Image Formation and Camera Models Allen Y. Yang Berkeley EE 225b - PowerPoint PPT Presentation

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Image Formation and Camera Models Allen Y. Yang Berkeley EE 225b Feb 28th, 2007 Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Images and Projection Models 1 Introduction Perspective Projection Orthographic Projection Camera Models 2 Imaging through a Pinhole Camera Intrinsic Parameters 3 Lense Distortions Modeling Camera Parameters Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Representation of Images I : Σ ⊂ R 2 → R + ; ( x , y ) �→ I ( x , y ) . This Lecture How are images captured from 3-D world to 2-D? 1 Camera projection model? 2 Image formation? 3 Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Perspective Projection A modern camera projects 3-D world into 2-D image plane through perspective projection : Properties of perspective projection: Foreshortening: Distance objects are smaller. Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Perspective Projection Horizon line: Vanishing points: Parallel lines in 3-D intersect at a point in the image plane. How many vanishing points in these images? Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Correct perspective projections are visible in paintings. (a) 1st Century B.C., Pompeii (b) “School of Athens”, Raphael, 1518 More reading: “Perspective” in wikipedia.com . Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Perspective Projection and Illusions (c) Necker cube (d) Escher waterfall (e) Ames room Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Orthographic Projection Model Orthographic projection: The difference between perspective and orthographic was illustrated in Christian artwork: (f) Perugino Fresco, Vatican (g) Birth of the Virgin, Ukraine Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Pinhole camera model ( Camera Obscura ) Mo-Zi (5th century BC) − → Aristotle (300 BC) − → Da Vinci (1490) − → Kepler (17th century) Figure: Pinhole camera model. Let p = [ X , Y , Z ] T ∈ R 3 , and its image x = [ x , y ] T ∈ R 2 : x : X = y : Y = − f : Z Hence, x = − f X y = − f Y Z . Z , Question: What is the projection model for orthographic projection? Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Frontal Camera Model The pinhole camera model is inconvenience that the focal length f is negative. Figure: Frontal pinhole camera model. � x = f � X � � x = . y Y Z In homogeneous coordinates:  X      x f 0 0 0 Y  =   Z y 0 f 0 0  .      Z 1 0 0 1 0  1 Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Distortions from Physical Lenses Pinhole camera model assumes Perfect pinhole camera lenses. 1 Image x = I ( p ) ∈ R 2 be measured in infinite accuracy. 2 Principal point is at the center of the image. 3 Physical camera lenses give us Distorted imaging projections. 1 (a) Fish- (b) Nor- (c) Tele- eye mal/Portrait photo Finite resolutions defined by the sensing devices in digital cameras. 2 Offset between image center and optical center. 3 Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters Camera Intrinsic Parameters Figure: Transformation from image coordinates to pixel coordinates. Modeling the camera distortion Transform from image coordinates (e.g., in metric units) to pixel coordinates. 1 � s x 0 � x s � � x � � = . y s y 0 s y Translate the image origin. 2 x ′ = x s + o x ; y ′ = y s + o y . Allen Y. Yang Image Formation and Camera Models

Outline Images and Projection Models Camera Models Camera Intrinsic Parameters In homogeneous coordinates:       x ′ s x 0 o x x x ′ =  =  . y ′ 0 s y o y y     1 0 0 1 1 Camera Intrinsic Matrix:   s x 0 o x K .  . = 0 s y o y  0 0 1 Complete transformation from 3-D to 2-D pixel coordinates:   X  x ′   s x 0 o x   f 0 0 0  Y   Z y ′ = 0 s y o y 0 f 0 0         Z 1 0 0 1 0 0 1 0   1   X     fs x 0 o x 1 0 0 0 Y   = 0 fs y o y 0 1 0 0  .       Z  0 0 1 0 0 1 0 1 In short hand, λ x ′ = K Π 0 X . K is called calibration matrix , Π 0 is called (perspective) projection matrix . Allen Y. Yang Image Formation and Camera Models