Chapter 3 Image Processing: Basic Transformation
- 1. CHAPTER3 BASIC TRANSFORMATION Dr. Varun Kumar Ojha and Prof. (Dr.) Paramartha Dutta Visva Bharati University Santiniketan, West Bengal, India
- 2. Basic Transformation Basic Transformation- Translation, Rotation & Scaling in 2D and 3D Inverse Transformation Perspective Transformation Relationship Between Cartesian Co-ordinate System and Homogeneous Co-ordinate System Back to Course Content Page Click Here
- 3. Basic Transformation- Translation 2D y x p p’ (x,y) (x’,y’) Translation from point p to p’ x’ = x + x0 y’ = y + y0 Where x0 & y0 are translation vector Asymmetric Form Unified Expression
- 4. Basic Transformation- Rotation 2D y x p(x,y) Rotation from point p to p’ x = r cos α y = r sin α Hence x’ = r cos (α – θ) y’ = r sin (α – θ) θ α x’ y’ r P’(x’,y’) x’ = r cos (α – θ) = r cos α cos θ + r sin α sin θ = x cos θ + y sin θ y’ = r sin (α – θ) = r sin α cos θ - r cos α sin θ = y cos θ - y sin θ (cos (A – B) = cosA cosB + sinA sinB) (sin (A – B) = sinA cosB – cosA sin ) B
- 5. Basic Transformation- Scaling 2D Scaling is simple process Rotation around arbitrary point q It is quite complex so one solution by translation point q to assign by vector (x0,y0) and translation of point p by same vector y p(x,y) θ r q x Rotation around arbitrary point q T(-r) (Rθ ( Tr(p)))
- 6. Basic Transformation – Translation 3D Cartesian coordinate system (x,yz) A new point (x,y,z) is translated to a coordinate (x*,y*,z*) using a displacement vector (x0,y0,z0)x* = x + x0 y* = y + y0 z* = z + z0 Asymmetric Form Unified Expression Unified matrix representation is of the form V* = A V Where A is 4x4 Transformation matrix V is the column vector of original co- ordinate V* is the column vector of transformed co-ordinate
- 7. Basic Transformation – Rotation 3D θ β α x y z There are three rotation in 3coordinate system, θ, β and α rotation In θ rotation z co-ordinate remains fixed and rotation is only viewed in the x,y plane as indicated by arrows Similarly in β rotation y axis remains same and rotation is only viewed in the x,z plane In α rotation the the x axis remains same and the rotation is viewed in y,z plane only
- 9. Concentration Several transformation can be represented by a single 4 x 4 transformation matrix Translation, scaling, rotation about a point z axis of a point V can be represented as V* = Rθ(S(T(V)) = A V Where A = RθS T ( is a 4 x 4 matrix) Matrix multiplication order is important because matrix operation are not commutative
- 10. Concentration Cont.. Concentration in non commutative z y x V V1 V2 About the point V we first do Translation and then Rotation then we get the point V1 V1 = R(T(V)) About the point V we do first rotation and then Translation we get the point V2 V2 = T(R(V)) From the picture it is clear that V1 & V2 are not the same point so concentration is non commutative
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- 12. Inverse Transformation The Inverse Transformation can be obtained by the following observation Inverse Translation Inverse Rotation
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- 14. Perspective Transformation A perspective transformation is also called imaging transformation A imaging transformation project 3D point onto a plane A perspective transformation is the approximation of the image formation.
- 15. Image Formation Process y,Y x,X z,Z λ Image Plane Lens Center (0,0,λ) (x,y,z) → Camera Co-ordinate System (0,0,λ) → Center of lens (X,Y,Z) → World Co-ordinate System aligned with camera co-ordinate system in 3D scene (X,Y,Z) (x,y) We are interested in (x,y) which is projection of (X,Y,Z) onto image plane For both camera & 3D image z axis are aligned and have same origin
- 16. Image Formation Cont.. By using similar triangular x/λ = – X/(Z – λ) = X/(λ – Z) y/λ = – Y/(Z – λ) = Y/(λ – Z) Now X = λX/ λ – Z Y = λY/ λ – Z It can be expressed as matrix expression in Homogeneous Co-ordinates Cartesian coordinate system to homogeneous coordinate system (X,Y,Z) → (K.X, K.Y, K.Z, K) K is an arbitrary nonzero constant
- 17. Image Formation Cont.. Conversion from Homogeneous to Cartesian coordinate system is a simple process In Vector Form A perspective transformation matrix P is defined as The element of Ch are the camera co-ordinate in homogeneous form corresponding to Cartesian Co- ordinate
- 18. Image Formation Cont.. The camera co-ordinate in Cartesian Co- ordinate System is as follows Finally x = λX/(λ-Z)) and y = λY/(λ-Z) coordinate in the image plane of projected 3-D point (X,Y,Z)
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