投影技術可擴展一般螢幕顯示器之外的顯示應用，像是空間藝術、擴增實境，以及死角減消應用等。一般投影時，為了不失真，需嚴格要求投影機正對投影幕。受限於擺設投影機與顯示幕之空間條件限制時，投影之前就必須先經過校正的前置處理，否則會有失真的現象。對於簡單斜面投影情境，電腦圖學的前置處理，可矯正非理想佈局下的失真狀況，然而對於更複雜的曲面投影，利用方格近似的作法就有精確度與計算量上的挑戰。據此，本論文採用投影影像可逆性原理發展應用到斜面與曲面投影平面上的校正技術，可逆性原理指的是：給定一任意投影幕P與任兩點：A為投影點、B為觀測點。當由A點將一圖像I投向P而被B點觀測為呈像I’如果置換A與B上投影與觀測的角色，即由從B點逆向投影I’到原投影幕上，則在A點將觀測到其原先投影出去的圖像I。在此可逆性原理的基礎上，本論文分別就斜向以及曲面投影面的校正進行探討；相較於文獻上其他利用方格分割再取逆投影矩陣的作法，本「利用投影影像可逆性原理之斜面與曲面投影校正技術」方法所取得的校正模版直接反應投影點到投影面再到觀測點之間的關係，而免除計算時累進的誤差，以及在曲面投影的狀況下，相鄰方格可能無法緊密貼合的問題。除了以上理論基礎之外，本論文並實現「自動化斜面校正與曲面投影校正技術」的裝置與自動校正程式，能夠在任曲面投影幕上展現無失真的投影效果。當作成果展現的一部份，我們利用此校正方法實現一牆壁上之斜面投影應用：隨視角調整之互動式虛擬實境投影技術。

Projection extends much more display possibilities beyond what traditional monitors can offer. However, strict condition requiring the projector directly facing a plane screen needs be satisfied; otherwise, distortion will occur. Distortion occurs in incline plane projection where the projector is not facing in normal direction and more complicated in the cases when the screen is a curved surface. To compensate, a rectified pattern is needed as a pre-processing before an intended image is projected to ensure no distortion. Esisting solutions for the inclined planes involve warp perscpective operation; solutions for curved surfaces are by approximating the curve surface as a collection of small inclined planes or grids, rendering the rectification for curved surface a repetitive rectification process on each of these constituent inclined plane. Alternatively, this study proposes an auto-calibration approach based on the reversibility principle of projection image. To wit, the principle states that, if an image Im from point A is projected to a screen and observed as Im’by point B, then, by reversibility, when Im’is projected from B to the same projection screen, the original image Im will be observed at A. With the original image being a checkboard and A the point of observation and B the position for projection, we are able to calibrate rectification patterns for projections on both inclined planes and curved surfaces. In addition to mathematical support for this approach, this study also presents an auomatic calibration solution that together with projector and camera sertup calibrates rectification patterns for both inclined planes and curved surfaces. The auto-calibration process requires no repetitive matrix manipulations which are prone to numerical errors and ensures no overllaps or gaps among approximated grids. To demonstrate, rectified projections to inclined wall, corner of ceilings, and cylinderal surfaced buck are shown. The technique prsented in this paper can also be extended to applications in interactive virtual reality, as is shown by the demonstration of Wallscape, an interactive viewer dependent wall projection technique.

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