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 surface): 𝑆 𝑞, 𝑟 =

1 𝑞2+𝑟2+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

• riginal 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 = 𝐼0 + 0.5(𝑏𝑦2 + 𝑐𝑦𝑧 + 𝑑𝑧2) 𝑞 = 𝜖𝐼 𝜖𝑦 = 𝑏𝑦 + 𝑐𝑧, 𝑟 = 𝜖𝐼 𝜖𝑧 = 𝑑𝑧 + 𝑐𝑦 𝐹 𝑦, 𝑧 = 1 2𝐽(𝑦, 𝑧)2 = 0.5 𝑞2 + 𝑟2 + 1 = 0.5 𝑏2 + 𝑐2 𝑦2 + 𝑏 + 𝑑 𝑐𝑦𝑧 + 0.5 𝑑2 + 𝑐2 𝑧2 + 1

𝐹𝑦𝑦 = 𝑏2 + 𝑐2 𝐹𝑧𝑧 = 𝑑2 + 𝑐2 𝐹𝑦𝑧 = 𝑏 + 𝑑 𝑐

 Solution

𝑐4 𝐹𝑧𝑧

2 − 2𝐹𝑧𝑧𝐹𝑦𝑦 + 𝐹𝑦𝑦 2 + 4𝐹𝑦𝑧 2

+ 𝑐2 −2𝐹𝑧𝑧𝐹𝑦𝑧

2 − 2𝐹𝑦𝑧 2𝐹𝑦𝑦 + 𝐹𝑦𝑧 4 = 0

 Only solutions with a<0,c<0 are relevant.

 Identify “clusters” – areas of local

maxima/minima. All points within certain %

• f 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 𝑏𝑗, 𝑐𝑗

is current and expected height of point i accordingly, find relative height x, such that error will be minimal: 𝑓 𝑦 = (𝑏𝑗 + 𝑦 − 𝑐𝑗)2→ 𝑛𝑗𝑜

𝑂 𝑗

𝑦 = 1 𝑂 (𝑐𝑗 − 𝑏𝑗)

𝑂 𝑗

 Find direction v, in which H is constant =>

derivative is 0. ∀𝑗: 𝑕𝑗 ∗ 𝑤 = 0, 𝑕𝑗 = (𝑞𝑗, 𝑟𝑗)

 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.