Multimedia Programming 06: Image Warping Departments of Digital Contents Sang Il Park

Outline Review Program Assignment #2 Image Warping –Scaling –Rotation –2x2 Transformation matrix

Review Point Processing –Brightness –Contrast –Gamma –Histogram equalization –Blur filter –Noise removal –Unsharp filter

Review Filter: Blur (smooth) filter –Mean Filter (Box Filter) –Gaussian Filter –Median Filter

OpenCV smoothing function cvSmooth ( input, output, type, param1, param2 ) –Type CV_BLUR (simple blur) – Mean Filter CV_GAUSSIAN (gaussian blur) – Gaussian Filter CV_MEDIAN (median blur) – Median Filter –param 1: size of kernel (width) : integer –param 2: size of kernel (height) : integer

Programming Assignment #2 “ 뽀샤시 필터 ” 디자인

Programming Assignment #2 뽀샤시 필터 디자인 결과의 예

Program Assignment #2 뽀샤시 필터 디자인의 주안점 – 눈이 부신 듯한 느낌 –No right answer! –Image Processing(filter) + Creativity 숙제 마감 : 9 월 20 일 수업 시간 전까지 –Use your own picture (smaller than 600*400) –Show before and after –2 인 1 조 ( or 1 인 1 조 ): 전과 달라도 상관 없음 – 리포트 (hardcopy) – 이미지파일 + 소스파일 + 실행파일 ( ) –

Image Processing 2 Image Warping Alexei Efros

Image Warping : Computational Photography Alexei Efros, CMU, Fall 2006 Some slides from Steve Seitz

Image Warping image filtering: change range of image g(x) = T(f(x)) f x T f x f x T f x image warping: change domain of image g(x) = f(T(x))

Image Warping TT f f g g image filtering: change range of image g(x) = h(T(x)) image warping: change domain of image g(x) = f(T(x))

Parametric (global) warping Examples of parametric warps: translation rotation aspect affine perspective cylindrical

Parametric (global) warping Transformation T is a coordinate-changing machine: p’ = T(p) What does it mean that T is global? –Is the same for any point p –can be described by just a few numbers (parameters) T p = (x,y)p’ = (x’,y’)

Scaling Scaling a coordinate means multiplying each of its components by a scalar Uniform scaling means this scalar is the same for all components:  2 2

Non-uniform scaling: different scalars per component: Scaling X  2, Y  0.5

Scaling Scaling operation: Or, in matrix form: scaling matrix S What’s inverse of S?

Inverse transformation T -1 p = (x,y)p’ = (x’,y’) p = T -1 (p’ ) Inverse transformation T -1 is a reverse process of the transformation T

Inverse of a scaling S Scaling operation S: Inverse of a S: In a matrix form: scaling matrix S -1

2-D Rotation  (x, y) (x’, y’) x’ = x cos(  ) - y sin(  ) y’ = x sin(  ) + y cos(  )

2-D Rotation x = r cos (  ) y = r sin (  ) x’ = r cos (  +  ) y’ = r sin (  +  ) 코사인 법칙 / 사인법칙 적용 x’ = r cos(  ) cos(  ) – r sin(  ) sin(  ) y’ = r cos(  ) sin(  ) + r sin(  ) cos(  ) 치환 x’ = x cos(  ) - y sin(  ) y’ = x sin(  ) + y cos(  )  (x, y) (x’, y’) 

2-D Rotation This is easy to capture in matrix form: Even though sin(  ) and cos(  ) are nonlinear functions of , –x’ is a linear combination of x and y –y’ is a linear combination of x and y R

Inverse of a 2-D Rotation Rotation by – 

2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?

2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Rotate around (0,0)? 2D Shear?

2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?

2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Translation? Only linear 2D transformations can be represented with a 2x2 matrix NO!

All 2D Linear Transformations Linear transformations are combinations of … –Scale, –Rotation, –Shear, and –Mirror Properties of linear transformations: –Origin maps to origin –Lines map to lines –Parallel lines remain parallel –Closed under composition

Homogeneous Coordinates Q: How can we represent translation as a 3x3 matrix?

Homogeneous Coordinates Homogeneous coordinates –represent coordinates in 2 dimensions with a 3-vector ( 동차좌표 )

Homogeneous Coordinates Q: How can we represent translation as a 3x3 matrix? A: Using the rightmost column:

Translation Example of translation t x = 2 t y = 1 Homogeneous Coordinates

Add a 3rd coordinate to every 2D point –(x, y, w) represents a point at location (x/w, y/w) –(x, y, 0) represents a point at infinity –(0, 0, 0) is not allowed Convenient coordinate system to represent many useful transformations (2,1,1) or (4,2,2)or (6,3,3) x y

Basic 2D Transformations Basic 2D transformations as 3x3 matrices Translate Rotate Scale

Affine Transformations Affine transformations are combinations of … –Linear transformations, and –Translations Properties of affine transformations: –Origin does not necessarily map to origin –Lines map to lines –Parallel lines remain parallel –Ratios are preserved –Closed under composition –Models change of basis

Projective Transformations Projective transformations … –Affine transformations, and –Projective warps Properties of projective transformations: –Origin does not necessarily map to origin –Lines map to lines –Parallel lines do not necessarily remain parallel –Ratios are not preserved –Closed under composition –Models change of basis

Matrix Composition Transformations can be combined by matrix multiplication p’ = T(t x,t y ) R(  ) S(s x,s y ) p Sequence of composition 1.First, Scaling 2.Next, Rotation 3.Finally, Translation

Inverse Transformation Translate Rotate Scale

Inverse Transformation p’ = T(t x,t y ) R(  ) S(s x,s y ) p P = S -1 (s x,s y ) R -1 (  ) T -1 (t x,t y ) p’

2D image transformations These transformations are a nested set of groups Closed under composition and inverse is a member