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 CryoSat-2 was approved by ESA, now scheduled for launch in 2009. The Cryosat orbit will have high-inclination (92°) and a long-repeat period (369 days, with a 30-day sub-cycle), 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 (pulse-limited) 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 Log-Spiral Antennas

A very interesting and useful phenomenon occurs when the arms of a log-spiral 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 half-space in the direc­tion the tip of the cone is pointing. The measured patterns displayed in Fig. 10 show that, when 60 < 15°, the front-to – back ratio is very high. This unidirectional pattern is needed for many applications and is the principal contributing factor to the popularity of conical log-spiral antennas.

The conical log-spiral can be fed by bonding the feed cable to one arm in a manner similar to the feed of the planar log-

Conical Log-Spiral Antennas

Figure 9. Parameters of a conical log-spiral antenna and the coordi­nates used to describe radiation patterns. After J. D. Dyson, The characteristics and design of the conical log-spiral antenna, IEEE Trans. Antennas Propag., AP-13: 7C, 1965.

E9

Figure 8. Measured radiation patterns of a two-arm, balanced, log – spiral antenna, r = axial ratio. After J. D. Dyson, The equiangular spiral antenna. IRE Trans. Antennas Propag., AP-7; 2C, 1959.

Conical Log-Spiral Antennas

Conical Log-Spiral Antennas

ф = 0

Conical Log-Spiral Antennas

ф = 90°

Conical Log-Spiral Antennas

Ев

Еф

Conical Log-Spiral Antennas

Conical Log-Spiral Antennas

/7

u

V

V

j)

//

/]

1 V

У

Figure 10. Radiation patterns of a conical log-spiral antenna show­ing increasing front-to-back ratio with decreasing cone angle, в0. After J. D. Dyson, The unidirectional equiangular spiral antenna. IRE Trans. Antennas Propag., AP-7: 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 near-field 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 log-spiral. 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 log-spiral 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.

Conical Log-Spiral Antennas

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.

Conical Log-Spiral Antennas

Figure 11. Relative amplitude and phase of magnetic fields (cur­rents) measured along the surface of conical log-spiral antennas (2в0 = 15°, S = 90°) and corresponding radiation patterns. After J. D. Dyson, The characteristics and design of the conical log-spiral an­tenna. IEEE Trans. Antennas Propag., AP-13: 7C, 1965.

Since the arms of a conical log-spiral 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 log-spiral with в0 = 10° and a = 73° was observed to have half-power 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 log-spirals 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 so-called 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 log-spiral 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 front-to-back 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 self-complementary. Although the argument for constancy of impedance applies only to such planar cases, evi­dently there are other advantages to be gained by applying the self-complementary 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

Modalities

Subspecialities

Studies

X ray

Chest, bone, gastrointestinal,

157,958

Computer tomography

genitourinary, neurological, neurosurgical, angiography, mammography Body, neurological (head)

14,343

Nuclear medicine

10,772

Magnetic resonance

12,380

imaging

Ultrasound

15,844

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 post-MD 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 X-ray 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.

Floyd-Warshall 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 Bellman-Ford 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 Floyd-Warshall 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.