Meteorology and Physical Oceanography (MPO) [3, 7]. Typically the data is collected daily (or every three days) for 6 or 12 months at a time. In this research attention is confined to the Atlantic Ocean (from latitude 0º- 60º, longitude -95º-16º). The high resolution data is available every 0.6 degree along the longitude and approximately about 0.8 degree in the latitude [15] In MICOM, measurements are available for 16 layers, for each layer, four variables have been investigated in this study, namely depth of the layer (also known as layer thickness), temperature, eastward velocity component and northward velocity component, for each point on the grid. Since depth of the layer is an independent variable, the third velocity component is represented by the change in layer thickness. After converting the MICOM numerical data from its native format to a matrix of longitude, latitude, depth, temperature and U-V velocity components the Poleward and Eastward heat flux are computed. The fluxes provide a measure for heat transport for each Latitude and Longitude and are important metrics. Using these metrics, mesoscale eddies are identified and their three dimensional shape is determined using visualization techniques and they are tracked spatially.
2. Previous work While tremendous progress has been made in the general area of Scientific Visualization and 3D visualization in specific, we confine our attention to the visualization issues of eddies in this section. Eddy motion over the spatial and temporal domains is very important for the research of an unsteady flow field, because it can tell us how many moving eddies there exist and which eddy will appear or disappear thus leading to a net heat transport. The sum over annual period provides the total amount of heat transferred poleward in a year. A lot work has been done by
using the satellite images, but a very little have been accomplished using visualizations of the mathematical numeric oceanographic models. There are many 2D techniques available for oceanic visualizations. Two-dimensional visualizations include basic plots like time-parameter and parameter-parameter. Maps are included in twodimensional. Tools like Live Access Server, Java Ocean Atlas, and Ocean Data View provide excellent plots and maps [8]. A number of researchers have developed 3D visualization tools, and 3D volumetric approach which models the ocean as a solid with variations in density and temperature. 3D volume rendering visualization of oceanographic include template matching and time tracking applied to NOAA/Levitus ocean data [16], color mapping of volumetric data [5], POP and MICOM data [9], VRML for volume visualization [2], OVIRT [12], motion estimation in volumetric data [10], and ParVox [11]. However most of them require servers, mainframes, supercomputers, special-purpose workstations, immersive virtual reality environments, or custom application software [6]. Detection and temporal segmentation of eddies have been done earlier [1, 20]. In these studies, Eddy is detected by pattern recognition techniques. This pattern matching technique is based on minimizing false negatives in the results without allowing the found patterns to be highly deviant from the structuring element [1, 20]. Once the eddy centers have been detected, the second step towards the tracking is to segment the eddy structures to mark their boundaries. The two approaches that were considered before were to draw an ellipse or a circle around an eddy [1, 20]. This approach segments eddies in either a circle shape or an elliptical shape. Figures and data from MICOM show that the oceanic eddies while being coherent structure, are not regular in shape. Therefore these shapes will not capture the volume and heat transfer. Hence, an exact method to segment eddies accurately is important for accurate volume and heat transfer estimation. Eddies move slowly in the ocean, and their path is of interest to oceanographers. Previous work has been done in tracking eddies non-spatially; i.e. considering movement of eddies in a particular layer. This gives an idea of motion of eddies only in that particular layer, whereas the motion for the same eddy could be different in other layers. To find out the total heat transported, eddies need to be tracked through their life cycle. In this period, new eddies are created, travel and change in size, and finally dissipate. 3D shape analysis and tracking of eddies spatially will reveal important information which can be of great interest to many climate researchers and oceanographers.
3. Motivation Mesoscale eddies and jets in the ocean are important contributors to poleward heat transport: the amount of heat the ocean moves from the equatorial regions, to the north and south poles. In addition, biological communities are also strongly affected by ocean circulation and mesoscale eddies. Nutrients exist abundantly deep in the ocean. These nutrients need to be drawn up to the bright surface in order to produce chlorophyll rich phytoplankton. However, constant density layering in the ocean provides a powerful barrier to vertical movement of water. The processes that lift deep water nearer to the surface, are likely to promote life and cold cyclonic eddies do the same [13]. Nutrient rich cold waters promote growth in the entire food chain. Eddy edges act as ocean fronts where biological communities aggregate. Eddies act to exchange water between shallow continental shelf and deeper parts of the ocean. Visualization of ocean circulation also helps cable laying vessels and offshore oil operations, avoiding and minimizing the impacts of strong currents. In addition, volume rendering of ocean assists offshore engineering projects worldwide [4]. The overall effect of mesoscale structures is important for climate. The understanding of ocean circulation is very important for diagnosing and predicting climate changes and their effects. The earth's weather is controlled by the ocean, as they heat and cool, humidify and dry the air and control the direction as well as speed of the wind. Visualizing and modeling changes in the distribution of heat of the ocean, enable researchers and oceanographers, to have ability of projecting future climate and their effect [14]. Representation and analysis of shape is considered a difficult and challenging problem in computer vision and image analysis. Main motivations for shape characterization in vision are extraction of characteristic features for object recognition and retrieval in image databases, building shape models for model-based segmentation, and object tracking for navigation and surveillance. The 3-D shape analysis and tracking of eddies provides a volumetric estimation which can be further used to determine the heat trapped in eddies. This is important for climate studies relating to poleward heat transport, and to oceanographers studying the biological community of the ocean. The spatial tracking method considers eddies in all the available layers of the ocean simultaneously, and provides a three-dimensional view for tracking eddies. Thus, the complete life span of eddies can be tracked spatially. This can provide answers to many important
questions such as: how much heat is trapped in an eddy? How much heat was transferred by the eddy in its complete life-span, etc? With a technique available to track eddies spatially, researchers and oceanographers would be able to quantify the contribution of eddies to poleward heat transport, and the influence on the biological community. Volume rendering techniques can be used to determine ocean properties that may explain fish and mammal behavior [4].
4. Methodology This section describes a methodology used to obtain the 3D shape analysis and modeling of mesoscale eddies. For shape modeling of eddies; specific eddies are considered. For this paper a fast moving eddy-rich region is used (20 N to 8 S and 74 E to -65 W). The MICOM data is available for 6 years (15-20) and for 360 days (000-360). This research uses year 15 for 10 (between day 036 and day 063) days.
4.1 Eddy Segmentation Visualizations with 2D plots of velocity, temperature, depth etc were considered to find the general trends in this oceanic feature. A quiver plot is a vector flow field that displays velocity vectors as arrows with components (u,v) at the points (x,y). Quiver plots are very useful in representing data displaying vector quantities with arrows indicating both direction as well as the magnitude [8]. Segmentation is an important step for accurate shape modeling of the mesoscale eddies.
of considering an eddy in circle or ellipse shape. The eddy detection algorithm, as presented in [1, 17] was used to locate eddies. Pre-developed tracing MATLAB algorithms were used to segment the boundaries of eddies accurately [19]. Thus, a trace can be drawn on the pseudo color map (with the color representing the intensity of the parameter) to segment the region of interest. The boundary points are carefully chosen based on the direction and magnitude of the quiver plots. The segment inside the boundary points is considered as region of interest and the boundary points are saved. This provides with the segmentation of the desired eddy for a particular layer on a specific day. Figure 1 shows a snap shot of the trace program developed to segment eddies manually.
4.2 Voxel Visualization Segmented samples are collected for the desired eddy for all 16 available layers, one per layer. The segmented samples are converted into binary images using polygonal region of interest. A MATLAB function called ROIPOLY was used for this purpose. ROIPOLY returns a binary image that can be used as a mask for masked filtering. Thus for each segmented sample a binary region is obtained. All the binary images are stacked on each other layer-wise to provide a spatial shape of the considered eddy. Layer thickness is an important contribution in the spatial shape analysis. In the ocean, each layer has some associated thickness and in order to represent that layer thickness accurately, several sheets of layers are interpolated between two consecutive layers of the ocean. The interpolation is done in proportion to the density layer thickness. Interpolation also helps in smoothening of the three-dimensional shape of the eddy. All the binary images are stacked to provide a three-dimensional view.
Figure 1 Manual marking of an eddy for segmentation This work uses manual segmentation approach in order to get the shape of the oceanic features, instead
Figure 2 Voxel Representation- Day 36 (Eddy Latitude:-68.25; Longitude: 13.53)
Figure 2 shows the voxel image of an eddy. It gives a good representation of the eddy about its total surface area and volume. Pre-developed algorithms to visualize a binary object are used to provide the voxel representation of the three-dimensional data [18]. A voxel represents a quantity of 3D data just as a pixel represents a point or cluster of points in 2D data. In our case considering the three dimensional data, the x and y coordinates are the latitude and longitude of the eddy in the ocean and the z coordinate is the depth (layer thickness). The input to the algorithm [18] is the 3D binary data and output is the voxel visualization of the binary data.
a
b
4.3 Three-Dimensional Surface Visualization A smoothed three-dimensional visualization of the considered eddy is shown in Figure 3.
c
d
Figure 4: Wireframe Visualizations of Eddy2 (Lat: -65.3, Long: 15.94). a) All Layers, actual depth, b) 16 layers, actual depth, c) 16 layers equidistant, d) all layers
Figure 3 Three-dimensional Surface Visualization- Day 36 (Eddy Lat: -70.85; Long: 13.8) Here the x and y coordinates represent longitude and latitude respectively and the z coordinate is the depth. MATLAB built-in functions smoothes the input data and returns the smoothed data in an array. The original unsmoothed data can be used to view details of the interior. The wire-frame model shows the colored parametric surfaces in Figure 4. Color of the mesh plot is proportional to surface height. Figure 4 shows a wire-frame model of the eddy with interpolated layers along with actual thickness of the layer at that point. Here the z coordinate represents the actual depth value of the ocean layer. Hence this figure shows the actual shape of the eddy via wire frame visualization. Interpolations of layers are in proportion to the actual thickness values. Figure 4 shows various wireframe visualizations of an eddy at latitude -65.36 and longitude 15.9499 on day 63.
Surface plot shows the colored parametric surfaces. Color of the surface plot is proportional to surface height. Surface color can be specified in two different ways: at the vertices or at the centers of each patch. In this general setting, the surface need not be a single-valued function of latitude (x) and longitude (y). Moreover, the four-sided surface patches need not be planar. Surface visualization is primarily a surface plot on the wire frame plot. Figure 5 shows all the surface plot visualizations of another eddy.
a
b
c
d
Figure 5: Surface Visualizations of Eddy3 (Lat: -2.56, Long: 16.18). a) All Layers, actual depth, b) 16 layers, actual depth, c) 16 layers equidistant, d) all layers
4.5 Volume Estimation
The voxel representation is a three-dimensional representation of the binary segmented eddies. The total volume of the eddy is the total number of voxels in all the layers. The Volume estimation of the above considered eddy is the number of 1’s in the threedimensional binary data set. The spatial resolution along the latitude (1/12th of a degree) and longitude (1/8th of a degree) provide a limit on how accurately the volume estimation can be made. Table 1 shows the volume estimation for an eddy, tracked temporally. Here, 1 Voxel = δ-lat * δlong * depth-of-layer = (1/12) * (1/8) * (total depth/73); where: lat is the latitude and long is the longitude for the considered eddy, δ signifies the change in latitude and longitude and 73 represents the total number of layers including the interpolated layers. The variable total depth in the formula is the total sum of individual layer thickness values. Table 1 Volume Estimation Results (Eddy1 Lat:-68 to -66; Long: 15 to 13)
36
Volume (in voxels) 1515082
39
1699010
42
1759606
45
1723301
48
1646723
51
1699181
54
1773376
57
1746581
60
1699601
63
1680259
Day
Change (in voxels)
Displacement X(in degrees)
model is needed. The temporal and spatial tracking of eddies is done with the parametric representation of this three-dimensional curve and independent of the volume. A curve (or skeleton, medial axis) representation of eddies was deemed best as tracking of the curve is easier and more convenient than tracking of the complete shape. Curve is easy to represent and track as it constitutes less number of points and hence makes motion tracking relatively easy. A 3D curve representation of an eddy with a line joining the centroids of all the 16 layers in the segmented eddy was determined the best (justification later). In order to represent a spatial model in 3D curve, first of all centroids are calculated for each layer in the model. The centroid in a particular layer will be a three dimensional value where x and y coordinate will be the longitude and latitude respectively and the z coordinate will be the thickness value of that layer. The centroids can be joined by a 3D line. Thus, a 3D curve representation was generated for the considered eddy. Figure 6 shows the three-dimensional curve representation of the eddy considered.
Y(in degrees) Z(in km)
183928
0.2739
0.2164
0.0256
60596
0.2150
-0.0295
-0.3400
-36305
0.2809
-0.1199
-0.2051
-76578
0.1661
-0.0960
0.3563
52458
0.1418
-0.0329
-0.2774
74195
0.3200
-0.1141
0.6414
-26795
0.2555
0.2352
0.8685
-46980
0.3702
-0.0861
-0.3217
-19342
0.2074
-0.0383
0.1276
The change in the volume of an eddy can be negative or positive. In this method the volume estimated is depended only on the shape of the eddy, so if the eddy breaks into two eddies the volume is reduced significantly. When the eddy breaks, volume of the eddy with centers closer to the original eddy is considered for correspondence. Similarly, as soon as any other eddy unites with the original eddy, the total volume increases. The volume can change even without the break-up or merging of eddies
4.6 3D Line or Skelton representations In order to track the motion of eddies spatially a three-dimensional curve representation of the spatial
Figure 6: Surface Visualization of eddy and respective three-dimensional curve. (Eddy1 Lat: -66.48, Long: 13.5518) 4.7 Why Centroids and not the detected centers? For three-dimensional curve representation, centroid for each eddy was chosen as the connecting point to generate the three dimensional curve for several reasons. Algorithms were also implemented to consider the center of an eddy at each layer to generate the curve, as centers are detected automatically for all eddies in the considered region. Centers of eddies are automatically generated by the center detection algorithm [1, 20]. Since the centers are automatically detected for each layer prior to the
segmentation of eddies; center is independent of the shape of eddies. Table 2: Volume vs. Displacement (Lat:-68 to -66; Long: 15 to 13) Day
Volume (in voxels)
36
1515082
39
1699010
183928
0.2429
0.1435
42
1759606
60596
0.2739
0.2764
0.2118
45
1723301
-36305
0.2809
0.1205
-0.2051
48
1646723
-76578
0.1661
-0.0960
0.3563
51
1699181
52458
0.1418
-0.0329
-0.2774
54
1773376
74195
0.3200
-0.1141
0.6414
57
1746581
-26795
0.2555
0.2352
0.8685
60
1699601
-46980
0.3702
-0.0861
-0.3217
63
1680259 -19342
0.2074
-0.0383
0.1276
Change
Displacement X(in Y(in degrees) degrees) Z(in km)
4.9 Motion Analysis Motion tracking deals with tracing the motion of the considered eddy in a given time frame. Relative difference in the positioning of the eddy between any two days provides the tracking temporally. Motion tracking gives a three-dimensional displacement vector. Figure 7 shows the displacement of an eddy for 10 days. The black curve shows the eddy at day 36 and the blue curve shows the eddy at day 63.
0.2021
For three-dimensional curve representation a parameter which should be dependent on the shape of eddies is needed, as the shape can vary even between layers. Thus, in order to represent the threedimensional curve for an eddy in accordance to its three-dimensional shape, centroid was used. The other reason to consider a centroid was its ease of computation and its attribute to be represented as the "centre of gravity" or the "center of mass” for any given polygon.
Figure 7: Motion Visualization (Lat:-68 to -66; Long: 15 to13) For motion tracking, each eddy is represented by a parametric equation. The coefficients of parametric equations of curves are compared to provide the x, y and z displacements. The temporal displacement computed using the following equation -
4.8 Parametric Representation Next step towards motion tracking was the calculation of a parametric representation for the generated three-dimensional curve. In order to track the motion of an eddy between two days, the coefficients of their mathematical equations can be compared which can further provide information about the displacement of the eddy from one day to the next. Algorithms were developed to generate parametric equations using 3 degree, 4 degree, 5 degree, 6 degree and 7 degree polynomials. Various curve fitting algorithms as Bezier curves were also considered. All the centroids, including the ones for the interpolated layers were considered for the curvefitting, to avoid any piece-wise fitting. For each polynomial degree a system of equations was set up and minimized over the error-of-fit. The study showed that a 5th degree polynomial (with 18 coefficients) was sufficient. Coefficients of the parametric representation are then used for motion tracking of eddies.
⎛ ∫ ⎜⎝ ⎡⎣( X
1
2 2 2 − X 2 ) + (Y1 − Y2 ) + ( Z1 − Z 2 ) ⎤ ⎟⎞ dt ⎦⎠
(1)
Here X, Y and Z are the x, y and z coordinates of an eddy at a particular time t for day m and day n. Here t varies from t = 0 to t = 1. The relative difference in the positioning of the eddy in any two consecutive days provides the shift of eddy from one day to another. The algorithm uses the difference in coefficients of the two parametric equations to estimate the displacement. Thus, the displacement of the eddy from a day (day m) to the day (day n) in the frame is determined. The result is the distance value representing the displacement of the eddy from one day to another.
5. Results Table 2 shows the change in volume when the considered eddy moves from a particular day to
another. It also shows the displacement of the eddy in its X, Y and Z coordinates. The displacement calculations were done by two methods. First was computed through equation (1), mentioned in the previous section.
segmentation, which will directly feed into this system for shape modeling and volume estimation.
Table 3: Temporal Distance covered by three eddies in 10 days Day 36-39
Eddy1 (in degrees) 0.3097
Eddy2(in degrees) 0.1550
Eddy3 (in degrees) 0.3332
39-42
0.3669
0.1762
0.2087
42-45
0.3132
0.9627
0.3460
45-48
0.1928
1.1814
0.0710
48-51
0.1738
0.1305
0.0791
51-54
0.3468
0.1743
0.2081
54-57
0.3611
0.0880
0.2255
57-60
0.3482
0.2109
0.3255
60-63 Total
0.2180 2.6305
0.0989 2.1779
0.3047 2.1018
In the second method each point on the 3D curve was considered. Difference of respective points for each coordinated in both the curves was computed. The average value gave us an idea about the average displacement in X, Y and Z coordinate. The results from the two methods were about the same and they were further used in the distance calculation of eddies. The displacement shows the movement of the eddy from one day to another in X, Y and Z coordinates. Temporal distance indicates the distance covered by the eddy from one day to another in X and Y coordinates and can be expressed in degrees. Table 3 shows the temporal distance covered by three eddy marked as eddy1, eddy2 and eddy3 over 10 days. Figure 8 shows a pseudo color plot for temporal displacement of three eddies from day 36 to day 63.
6. Conclusion and future work Techniques presented here provide a successful way to analyze three-dimensional shape of mesoscale eddies. Various visualization techniques are discussed and results are shown, which provides very detailed understanding of an eddy structure. Methods for Volume estimation and displacement calculation are also developed. In this research, a spatial eddy tracking method was presented. The future will aim to make the three–dimensional tracking method automated. We have automated methods for accurate
Figure: 8 Temporal Displacements of Eddy1, Eddy2 and Eddy3
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