在近幾年來，快速響應矩陣碼(QR碼)被廣泛地應用在商業及工業，受歡迎的程度與相關的技術也都成長地非常快速，而在相關的技術中，QR碼的影像處理是一個很重要的課題，在本論文中，我們嘗試解決在QR影像解碼中會遇到的一種問題，就是當QR影像被歪斜地貼在柱面上，然後我們對著QR碼拍下一張QR影像，這張QR影像已經不是預想中QR影像會被拍成矩形的形式，而是由於相機截取影像的方式而使QR影像變得扭曲，這種扭曲造成原始QR影像嚴重的失真，並且導致QR碼的解碼失敗，而本論文將探討並處理這樣的問題。
在一般認知中，QR碼是由黑白方塊所組成的矩陣，而這些方塊又被稱為模組(module)，本論文嘗試把柱面QR影像的扭曲模組切割出來，並且準備好另一張方正的空白QR影像，接著將每一個扭曲模組的顏色(或稱亮度，在QR影像只有黑白)複製到空白QR影像中相對應的位置，直到這張方正的空白QR影像被填滿之後，其結果就會如一般的QR影像沒有任何的扭曲，雖然有時候模組上的顏色可能會因為切割失誤而造成顏色不準確。本段落所敍述這種將扭曲QR影像轉換成方正QR影像的方法又被稱為QR影像的回正。
仔細觀察QR影像可以很清楚地發現到，QR影像的架構是以水平以及垂直的直線所分割而成，這些直線很明確地將QR影像分隔成一塊一塊的模組，而由於被貼在柱面上，這些直線會被扭曲而成曲線，QR影像回正的基本概念就是把這些曲線套進圓錐曲線(conic)的模型，然後用圓錐曲線來分割柱面QR影像的模組，也是我們的方法被稱為”圓錐分割”的原由，更進一步說明，我們在QR影像的前處理中使用索貝爾邊緣檢測(Sobel edge detection)尋找邊緣點，這個過程中會應用到數位影像處理(DIP)中數學型態學(mathematical morphology)的影像膨脹(dilation)與影像侵蝕(erosion)，並且將QR影像的水平與垂直曲線分成一段段的連結區塊(CC)，而我們會將屬於同一群曲線的CC套進圓錐曲線的模型中使用最小平方誤差解(LSE solution)做圓錐曲線擬合(conic fitting)，最後擬合出來的圓錐曲線將可以把整張QR影像的扭曲模組一塊一塊切割出來，最後在方正QR影像相對應的位置填上扭曲模組的顏色，就能完成QR影像回正的動作，經多次的實驗，我們用上述的方法將大量解碼失敗的QR影像成功回正，證明這個方法的確有非常好的效果。

In recent years, QR codes have found a wide range of applications in business and industry. Moreover, their popularity and function are still growing (fast). Therefore, the processing of the images of QR codes is an important task. In this thesis, we try to solve a problem encountered in QR decoding when the QR code is pasted on the surface of a cylindrical container. The pasting can even be tilted. When a picture is taken on such a QR code, the picture is no longer in a rectangular shape, which is supposedly the “right” shape that any QR code should take, due to warping from the image capturing mechanism by the camera. This warping translates into serious distortion of the original QR-code image (also referred to as QR image, for short) and thus causes failure in its decoding. In this thesis, we try to deal with this kind of problem.
Realizing that originally a QR code is a square array of black and white modules, each of which is a square-shaped cell, we try to segment the distorted QR image into separate cells, which are typically non-rectangular. Prepare a blank template alongside. Then, the color (more precisely speaking, intensity (black or white)) of each cell is duplicated into the corresponding cell in the template. After all cells in the template are thus painted (to be black or white), we have a square array of black and white modules. The result looks like a normal QR image, which is not distorted in shape, although some of its modules may be wrong in color as compared to the original QR code. The transformation of the distorted QR image into a square array of black and white modules, as described above, is referred to as QR image rectification.
It is obvious that a QR code contains a set of horizontal lines and vertical lines. It is exactly those lines that segment the QR code into modules. When those lines are warped due to the cylinder-pasting, they of course are no longer straight lines. Instead, they become curves. The basic idea for our QR image rectification is to model those curves as conic sections and then use those conic sections to segment the QR image into cells. It is for this reason that we call our method by the name of “conic segmentation”. More specifically speaking, we pre-process the QR image to find its edge points (by Sobel edge detection). During the process, we also need to perform dilation and erosion, etc., to decompose the QR image into some connected components. Then, we will fit the edge points that are determined to have belonged to the same curve into a conic section, by the least-square-error fitting. Finally, those conic sections would naturally segment the whole QR image into many small cells, wherein each cell corresponds to a QR module. In this way, the rectification is completed.
Experiments show that the proposed scheme is effective, in the sense that lots of failed QR decoding becomes successful after the rectification.

[1]“Information technology – automatic identification and data capture techniques – bar code symbology – QR code, ”International Standard ISO/IEC 18004, ISO/IEC, 2006
[2]LIAO Zhao-lai, HUANG Ting-lei, WANG Rui, ZHOU Xiao-yan, “A Method of Image Analysis for QR code Recognition,” Proc. IEEE Int. Conf. Intelligent Computing and Integrated Systems, Vol.1, pp.250 – 253 ,Oct. 2010
[3]Zou Xiong, He Cuiqun, Liu Guodong, Liang Zhijun,”A Binarization Method of Quick Response Code Image,” Proc. IEEE Int. Conf. Signal Processing Systems, Vol.3, pp.317 – 320, July. 2010
[4]Jiejing Zhou, Yunfei Liu, Peng Li “Research on Binarization of QR code Image,” Proc. IEEE Int. Conf. Multimedia Technology, Vol.1, pp.1 – 4, Oct. 2010
[5]Yue Liu, Ju Yang, Mingjun Liu “Recognition of QR code with mobile phones,” Proc. IEEE Int. Conf. Control and Decision Conference, Vol.1, pp. 203 – 206, July 2008
[6]Nobuyuki Otsu “A threshold selection method from gray-level histograms,” Proc. IEEE Trans. Sys., Man., Cyber. Vol.9, pp. 62 – 66, Jan 1979
[7]Irwin Sobel, “History and Definition of the Sobel Operator” , 2014
[8]Gonzalez, R.C. and Woods R.E., “Digital Image Processing, 3rd Edition,” Prentice-Hall, 2008.
[9]Hartshorne, Robin. “Foundations of Projective Geometry, 2nd ed.” Ishi Press, 2009
[10]Kolman, Bernard, Hill, David R., “Elementary Linear Algebra with Applications (9th ed.),” Prentice Hall, May, 2007
[11]http://unity3d.com/
[12]Michael Kass, Andrew Witkin, Demetri Terzopoulos, “Snakes: Active Contour Models,” Proc. Int. Journal of Computer Vision, Vol.1, pp. 321-331, 1988.
[13]Kalman, R. E. “A New Approach to Linear Filtering and Prediction Problems, “ Transactions of the ASME - Journal of Basic Engineering Vol. 82: pp. 35-45 , 1960
[14]Kalman, R. E., Bucy R. S., “New Results in Linear Filtering and Prediction Theory,” Transactions of the ASME - Journal of Basic Engineering Vol. 83: pp. 95-107 ,1961