Project by Arie Kozak Given photograph with sheet of paper with - PowerPoint PPT Presentation

Project by Arie Kozak

 Given photograph with sheet of paper with text only, infer shape of the surface and plot it in 3d.  Single (infinite) light source from above, using reflectance map (paper is nearly Lambertian 1 surface): 𝑆 𝑞, 𝑟 = 𝑞 2 +𝑟 2 +1  The surface is assumed to be constant in one direction.

 Mark it using personal biological visual system.

 Divide the image into two connected sub- images divided by red border.

 Use thresholding twice: after high pass and original image. Text found in the intersection.

 Constant albido assumption for ink, doesn’t work, use (cubic) interpolation.  Smooth image with Gaussian kernel before to reduce “sharpening effect” (lateral inhibition), and also after.

 Maximum intensity point in image => p = q = 0. Use parabolic approximation according to B.K.P. Horn's chapter 11: 𝐼 𝑦, 𝑧 = 𝐼 0 + 0.5(𝑏𝑦 2 + 𝑐𝑦𝑧 + 𝑑𝑧 2 ) 𝑞 = 𝜖𝐼 𝜖𝑦 = 𝑏𝑦 + 𝑐𝑧, 𝑟 = 𝜖𝐼 𝜖𝑧 = 𝑑𝑧 + 𝑐𝑦 1 2𝐽(𝑦, 𝑧) 2 = 0.5 𝑞 2 + 𝑟 2 + 1 𝐹 𝑦, 𝑧 = = 0.5 𝑏 2 + 𝑐 2 𝑦 2 + 𝑏 + 𝑑 𝑐𝑦𝑧 + 0.5 𝑑 2 + 𝑐 2 𝑧 2 + 1

𝐹 𝑦𝑦 = 𝑏 2 + 𝑐 2 𝐹 𝑧𝑧 = 𝑑 2 + 𝑐 2 𝐹 𝑦𝑧 = 𝑏 + 𝑑 𝑐  Solution 2 − 2𝐹 𝑧𝑧 𝐹 𝑦𝑦 + 𝐹 𝑦𝑦 2 + 4𝐹 𝑦𝑧 𝑐 4 𝐹 𝑧𝑧 2 2 − 2𝐹 𝑦𝑧 4 = 0 + 𝑐 2 −2𝐹 𝑧𝑧 𝐹 𝑦𝑧 2 𝐹 𝑦𝑦 + 𝐹 𝑦𝑧  Only solutions with a<0,c<0 are relevant.

 Identify “clusters” – areas of local maxima/minima. All points within certain % of highest intensity values.

 Start with H = 0, perform for each cluster separately.

 Find closest clusters A and B; B with known height.  For points in A close to B, calculate expected height according to B.  Find closest points using Voronoi diagram.

 Find relative height between A and B. If 𝑏 𝑗 , 𝑐 𝑗 is current and expected height of point i accordingly, find relative height x, such that error will be minimal: 𝑂 𝑓 𝑦 = (𝑏 𝑗 + 𝑦 − 𝑐 𝑗 ) 2 → 𝑛𝑗𝑜 𝑗 𝑂 𝑦 = 1 𝑂 (𝑐 𝑗 − 𝑏 𝑗 ) 𝑗

 Find direction v, in which H is constant => derivative is 0. ∀𝑗: 𝑕 𝑗 ∗ 𝑤 = 0, 𝑕 𝑗 = (𝑞 𝑗 , 𝑟 𝑗 )  Find least square line, its directions is perpendicular to v.

 If v is new x-axis, calculate projection of all points to YZ plane.

 Use polyline approximation. Given number of desired points = number of clusters + 2, the desired error can be approximated using binary search.  Example – 5 points:

 Finally, use spline, on polyline edge points.

 Not perfect, usually works sufficiently.

 Detect sheet of paper automatically.  Relax assumptions (light direction, H is constant in one direction).  Improve clusters search.  Replace/improve polyline approximation.  Use this for text recognition.