Monthly Archives: May 2014
FUTURE DIRECTIONS CryoSat
CryoSat [30] is the first satellite of the European Space Agency’s Living Planet Programme to be realized in the framework of the Earth Explorer Opportunity Missions. The mission concept was selected in 1999 with launch originally anticipated in 2004. Unfortunately, the launch (October 2005) failed. A rebuild of CryoSat2 was approved by ESA, now scheduled for launch in 2009. The Cryosat orbit will have highinclination (92°) and a longrepeat period (369 days, with a 30day subcycle), designed to provide dense interlocking coverage over the polar regions. Its aim is to study possible climate variability and trends by determining the variations in thickness of the Earth’s continental ice sheets and marine sea ice cover.
The Cryosat altimeter will be the first of its kind: SAR/Interferometric Radar ALtimeter (SIRAL), whose advanced modes are patterned after the D2P altimeter [31], and whose flight hardware has extensive Poseidon heritage. Unlike previous radar altimeter missions, Cryosat will downlink all altimetric data. These data will support three modes: conventional, interferometric, and synthetic aperture. The conventional (pulselimited) mode will be used for open ocean (for calibration and sea surface height
reference purposes) and the central continental ice sheets that are relatively level. The interferometric mode will be used for the more steeply sloping margins of the ice sheets. The synthetic aperture mode will be used primarily over sea ice, where its sharper spatial resolution and better precision will support measurement of the freeboard for floating sea ice. These measurements can be inverted to estimate ice thickness.
Minimum Spanning Tree (MST)
The minimum spanning tree is the “best” tree one can identify in a given graph with edge weights. Recall that edge weights represent some “cost” of communicating on that edge. The cost may be delay or expense in terms of real dollars to use the link.
The MST problem is to find a set of edges with a total minimum cost so that the nodes in the graph remain connected. A greedy algorithm can be used to find this set of edges, called MSTE. The algorithm starts with one edge with minimum weight. Then it finds an edge “e,” the best candidate that has not yet been considered and adds it if it is feasible. An edge can only be added to this set if it does not create a cycle in the graph with the same set of nodes as the original graph and set ofedges MSTE. MSTE is complete when it contains N — 1 edges in an N node graph. It is known that a greedy algorithm indeed finds an MSTE.
Several algorithms are available to find an MST. We will consider two algorithms here based on the greedy approach, but their complexities may differ slightly.
Kruskal Algorithm. This algorithm essentially requires all edges to be sorted, shortest first. Then the edges are included in set MSTE, one at a time, in an order such that the edges do not form a cycle. The test for forming a cycle can be efficiently made by maintaining a proper data structure of edges included thus far. The complexity of sorting is O(M log M)), the test is of complexity O(M + N) as suggested by Tarjan (12). As the process terminates once the set MSTE includes N — 1 edges, one may not have to sort all edges (the first few may be sufficient). This result be achieved by putting all edge weights in a heap that can be created in O(M) time. An edge with the smallest weight can be removed from the heap in O(log M) time. If k edges have to be considered to select N — 1 edges for inclusion in MSTE, then the complexity of the selection process is O(M + klog M). Therefore, the total complexity is O(M + N + k log M). An example of execution of Kruskal’s algorithm is shown in Fig. 12. Each edge is labeled with its weight and its number (shown in brackets). In each pass, the selected edge and the included nodes are shown in the table.
Prim’s Algorithm. For a dense network, when M is of O(N2), an alternative method to find an MST is from Prim (13). This algorithm maintains a tree and adds additional nodes to the tree using minimum cost edges. For this pur
Krushal Algorithm Figure 12. Kruskal’s MST algorithm. 
pose, the minimum distance of each node that is out of the tree is maintained from the tree nodes. Each time a new node is added, the distances ofnodes that are not yet in the tree from the tree change. Therefore, these distances need to be revised. In fact, the distances of the nodes outside the partial tree from the newly inserted node only need to be considered as that is the only change in the tree. The algorithm has a complexity of O(N2). We need N passes, one each to select N nodes to be included in the tree. Each time we need to find a node with minimum distance (this is an O(N) procedure) and update distances of all other nodes after considering the new node (another O(N) procedure if the distances are maintained in the adjacency list). Both O(N) procedures can be performed in O(d) if the maximum degree of each node is only d because we only need to consider d neighbors of the new node introduced in the tree. Thus, the overall complexity of the procedure is O(dN).
Constrained MST. The MST computation may be constrained using some optimality criteria or requirements. In the case of constrained MST computation, the Selection of edges is constrained using appropriate selection criteria consistent with the specified constraints. For example, in the previous algorithm, it is assumed that the weight of an edge is the only criterion. But the new constraint may be that no node can have more than a certain number ofedges connected to it. In that case, the algorithm may have to decide on a selectable candidate differently. If a node already has a given number of edges originating from it, then no more edges connected to that node may become part of the solution.
Conical LogSpiral Antennas
A very interesting and useful phenomenon occurs when the arms of a logspiral antenna are developed on the surface of a cone, as shown in Fig. 9, particularly a cone with small apex angle (200). As the apex angle decreases, the radiation becomes confined more and more to the halfspace in the direction the tip of the cone is pointing. The measured patterns displayed in Fig. 10 show that, when 60 < 15°, the frontto – back ratio is very high. This unidirectional pattern is needed for many applications and is the principal contributing factor to the popularity of conical logspiral antennas.
The conical logspiral can be fed by bonding the feed cable to one arm in a manner similar to the feed of the planar log
Figure 9. Parameters of a conical logspiral antenna and the coordinates used to describe radiation patterns. After J. D. Dyson, The characteristics and design of the conical logspiral antenna, IEEE Trans. Antennas Propag., AP13: 7C, 1965. 
E9
Eф
Figure 8. Measured radiation patterns of a twoarm, balanced, log – spiral antenna, r = axial ratio. After J. D. Dyson, The equiangular spiral antenna. IRE Trans. Antennas Propag., AP7; 2C, 1959.
ф = 0 
ф = 90° 
Ев
Еф
/7 u V V 
j) 
// 

/] 

1 V 
У 
Figure 10. Radiation patterns of a conical logspiral antenna showing increasing fronttoback ratio with decreasing cone angle, в0. After J. D. Dyson, The unidirectional equiangular spiral antenna. IRE Trans. Antennas Propag., AP7: 4C, 1959.
fire. As frequency changes, the active region moves, maintaining constant size in wavelength measure.
When the antenna is wrapped more loosely (a = 60°), the active region is broader and includes turns that are phased away from backfire toward broadside. This accounts for the broader beams. This effect becomes even more pronounced as the wrap is loosened further (a = 45°). Fluctuation in the nearfield amplitude in Fig. 11 is caused by the probe passing close to conductors of the spiral. The dashed lines show smoothed data more indicative of the behavior of the near field along the arm rather than that along a radial line.
The plots of phase versus displacement in Fig. 11 illustrate how the phasing is affected by the increasing length of conductor in each successive turn of the logspiral. Appreciable slope is seen in the active region for a = 80°. As the slope in the active region decreases, more turns are included in the active region and some of these turns are phased in directions slightly removed from backfire. The bandwidths of antennas with wide active regions are smaller for a given physical size than those of antennas with more limited active regions.
Apparently because of the effective small wrap angle of conical logspiral antennas, the pattern performance does not depend greatly upon the width of the arms. In fact, for moderate bandwidths it is possible to eliminate the flat conductors altogether if a dummy cable is used on the second arm as recommended to preserve the symmetry.
spirals. However, because the cable so used may be much longer than for the planar antennas, cable loss may be unac – ceptably high, particularly at frequencies above 1 GHz. It is possible to bring a feed line along the interior axis of the cone with minor effect on the performance of the antenna. A singlefeed cable can be used with a broadband balun at the feed point, or twin cables can be excited with 180° phase difference by a hybrid network located at the base of the cone.
Figure 11. Relative amplitude and phase of magnetic fields (currents) measured along the surface of conical logspiral antennas (2в0 = 15°, S = 90°) and corresponding radiation patterns. After J. D. Dyson, The characteristics and design of the conical logspiral antenna. IEEE Trans. Antennas Propag., AP13: 7C, 1965. 
Since the arms of a conical logspiral appear to be wrapped more tightly than those of the corresponding planar antenna, the radiation patterns are more nearly rotationally symmetric. Even though the pattern rotates with frequency, the variation in beamwidth is hardly noticeable. A conical logspiral with в0 = 10° and a = 73° was observed to have halfpower beamwidths of 70° for electric field polarized in the в direction and 90° for electric field polarized in the ф direction. The result is radiation that is very nearly circularly polarized over much of the beam.
Much of the behavior of the radiation pattern can be understood in light of knowledge of the near field. Figure 11 shows measured amplitude and phase of the magnetic field close to one arm of conical logspirals that differ only in angle of wrap, a. For the tightly wrapped case (a = 80°) the near field displays a single dominant peak beyond which occurs a precipitous drop. The measured phase is smooth and indicative of the phase progression of a wave traveling toward the feedpoint (i. e., radiation in the backfire direction). Most of the radiation takes place in the vicinity of the peak of the near field, the socalled active region. The radiation in the backfire direction results in the rapid decay in the near field. When the active region is relatively small, encompassing only a few turns of the spiral, most of the radiation is phased for back
Several parameters are required to describe a single conical logspiral antenna. The pattern shape is most dependent upon the apex angle, 2t0, of the cone. However, the arm width, S, and the wrap angle, a, can also have an effect. Radiation patterns for several values of 2t0, S, and a are shown in Fig. 12. As to be expected, tightly wrapped antennas have the smoothest, most symmetrical patterns for a wider range of 2в0 and S. However, the fronttoback ratio is best for the smallest apex angle, 2в0 = 15°. Antennas with arm widths given by S = 90° have patterns superior to those with either wider or narrower arms. Infinite planar spiral structures with S = 90° are selfcomplementary. Although the argument for constancy of impedance applies only to such planar cases, evidently there are other advantages to be gained by applying the selfcomplementary condition.
The measured input impedance does not change much over the frequency band of good patterns, a span of 20 : 1 or more in frequency is possible. Table 1 gives data for the mean impedance and SWR as a function of the cone angle, t0.
ECOLOGY OF MEDICAL IMAGING
What is an ideal medical imaging instrument? Cost, convenience, and safety are important considerations. However, the most important issue is whether the imager can provide definitive information about the medical condition of the patient. To have a better appreciation of these issues, it is necessary to review the conditions under which medical imaging is performed.
Medical imaging may be used as part of therapeutic procedures, to detect disease where no symptoms are present (screening), or to monitor the process of healing. The most common use of medical imaging is for differential diagnosis, where a patient exhibits symptoms that can be caused by one
Table 1. Modalities, Subdivisions, and Annual Case Load in a Large Teaching Hospital

or more diseases, and there is a need to ascertain the specific condition so that proper treatment can be applied. The physician orders a specific imaging procedure, which may require patient preparation (for example, ingestion of a contrast material), the use of specific imaging instruments and settings (X rays of a certain voltage, intensifying screens, and films), and proper positioning of the patient. Most procedures are performed by properly trained technicians, but some very dangerous or invasive procedures, such as injecting contrast agents directly into the heart, are performed with direct participation of a medical doctor. The majority of imaging procedures are administered by a radiology department of a hospital, and images are interpreted (read) by radiologists, medical doctors who undergo postMD training (residency) in this specialty. The choice of procedure for a given medical problem is based on medical knowledge and on guidelines promulgated by medical and/or health insurance organizations. We see that most medical imaging tasks are highly specialized and are performed in a very structured setting.
Table 1 shows the divisions of a radiology department of a large teaching hospital. The divisions have been grouped by modality. It also shows the number of studies performed in one year in each modality. It is clear that Xray imaging is still the predominant technique. Other modalities are growing more rapidly, but none are likely to displace X rays in the near future.
Medical imaging procedures and modalities exist in an extremely competitive environment. For example, heart disease may be diagnosed either with angiography or nuclear medicine. Each procedure, to survive, must fit an ecological niche: it must satisfy a need where it has advantages over other methods. The same issue arises when new image processing techniques are introduced: images produced with these procedures must have an advantage over other imaging procedures. The advantage will probably be application dependent. An image processing technique that is effective for mammography may not be of much value for cardiology, and vice versa.
FloydWarshall method
Another method to compute the shortest paths between all node pairs is from Floyd and Warshall with a computational complexity of O(N3). In this method, udefines the length of the shortest path from node i to j such that it does not pass through nodes numbered greater than m — 1 except nodes i and j. Then
1
uii = a
and
П+1 
u 
= min{u",u" + umm]}
ufj+1 is the shortest path length matrix. Also, um+1 = 0 for all i and for all m.
This procedure has N(N — 1)(N — 2) equations, each of which can be solved by using N(N — 1)(N — 2) the additions and N(N — 1)(N — 2) comparisons. This order of complexity is the same as that for Bellman’s method (also known as the BellmanFord method as it was independently discovered by two researchers), which yields the shortest path only from a single origin. The Dijkstra method can also be applied N times, once from each source node, to compute the same shortest path length matrix. This process takes only N(N — 1)/2 additions for each pass, for a total of N2(N — 1)/2 additions, but again housekeeping functions in Dijkstra’s method make it noncompetitive.
The computation in the FloydWarshall method proceeds with u1 = A and Um+1 is obtained from Um by using row m and column m in Um to revise the remaining elements. That is, uij is compared with uim + umj and is replaced if the latter is smaller. Thus, the computation can be performed in place and is demonstrated in the following for the graph in Fig. 6a.
/ 0 
100 
40 
30 
ж 
0 
100 
40 
30 
ж 
0 
100 
40 
30 
120 

100 
0 
ж 
ж 
20 
100 
0 
140 
130 
20 
100 
0 
140 
130 
20 

40 
ж 
0 
20 
30 
А1 = 
40 
140 
0 
20 
30 
А2 = 
40 
140 
0 
20 
30 
30 
ж 
20 
0 
ж 0 
30 
130 
20 
0 
ж 0 
30 
130 
20 
0 
ж 

оо 
20 
30 
00 
V 00 
20 
30 
00 
120 
20 
30 
00 
0 
A = 
0 
100 
40 
30 
70 
0 
100 
40 
30 
70 
0 
90 
40 
30 
70 

100 
0 
140 
130 
20 
100 
0 
140 
130 
20 
90 
0 
50 
70 
20 

40 
140 
0 
20 
30 
А4 = 
40 
140 
0 
20 
30 
А5 = 
40 
50 
0 
20 
30 
30 
130 
20 
0 
50 
30 
130 
20 
0 
50 
30 
70 
20 
0 
50 

70 
20 
30 
50 
0 
70 
20 
30 
50 
0 
70 
20 
30 
50 
0 
A = 
Multiple Shortest Paths
Many times it is useful to be able to compute additional shortest paths between a node pair, which may be longer than the first shortest path but still short in case the first shortest path is not available. The first path may be congested or may have a failed link or a node. The problem can be constrained by specific requirements such as allowing
or not allowing repeated nodes and links or specific nodes and/or links. Specific methods exist to compute alternative shortest paths for all cases (see reference 14). One specific case with respect to fault tolerance is nonavailability of a node or a link. Such a path can be computed by removing the specific node or link in the original graph (removal of a node also removes all associated links) and then using the same shortest path algorithm. In another scenario, we may want another path that is mutually exclusive of the first path. In that case, all nodes and links have to be removed from the original graph before computing another shortest path. The algorithm to be used in these cases is the same as already stated.