Geometric Image Transformations

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1 Geometric Image Transformations Part One 2D Transformations Spatial Coordinates (x,y) are mapped to new coords (u,v) pixels of source image -> pixels of destination image Types of 2D Transformations Affine Scales Translations Rotations Shears Projective Projections Homographies or Collineations

2 Scale and Translation Scale u = s_x * x v = s_y * y Translation u = x + t_x v = y + t_y Rotation and Shears Rotation u = x*cos θ - y * sin θ v = x*sin θ + y * cos θ Shear u = x + Sh_x * y v = y + Sh_y * x 2

3 Shear Consider Sh_x only u = x + Sh_x*y; v = y; (Sh_y=) Homogenous Coords Translation is just an addition We can make it a function of multiplication by using homogenous coords p p Homogenous coords are equivalent up to a scale factor p p u v = tx ty x y wu wv w = tx ty wx wy w 3

4 Homogenous Coordinates Transform works on scale (w) * w w= Dividing by the scale w puts you in Cartesian coords Matrix Notation p Affine p u v = a b c d e f x y 4

5 Matrix notation of transforms Translation Scale p p p p u v = tx ty x y u v = Sx Sy x y Rotation p p Shear p p u v = cosθ -sinθ sinθ cosθ x y u v = Shx Shy x y Concatenation You can concatenate several transforms into one matrix A = R S Sh T rotate scale shear translate 5

6 Affine Transformations Affine is a linear mapping plus a translation: A function f(x) is linear iff f(x+y) = f(x) + f(y) a*f(x) = f(a*x) A function T(x) is Affine if there exists a linear mapping L(x) and a constant c Such that: T() = L(x) + c (for all x) Affine Transforms Properties Preserves parallel lines Preserves equispaced points along lines equally spaced points on a line in the source space will produce equally spaced points on a line in the destination space (although the scale may be different) Preserves incident Points of intersection hold 6

7 Preserves Source Destination Source Destination 6 degrees of freedom u v = a b c d e f Y We can define an affine transform by specifying 3 point correspondences 3 (x,y) from the source space that map to 3 (u,v) in the destination space

8 Equations to solve for Affine Transform u = a*x + b*y + c u2 = a*x2 + b*y2 + c u3 = a*x3 + b*y3 + c v = d*x + e*y + f v2 = d*x2 + e*y2 + f v3 = d*x3 + e*y3 + f More commonly You can build an Affine transform by concatenating several transforms together translate to the origin Rotate by 2 degrees translate back from the origin scale by 5 GUI Interface tool that allows you to rotate, translate, scale, etc.. 8

9 Example Affine Transform Limitation Can map a triangle in source space To a triangle in destination space (or two parallelograms) 9

10 Affine Transform Limitation Can map a triangle in source space To a triangle in destination space (or two parallelograms) What about a rectangle to a general quadrilateral? Projective Transform Projective transform can transform general quadrilaterals between source and destination space Does not preserve parallel lines, or lengths Does not preserve equispaced points

11 Projective transform uses homogenous coords su sv s = a b c d e f g h x y u = u/s v = v/s Maps us back to Cartesian space Projective Transforms Very common in computer graphics Texture mapping a 3D polygon polygon has been projected onto a plane 3D polygon 3D perspective projection Texture map 2D polygon

12 Projective Transform Does not preserve length, equispacing Does Map lines to lines Preserve incidents Preserve cross ratio Cross Ratio Given 4 points on a line in source space and destination space D C B A A B C D Source Space Destination Space AC AD BC BD = A C A D B C B D Cross Ratio where Y is the Euclidean distance between point and Y 2

13 Solving for a Projective Transform 8 degrees of freedom su sv s = a b c d e f g h x y We need 4 point correspondences between source and destination image su = ax + by + c sv = dx + ey + f s = gx + hy + source destination Remember, with projective transform destination points are: dest_u = su/s dest_v = sv/s Solving for a Projective Transform su = ax + by + c sv = dx + ey + f s = gx + hy + u =su = ax + by + c s gx + hy + v =sv = dx + ey + f s gx + hy + (gx+hy+)u = ax + by + c (... ) gx + hy + gxu + hyu + u = ax + by + c u = ax + by + c gxu hyu 3

14 8x8 System of Equations x y -x u -y u a u x y -x u -y u b u x 2 y 2 -x 2 u 2 -y 2 u 2 c u 2 x 3 y 3 x y -x 3 u 3 -x v -y 3 u 3 -y v d e = u 3 v x y -x v -y v f v x 2 y 2 -x 2 v 2 -y 2 v 2 g v 2 x 3 y 3 -x 3 v 3 -y 3 v 3 h v 3 Solve the system Matrix is in form: Ax = b Solve for x Gaussian elimination (LU decomposition) QR decomposition Entries of vector x are the coefficients for the projective transform 4

15 Properties of Transforms euclidean similarity affine projective Transformation translation rotation uniform scale non-uniform scale shear projection combination (w/ projection) Properties of Transforms euclidean similarity affine projective Invariant length angle ratio of lengths parallelism incident cross ratio 5

16 Transforming an Image origin Transforms and Images Coordinates x Y f(x,y) 6

17 Image Coords vs. Cartesian Coords Image Coords are generally defined by raster alignment Y [,Height], [,Width] Affine, projective transforms are converted into Cartesian coords Transforms and Images Coordinates (-y value) origin x Y f(x,y) 7

18 Cartesian space to image space From Cartesian space to Image Space find (x_min, xmax) find(y_min, ymax) new size dimensions w = x_max x_min h = y_max y_min create newimage size (w, h) Translate transformed points, such that: T * (x,y) = (u,v) newimage( u + abs(x_min), v + abs(y_min) = I(x,y) Converting to an Image New Image Dimensions origin x Y 8

19 The transformed image origin New Image Dimensions x Y Creating the new image Forward Mapping Inverse Mapping Sampling 9

20 Mapping Pixels (,) (,N) 2 Forward Mapping (,) (,N) (,M) (M,N) (,M) (M,N) Source Image [u,v,s] T = A [x,y,] T Destination Image Transform Draw backs Forward Mapping Source pixels do not map directly to a single pixel in the destination space Possibility for holes in the destination image We can map the other direction 2

21 Reverse Mapping Reverse Mapping 2 2 (,) (,N) (,M) (M,N) black [x,y,s] T = A - [u,v,] T Inverse Mapping Advantages We assign an intensity to each pixel in the destination (no holes) Affine/projective transforms have inverses (not a problem) just reverse direction of the point correspondences We still don t have pixel to pixel mapping 2

22 Sampling the source How do we sample the source to determine the intensity for the destination? Sampling the source How do we sample the source to determine the intensity for the destination? 22

23 Mapping Source 2 x 2 pixels Option : Pick the pixel nearest to our center. Mapping Source 2 x 2 pixels Small change results in big difference Option : Pick the pixel nearest to our center. 23

24 Try different sampling Source 2 x 2 pixels 2 What if we assign an intensity to each vertex and then average? Pick the intensity which the vertex lies New Sample = Sampling Example Source 2 x 2 pixels 2 Move the destination slightly. Pick the intensity which the vertex lies New Sample = 24

25 No Difference Source 2 x 2 pixels Source 2 x 2 pixels Try different Sampling Source 2 x 2 pixels 2 3 What if we had more samples? 4 25

26 Try different Sampling Source 2 x 2 pixels 2 What if we sampled a larger area? 4 3 Try different Sampling Source 2 x 2 pixels How should we sample? 26

27 Common Sampling Approaches Nearest Neighbor Sample Take closest pixel value Bi-linear Interpolation 2x2 (4) Samples Interpolate from these samples Slower Bi-Cubic 4x4 (8) samples Construct a new sample using a non-linear interpolation Slower Common Sampling Approaches Nearest Neighbor Bi-linear Interpolation Bi-Cubic 27

28 Common Sampling Approaches What do these approaches mean? How can evaluate what they are doing? Nearest Neighbor Bi-linear Interpolation Bi-Cubic Sampling and Signal Processing Proper sampling is a classic reconstruction problem Given a continuous signal f(x) How do you take discrete samples such that you can properly reconstruct the signal f(x) from the samples? f(x) x 28

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