Image Transforma1ons image filtering : change range of image Image - PDF document

Image Transforma1ons • image filtering : change range of image Image Processing : • g(x) = h(f(x)) Transforma1ons, Warping and Morphing h Last Week Courtesy: Irfan Essa, Georgia Tech James Hayes, Georgia Tech Alexei Efros, UC Berkeley Steve Seitz, U Washington Image Transforma1ons • image filtering : change domain of image • g(x) = f(h(x)) h • x Parametric (global) warping Parametric (global) warping Original Aspect Skew Original • Examples: – Original – Transla1on Translate Scale Others, – Rota1on Translate Cylindrical Perspec4ve Rotate Rotate

Image Scaling Parametric Global warping • Mul1ply each T components by a scalar p = (x,y) p � = (x � ,y � ) • Transformation T is a coordinate -changing machine • Uniform scaling : Scalar of pixels at x, y. × 2 same for x, y p � = T (p) • Global and parametric T: • Non-uniform : Not – Is the same for any point p same – Described parameters • T can be represented as a matrix: x ' x ⎡ ⎤ ⎡ ⎤ p � = M p x ' = ax + by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ M x ' a b x = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ y ' y y ' y c d y ' = cx + dy ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Image Scaling Image Scaling • Mul1ply each • Mul1ply each components by a components by a scalar scalar • Uniform scaling : Scalar • Uniform scaling : Scalar × 2 × 2 same for x, y same for x, y (s x =s y ) • Non-uniform : Not • Non-uniform : Not same same (s x ≠s y ) ⎡ ⎤ x ' = ax + by x ' = s x x ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ s x 0 x ' a 0 x x ' x ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ y ' y y ' y 0 d y ' = s y y ⎢ 0 s y ⎥ y ' = cx + dy ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Image Scaling 2D Image Transforma1on • Mul1ply each • Scaling: ⎡ ⎤ components by a x ' = s x x ⎡ ⎤ ⎡ ⎤ s x 0 x ' x ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ scalar y ' 0 s y y y ' = s y y ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ • Uniform scaling : Scalar same for x, y (s x =s y ) X × 2 Y × 0.5 • Non-uniform : Not • Mirror same (s x ≠s y ) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x ' x − 1 0 ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ y ' y 0 − 1 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ x ' = s x x ⎡ ⎤ ⎡ ⎤ s x 0 x ' x ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ y ' y y ' = s y y ⎢ 0 s y ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2D Rota1on 2D Rota1on (x � , y � ) (x � , y � ) • X’ is a linear combination x = r cos ( φ ) of x and y (even if sin( θ ) y = r sin ( φ ) (x, y) (x, y) and cos( θ ) themselves x � = r cos ( φ + θ ) are non-linear) y � = r sin ( φ + θ ) Trig Iden1ty • Inverse transformation? θ θ φ φ x � = r cos( φ ) cos( θ ) – r sin( φ ) – Rotation by – θ sin( θ ) – For rotation matrices R -1 = y � = r sin( φ ) cos( θ ) + r cos( φ ) R T x � = x cos ( θ ) - y sin ( θ ) x � = x cos ( θ ) - y sin ( θ ) sin( θ ) y � = x sin ( θ ) + y cos ( θ ) y � = x sin ( θ ) + y cos ( θ ) Subs1tute x � = x cos ( θ ) - y sin ( θ ) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ cos( θ ) − sin( θ ) cos( θ ) − sin( θ ) x ' x x ' x ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ y � = x sin ( θ ) + y cos ( θ ) = = y ' y y ' y sin( θ ) cos( θ ) sin( θ ) cos( θ ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 2D Linear Transforma1ons (LD) Linear Combina1on ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x ' x − 1 0 x ' = − x ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ y ' y 0 − 1 y ' = − y ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ Linear Transforma1ons ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x ' x x ' x a b ⎢ ⎥ = M ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⇔ = ⎢ ⎥ x ' x x ' x = 1 0 • Mirror y ' y y ' y c d ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ y ' y ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = y ' y 0 1 ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ • Iden1ty x ' = s x x ⎡ ⎤ • Only Linear Combina1ons can be ⎡ ⎤ ⎡ ⎤ s x 0 x ' x • Scale ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = y ' = s y y represented by a 2 x 2 matrix y ' 0 s y y ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ – Mirror, Iden1ty, Shear, Scale, and Rotate • Shear – Origin maps to origin ⎡ ⎤ ⎡ ⎤ 1 sh x ⎡ ⎤ x ' = x + sh x * y x ' x • Rotate ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ – Lines map to lines y ' ⎢ sh y 1 ⎥ y y ' = sh y * x + y ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ – Parallel lines remain parallel – Ratios are preserved – Closed under composition ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x ' cos( θ ) − sin( θ ) x x � = x cos( θ ) - y sin( θ ) x ' = x + t x ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ • What about Translate? y � = x sin( θ ) + y cos( θ ) y ' sin( θ ) cos( θ ) y ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ y ' = y + t y Homogenous Coordinate Basic 2D Transforma1on ⎡ ⎤ • Idea : Add an extra 1 0 t x x ' 1 0 t x x t ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ + ⎤ ⎡ ⎤ ⎢ ⎥ x ' = x + t x x x x dimension ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ y ' 0 1 t y y t T ranslation = 0 1 t y = = + x ⎢ ⎥ ⎢ y ⎥ y ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ y ' = y + t y – In 2D use a 3-vector and 3x3 y ⎢ ⎥ ⇒ 1 ⎢ 0 0 1 ⎥ 1 1 ⎢ 0 0 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ matrices. y ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎡ ⎤ – In 3D use a 4-vector and 5x4 1 1 0 ⎣ ⎦ Identity = ⎢ ⎥ matrices. 0 1 ⎣ ⎦ • The extra coordinate (3 rd for 2D images) is an arbitrary t x = 2 y value w, added to each 2D t y = 1 2 point (x,y) � (x,y,w) (2,1,1) or (4,2,2) or (6,3,3) ) 1 • (x, y, w) � (x/w, y/w) 1 2 x • (x, y, 0) � infinity ed • (0, 0, 0) is not allowed