This paper presents some mathematical results concerning the measurement of motion of contours. A fundamental problem of motion measurement in general is that the velocity field is not determined uniquely from the changing intensity patterns. Recently Hildreth & Ullman have studied a solution to this problem based on an Extremum Principle [Hildreth (1983), Ullman & Hildreth (1983)]. That is, they formulate the measurement of motion as the computation of the smoothest velocity field consistent with the changing contour. We analyse this Extremum principle and prove that it is closely related to a matching scheme for motion measurement which matches points on the moving contour that have similar tangent vectors. We then derive necessary and sufficient conditions for the principle to yield the correct velocity field. These results have possible implications for the design of computer vision systems, and for the study of human vision.

Computational theories of structure-from-motion and stereo vision only specify the computation of three-dimensional surface information at points in the image at which the irradiance changes. Yet, the visual perception is clearly of complete surfaces, and this perception is consistent for different observers. Since mathematically the class of surfaces which could pass through the known boundary points provided by the stereo system is infinite and contains widely varying surfaces, the visual system must incorporate some additional constraints besides the known points in order to compute the complete surface. Using the image irradiance equation, we derive the surface consistency constraint, informally referred to as no news is good news. The constraint implies that the surface must agree with the information from stereo or motion correspondence, and not vary radically between these points. An explicit form of this surface consistency constraint is derived, by relating the probability of a zero-crossing in a region of the image to the variation in the local surface orientation of the surface, provided that the surface albedo and the illumination are roughly constant. The surface consistency constraint can be used to derive an algorithm for reconstructing the surface that "best" fits the surface information provided by stereo or motion correspondence.

In the last seven years a new computational approach has led to promising advances in the understanding of biological visual perception. The foundations of the approach are largely due to the work of a single man, David Marr at M.I.T. Now, after his death in Boston on November 17th 1980, research in vision will not be the same for the growing number of those who are following his lead.

We study the distribution of combinatorial statistics that exhibit a structure of crossings and nesting in various discrete structures, in particular, in set partitions, matchings, and fillings of moon polyominoes with entries 0 and 1. Let pi and y be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integers l, m >̲ 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to pi, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (pi, l) and T (y, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (pi, l) and T (pi, l) for l = 0; 1, then it coincides on T (pi, l, m) and T (y, l, m) for all l, m >̲ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. Moreover, we give a bijection between partially directed paths in the symmetric wedge y = +̲ x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of E. J. Janse van Rensburg, T. Prellberg, and A. Rechnitzer that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = +̲ x with k north steps is equal to the number of matchings on [2n] with k nestings. Furthermore, we propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s, A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations.

Thesis (M. Env. Des.)--University of Calgary, 1998. / Includes bibliographical references (leaves 36-40).

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The influences of the background experiences of five pre-service Science teachers on their beliefs about diversity were the focus for this study. These individuals were followed throughout their teacher preparation program. The data for this study consisted of interviews, conducted before and after entering the field. Data also consisted of any relevant written assignments. The data for this study were analyzed according to emerging themes, depicting initial beliefs and any changes in the beliefs occurring over time. The results of this study were organized into vignettes, telling each story from before and after the students entered the program. Three themes emerged from an analysis and interpretation of the vignettes: (1) Early life experiences shaped the pre-service teachers' sense of identity and influenced their beliefs on diversity, (2) Experiences with diversity influenced pre-service teachers' philosophy of teaching, and (3) Experiences with diversity during the teacher preparation program challenged or confirmed pre-service teachers' preexisting beliefs. The implications from this study suggest that pre-service teachers need challenging experiences in diverse classroom settings that will promote an expansion of their beliefs, enabling them to cross cultural borders. / Ph. D.

The impact of forest roads and skid trails on stream health is being increasingly scrutinized. Forest roads and skid trails have repeatedly been identified as forest operations having the greatest potential to produce sediment by way of non-point source pollution. The stream crossing portion of a skid trail is where sediment delivery is most likely to occur. Forestry Best Management Practices (BMPs) have been developed by most states to reduce both erosion and sedimentation. In general, BMPs have been proven to be effective. Few studies have quantified the impact of various levels of BMPs on sedimentation. In this study, three replications of three skid trail stream crossing BMP treatments were monitored following skidder bridge removal to determine their efficacy in reducing sedimentation: slash, mulch, and mulch plus silt fence. Water samples were collected upstream and downstream of each crossing daily for one year following timber harvesting. Samples were evaluated for total suspended solids. Results indicate that both slash and mulch treatments applied to the stream crossing approach after skidder bridge removal are effective at reducing stream sedimentation after harvest. The mulch plus silt fence treatment allowed the most sediment to enter the stream at the approach, perhaps due to silt fence installation disturbances. We do not recommend using silt fences directly adjacent to a stream bank, if other alternatives exist. / Master of Science

Stream crossings are an integral component of forest road systems that provide access for timber harvesting and silvicultural activities. Stream crossings and their associated approaches are often the most critical point of concern for water quality along forest roads. Several types of crossings are used for extracting timber, but limited studies actually compare different types of stream crossings with regard to their effect on water quality. The objectives of this study were to examine four different stream crossing structures: 1) steel bridges, 2) pole bridges (pipe with poles), 3) standard culverts, and 4) re-enforced fords (with GeoWeb or Geotextile) to determine the influence of stream crossing type on water quality and to evaluate erosion associated with stream crossing approaches. We also evaluated each site at four different time intervals to determine if water quality was more affected during different stages of the operations. Prior to operational timber harvests, we identified six replications for each type of crossings (4 fords) and collected data at four time intervals: 1) prior to reopening or installation of crossing, 2) after crossing installation, 3) during harvest operation, and 4) after road closure. Potential erosion rates from approaches to the crossings were estimated by collecting the road/site information necessary to estimate erosion with the Water Erosion Prediction Project for forest roads (WEPP) and the forest version of the Universal Soil Loss Equation (USLE). In-stream water samples were collected at fixed locations above and below each crossing and were evaluated for total dissolved solids (TDS), pH, conductivity, water temperature, and total suspended solids (TSS) or sediment concentration. Steel bridge crossings generally caused the least amount of water quality disturbance. Model-generated estimates of erosion demonstrated that culvert crossings were associated with the highest average soil loss potential. Although steel bridge crossings had the best overall results, pole bridges proved to be a viable option for ephemeral or intermittent streams due to low potential of soil loss. Ford crossings were found to impact water quality indicators, but showed a decrease in total dissolved solids (TDS) after installation, prior to harvest. Overall, the steel skidder bridges were generally the best crossing type, but any of the crossings can be used effectively with minimal impact under specific site conditions and with judicious installation, use, and closure. Road/skid trail location and adherence to existing road grade, water control, cover, and closure best management practices are critical for protection of water quality at stream crossings. / Master of Science