chapter2 image formation reading: szeliski, chapter 2

Chapter2 Image Formation

Reading: Szeliski, Chapter 2

What are we tuned to?

The visual system is tuned to process structures typically found in the world.

What is a natural image?

The structure of ambient light

What is a natural image?

The visual system seems to be tuned to a set of images:

What is a natural image?

The visual system seems to be tuned to a set of images:

Did you saw this image?

What is a natural image?

The visual system seems to be tuned to a set of images:

Demo inspired from D. Field

6 images

Not all these images are the result of sampling a real-world plenoptic function

http://www.rowland.harvard.edu/images/ModPurp.jpg

http://www.alexfito.com/

• Proposition 1. The primary task of early vision is to deliver a small set of useful measurements about each observable location in the plenoptic function.

• Proposition 2. The elemental operations of early vision involve the measurement of local change along various directions within the plenoptic function.

• Goal: to transform the image into other representations (rather than pixel values) that makes scene information more explicit

Cavanagh, Perception 95

What are “visual features”

Shape, color, texture, etc

2.1 Photometric Image Formation

• Discrete color or intensity values• Where do these value come from?

– Geometry, projection– Camera optics, sensor properties – Lighting, surface properties

Images as Functions

Images as Functions• We can think of an image as a function, f, from R2 to R:

– f( x, y ) gives the intensity at position ( x, y )

– Realistically, we expect the image only to be defined over a rectangle, with a finite range:

• f: [a,b]x[c,d] [0,1]

• A color image is just three functions pasted together. We can write this as a “vector-valued” function:

( , )

( , ) ( , )

( , )

r x y

f x y g x y

b x y

Images as functions

What is a digital image?

• We usually work with digital (discrete) images:– Sample the 2D space on a regular grid– Quantize each sample (round to nearest integer)

• If our samples are apart, we can write this as:• f[i ,j] = Quantize{ f(i , j ) }• The image can now be represented as a matrix of integer

values

Photometric Image Formation

Perspective projection Light scattering

Lens optics Bayer color filter array

Photometric Image Formation

2.2 Lighting• Point light source

– Single location (small light bulb)– Infinity: the sun --directional light

• Area light source – A finite rectangular area emitting light equally in all

directions

• Environment map • Light direction to color mapping

2.2.2 Reflectance and Shading

many models for reflectance and shadingBRDF: Bidirectional Reflectance Distribution Function

BRDF

• BRDF is reciprocal

• For isotropic surface, no preferred directions for light transport

BRDF

• Light existing a surface point:

Foreshortening factor

Diffuse Reflection

• Also called Lambertian or matte reflection– Light is scattered uniformly in all directions, i.e. – BRDF is constant:

Think about the inverse problem

Specular Reflection• Depends on the direction of outgoing light• Mirror surface:

– Specular reflection direction

rv

cos( ) ( )s r iv s

Specular Reflection• Amount of light

– Phone model – Micro-facet model

– Larger , , more specular surface with hightlights ; Smaller, softer gloss

Phone Shading Model

• Diffuse • Specular • Ambient:

– Does not depend on surface orientation– Color and both ambient illumination and

the object

Phone Shading Model

( ; ) ( ) ( )

( ) ( )( )

( ) ( )( ) e

r r a a

d i i ii

ks i r i

i

L v k L

k L v n

k L v s

The recent advent of programmable pixel shaders makes the use of more complex models feasible.

Example

Realistic Rendering

The recent advent of programmable pixel shaders makes the use of more complex models feasible.

Ioannis Gkioulekas, et al, Siggraph’13

Optics

• Lens, sensor• Ideal pinhole camera • More complex: focus, exposure, vignetting,

aberation,…,

Thin lens model Thin lens: low, equal curvature on both sides

Optical axis

Thin lens model

object

Focus plane

Thin lens model

object

Circle of confusion

Pinhole camera

Pinhole Camera Model

object

Pinhole Camera Model

object

Pinhole Camera Model

object

2.3 3D to 2D Projection

• 3D perspective: the most commonly used projection in computer vision and computer graphics

3D view of world perspective

book: pp32-60

Pinhole Camera Model

object

xfu

zyf

vz

Pinhole Camera Model

object

Pinhole Camera Model

• Using homogeneous (projective) coordinate –

0 0 0/

0 0 0/

0 0 1 01 1

0 0 0 1 1

f xu fx z

f yw v z fy z

z

wx = Kp

xfu

zyf

vz

Camera Intrinsics• Imperfect camera• image sensor

• s: possible skew between sensor axes• a: aspect ratio • : optical center• F : focal length

0

0 0 1

x

y

f s c

af c

K

. ., ( , ) ( / 2, / 2)x ye g c c W H

( , )x yc cFive intrinsic parameters

Camera Intrinsics

• Focal length• Actual focal length, e.g. 18~55mm, • Conventional sensor width: 35 mm• Digital Image: integer values, [0,W) x [0,H)

Focal length Sensor width

Field of view

Extrinsic Parameters

• World Coordinate system to Camera Coordinate system

[c wp R | t]pExtrinsic parameters

j

i

kj

i

k

O

C

1wC R t

Extrinsic Parameters

wx = Kpwith

[ ww x K R | t]p

[M K R | t]Camera Matrix

2.3 Digital Camera

• Process chart

2.3 Digital Camera

• Process chart

2.3.2 Color

• Light from different parts of the spectrum is somehow integrated into discrete RGB color values

[ ww x K R | t]p

2.3.2 Color

• Primary and Secondary Colors• Additive colors (projector, monitor)• Subtractive colors (printing, printing)

CIE color matching

• Commission Internationale d’Eclairage (CIE)• Color matching experiments

pure colors to the R=700.0nm, G=546.1nm, and B=435.8nm

XYZ Color Space

Y=1 for pure R (1,0,0)

XYZ Color Space

• Y=1 for (1,1,1)

XYZ Color Space

• Chromaticity coordinates

• Yxy (luminance plus the two most distinctive chrominance components)

Chromaticity Diagram

L*a*b* Color Space

• Human visual system is roughly logarithmic• Differences in luminance or chrominance are

more perceptually uniform• Non-linear mapping from XYZ to L*a*b* space

L*a*b* Color Space

Color Cameras

• Spectral response function

• Make sure to generate the standard color values

HDTV, new monitors, new standard ITU-R BT.709

Color Filter Arrays

• Separate sensors for three primary colors

Bayer RGB pattern: (a) color filter array layout; (b) interpolated pixel values

Bayer Pattern, 1976

• Green filters are twice as many as red and blue filters

• Human visual system is much more sensitive to high frequency detail in luminance than chrominance

• Luminance is mostly determined by green value

Color Balance

• Move the white point of a given image closer to pure white (R=G=B)– Multiply RGB values by a different factor – Color twist, general 3x3 transform matrix– Exercise 2.9 (optional)

Gamma

• CRT Monitor: non-linear relationship between the voltage and the resulting brightness is determined by gamma

• Pre-map the sensed luminance Y through an inverse gamma

2.2

10.45

Gamma Compensation

Noise added during transmission or quantization will be reduced in the darker regions of the signal where it was more visible

Other Color Spaces

• XYZ, RGB for spectral content of color signals• Others for image coding and computer

graphics– YUV, YCrCb, HSV

YUV Color Space

• YUV for video transmission– Luma

– Two lower frequency chroma channels

YCrCb Color Space

• Closely related to YUV• Different scale factor to fit within the 8-bit

range for digital signals

• Useful for careful image de-blocking, et al.

HSV Color Space

• Hue: direction around a color wheel

• Saturation: scaled distance from the diagonal

• Value: mean or maximum color value

More suitable for color picking

Color Ratios

• Suitable for algorithms that only affect the value/luminance and not saturation or hue

• After processing, scale rgb back by the color ratio Ynew/Yold

Color FAQ, http://www.poynton.com/ColorFAQ.html

2.3.3 Compression

• Converting signal into YCbCr (or related variant)

• Compress the luminance signal with higher fidelity than the chrominance signal