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This thesis addressed the problem of recognizing 3-D objects, using shape information extracted from range images, and parameterized volumetric models. The domains of the geometric shapes explored is that of complex curved objects with articulated parts, and a great deal of similarity between some of the parts. These objects are exemplified by animal shapes, however the general characteristics and complexity of these shapes are present in a wide range of other natural and man-made objects. In model-based object recognition three main issues constrain the design of a complete solution: representation, feature extraction, and interpretation. this thesis develops an integrated approach that addresses these three issues in the context of the above mentioned domain of objects. For representation I propose a composite description using globally deformable superquadratics and a set of volumetric primitives called geons: this description is shown to have representational and discriminative properties suitable for recognition. Feature extraction comprises a segmentation process which develops a method to extract a parts-based description of the objects as assemblies of defoemable superquadratics. Discontinuity points detected from the images are linked using 'active contour' minimization technique, and deformable superquadratic models are fitted to the resulting regions afterwards. Interpretation is split into three components: classification of parts, matching, and pose estimation. A Radical Basis Function [RBF] classifier algoritm is presented in order to classify the superquadratics shapes derived from the segmentation into one of twelve geon classes. The matching component is decomposed into two stages: first, an indexing scheme which makes effective use of the output of the [RBF] classifier in order to direct the search to the models which contain the parts identified. this makes the search more efficient, and with a model library that is organised in a meaningful and robust way, permits growth without compromising performance. Second, a method is proposed where the hypotheses picked from the index are searched using an Interpretation Tree algorithm combined with a quality measure to evaluate the bindings and the final valid hypotheses based on Possibility Theory, or Theory of Fuzzy Sets. The valid hypotheses ranked by the matching process are then passed to the pose estimation module. This module uses a Kalman Filter technique that includes the constraints on the articulations as perfect measurements, and as such provides a robust and generic way to estimate pose in object domains such as the one approached here. These techniques are then combined to produce an integrated approach to the object recognition task. The thesis develops such an integrated approach, and evaluates its perfomance inthe sample domain. Future extensions of each technique and the overall integration strategy are discussed.

The study investigated the strategies used in teaching geometry in primary six as well as the perception of teachers on geometry vocabulary teaching, how geometry vocabularies were taught and, lastly, how the teaching of geometry vocabulary influenced primary six learners’ performance in geometry. The Van Hiele Theory of geometrical thinking and the Constructivist Theory of learning guided the study. The study conveniently sampled 250 primary 6 learners and 7 primary 6 mathematics teachers from three privately-owned primary schools in the Greater Accra Region of Ghana. It combined quantitative and qualitative approaches, using O1–X–O2 design. Data collection instruments were 5-point Likert type scale questionnaires (one for teachers, one for learners), a pre-test and post-test of basic geometry, and a semi-structured one-on-one audio-recorded interview of a selected number of learners and all seven teachers. An intervention was carried out in-between the pre-test and post-test, where the researcher taught geometry vocabulary to participants. Quantitative data were analysed using tables, charts, and simple tests while the qualitative analysis involved the transcription of interviews that were coded, categorised and themed. The study found that geometry vocabularies were not taught and that the most commonly used strategy for teaching geometry was the drawing of 2-D shapes and models of 3-D objects on the board. The pre-test and post-test scores were analysed using a paired t-test and the results indicated that the intervention had a positive effect. The qualitative and quantitative results confirmed that the teaching of geometry vocabulary improved learners’ performance in geometry. The study developed a prototype lesson plan for teaching 3-D objects, a geometry vocabulary activity sheet, a sample assessment for prisms and pyramids and recommends a curricular reform to inculcate the teaching of geometry vocabulary in the curriculum with a geometry vocabulary list for learners in each year group, as contribution to knowledge in mathematics education. The study recommends further research to investigate the effect of geometry vocabulary teaching on learners’ performance in geometry across all year groups in the primary school. / Dyondzo a yi lavisisa maendlelo lawa ya tirhisiwaka ku dyondzisa geometry ya tidyondzo ta le hansi ta ka ntsevu, mavonelo ya vadyondzisi eka madyondziselo ya marito ya geometry, tindlela leti tirhisiweke ku dyondzisa marito ya geometry xikan’we ni ndlela leyi madyondziselo ya marito ya geometry ya khumbheke matirhelo ya vadyondzi va tidyondzo ta le hansi ta ka ntsevu. Dyondzo ya ndzavisiso yi leteriwile hi ehleketelelo ra Van Heile ra maehleketelelo ra ndlela ya geometry ni ndlela yo dyondzisa leyi pfumelelaka vadyondzi ku vumba vutivi ku nga ri ntsena ku teka vutivi ku suka eka mudyondzisi. Dyondzo ya vulavisisi yi hlawurile vana va 250 va tidyondzo ta le hansi ta ka ntsevu na 7 wa vadyondzisi va tnhlayo ta tidyondzo ta le hansi ta ka ntsevu kusuka eka swikolo swinharhu swo ka swi nga ri swa mfumo e Greater Accra etikweni ra Ghana. Yi hlanganisile qualitative na quantiutative aapproach, yi tirhisa O1–X–O2 design. Switirhisiwa swo hlengeleta data a swi ri swivutiso hi muxaka wa 5-point scale(yin’we ya vadyondizi, yin’we ya vadyondzi), xikambelwana xo rhanga na xo hetelela xa geometry ya masungulo, xikan’we na nkandziyiso wa mburisano wa vanhu vambirhi eka nhlayo ya vadyondzi ni vadzyondzisi hinkwavo va nkombo. Ntirho wo nghenelerisa wu endliwile exikarhi ka xikambelwana xo rhanga ni xo hetelela laha mulavisisi a nga dyondzisa marito ya geometry eka vanhu lava ngheneleleke. Quantitative data yi hleriwile hi ku tirhisa matafula, ti charts ni swikambelwana swo olova kasi vuhleri bya qualitative byi nghenise kutsariwa ka miburisano leyi hundzuluxiweke yi nyika tinhlamuselo leti tumbeleke. Leti vekiweke hi ku ya hi mintlawa ni maendlelo ya tona. Dyondzo ya ndzavisiso yi kume leswaku marito ya geometry a ya dyondzisiwanga ni leswaku maendlelo yo toloveleka ya ku dyondzisa geomeyry i ya drawing ya xivumbeko xa 2-D ni mfanekiso wa nchumu wa 3-D eka bodo. Mbuyelo wa Xikambelwana xo sungula na xo hetelela wu hleriwile hi ku tirhisa t-test (xikambelwana xa T) lexi hlanganisiweke naswona mbuyelo wu komba leswaku maendlelo himkwawo ya vile ni xiave lexinene. Mbuyelo wa Qualitative na Quantitative wu tiyisisile leswaku ku dyondzisiwa ka marito ya geometry swi antswisa matirhelo ya vadyondzi eka dyondzo ya geometry. Dyondzo ya vulavisisi yi antswisile kumbe ku kurisa prototype lesson plan ya ku dyondzisa 3-D objects, sheet ya migingiriko ya marito ya geometry na ku bumabumela circular reform ku dyondzisa madyondziselo ya marito ya geometry eka kharikhulamu leyi ng na nxaxamelo wa marito ya geometry ya vadyondzi eka ntlawa wa lembe na lembe, ta ni hi mpfuneto wa vutivi eka dyondzo ya tinhlayo. Dyondzo ya vulavisisi yi bumabumela leswaku vulavisisi byi ya emahlweni ku lavisisa xiave xa madyondziselo ya marito ya geometry eka matirhelo ya vadzyondzi eka geometry eka malembe ni mintlawa hinkwayo exikolweni xa le hansi. / Thuto ye e nyakisisitse ditsela tseo di somiswago go ruteng ga geometry go mphato wa bo tshelela, temogo ya barutisi go ruteng tlotlontsu ya geometry, tsela yeo ditlotlontsu tsa geometry di rutilwego ka gona go akaretswa le, sa mafelelo, ka mokgwa wo thuto ya tlotlontsu ya geometry e tutueditsego mabokgoni a barutwana ba mphato wa bo tshelela go dithuto tsa geometry. Thuto ya van Hiele ya geometrical thinking le ya constructivist theory of learning di hlahlile thuto ye. Thuto ye e somisitse ga bonolo mohlala wa barutwana ba 250 ba mphato wa 6 le barutisi ba dipalo ba supa ba go ruta mphato wa 6 go tswa dikolong tsa tlase tse tharo tsa go ikema seleteng sa Greater Accra Region of Ghana. Thuto ye e kopantse mekgwa ya bontsi/dipalopalo (quantitative) le boleng (qualitative), go somiswa tlhamo ya O1-X-O2. Didiriswa tsa kgobaketso ya boitsebiso e bile 5-point Likert Type Scale Questionnaire (ye tee ya barutisi, ye tee ya barutwana), moleko wa pele le moleko wa morago wa geometry ya motheo, le poledisano yeo e gatisitswego ya tlhamego ya sewelo (semi-structured) ya barutwana bao ba kgethilwego ga mmogo le barutisi ka moka ba supa. Thekgo e ile ya phethagatswa/fiwa magareng ga moleko wa pele le moleko wa morago moo monyakisisi a rutilego tlotlontsu ya geometry go batseakarolo. Boitsebiso bja bontsi (quantitative data) bo sekasekilwe ka go somisa ditafola, ditshate, le teko e bonolo mola ditshekatsheko tsa boleng (qualitative analysis) di akareditse go ngwalolla dipoledisano tseo di thulagantswego, tsa hlophiwa le go beakanywa ka sehlogo. Thuto ye e itullotse gore ditlotlontsu tsa geometry ga se tsa rutwa ebile mekgwana yeo e somisitswego ya setlwaedi go ruta geometry ebile go thala dibopego tsa 2-D le mehlala ya didiriswa tsa 3-D letlapeng. Dintlha tsa moleko wa pele le moleko wa bobedi di sekasekilwe ka go somisa mokgwa wa go phera moleko wa t (t-test). Dipoelo di supeditse gore thekgo yeo e filwego e bile le khuetso ye botse. Dipoelo tsa bontsi le boleng di netefaditse gore go ruta tlotlontsu ya geometry go kaonafatsa mabokgoni a barutwana dithutong tsa geometry. Nyakisiso ye e tsweleditse lenaneothuto la go dira diteko go ruteng didiritswa tsa 3-D le papetlatshomelo ya tlotlontsu ya geometry gape le go kgothaletsa mpshafatso ya lenaneo-thuto go tsenyeletsa thuto ya tlotlontsu ya geometry ka gare ga lenaneo-thuto gammogo le lelokelelo la tlotlontsu ya geometry ya barutwana go dihlopha tsa mengwageng ka moka. Se e tla ba e le tlaleletso ya tsebo go thuto ya dipalo. Thuto ye e kgothaletsa dinyakisiso tsa go ya pele go nyakolla mafelelo a go ruta tlotlontsu ya geometry go tiro ya, goba dipoelo tsa, barutwana go thuto ya geometry go dihlopha tsa mengwaga ka moka tsa sekolo sa tlase. / Mathematics Education / Ph. D. (Mathematics Education)