Computational Vision, Fall 1999
Class 13, Wednesday October 13
Image Brightness

Today's class

• Finish discussion of camera geometry, including intrinsic and extrinsic camera parameters.
• Lens models and focus.
• Reflectance and image brightness.

Reading options

• Jain, Kasturi and Schunck, Chapter 9, Sections 1-3 covers brightness.
• Geometric optics is covered Chapter 8.

Real Lenses

• Real lenses have an aperture (or opening) with a finite diameter, d, which controls the amount of light that enters the lens. Because sensors (including our eyes) can only handle certain ranges of intensities this aperture must be opened and closed depending on the amount of light in a scene.
• We model the optics of our lens using the ideal lens law of geometric optics.
  • It tells how well focused points in the image are and what the depth-of-field of the lens is.
  • More sophisticated models that account for distortions in the appearance of objects are sometimes needed, but we will not discuss them here.

The Ideal Lens Law

• The model describes the relationship between
  • the focal length of the lens, f,
  • the depth, Z, of a point ${\bf P} = (X, Y, Z)^T$ in the world, and
  • u, the distance that the image plane must be from the lens so that all light entering the lens from point P converges at a point on the plane.

• Assume the image plane and the plane of the lens are parallel to the X-Y plane and the center of the lens is at the origin of the coordinate system (the figure only shows the X-Z plane).

  
  
  
  

  
  

  
  
  
  

• The relation between u, v, and f is

  $$\frac{1}{u} + \frac{1}{Z} = \frac{1}{f}$$

• The position of the point on the image plane is governed by perspective projection because the light passes through the center of the image undeflected.
  • But, in the perspective model the value u from the ideal lens law should be used instead of f.
• For a point at a different depth, the light does not quite converge at the image plane. Instead the light from the point intersects the image in a circle called the blur circle.

  
  
  
  

  
  

  
  
  
  

  This could also occur with the light converging between the image plane and the lens.

  
  
  
  

  
  

  
  
  
  

• Assuming that the image plane is at position u₀ along the Z-axis, what is the diameter of the blur circle, d₀ as a function of D (the width of the aperture), f, u₀ and Z?
• The depth of field of a lens is the range of depths such that any point in the range has a blur circle radius that is less than the distance between pixels (picture elements) in the image.

Capturing Images on a Sensor Array

• Images are formed by recording the light falling on each location in an array of sensors.
• At each location the ``intensity'' is related to the number (flux) of electrons ``liberated'' from the imaging surface when the light strikes the surface that are ``caught'' by the sensor.
• This in turn depends on
  • $q(\lambda)$, the sensitivity of the device to light as a function of the wave length of the light, and
  • $b(\lambda)$, the flux of the incoming light as a function of wavelength.
• Thus, the number of electrons liberated at a given location is

  $$\int_{0}^{\infty} b(\lambda) q(\lambda) d\lambda$$

• The intensity values at each location in the array are then read into the computer through an A/D converter (called a digitizer board) and digitized, usually to the range 0..255.
• There is noise in each stage of this image formation process, so that there is a large amount of uncertainty in the intensity values that are recorded.
  • For example, on a CCD (charge-coupled device) array sensor sources of noise include ``bloom'', cross-talk, and geometric aberrations in the lay-out of the array.

Light Intensity Values

• Assume without proof that the gray level at a pixel is proportional to the brightness of the corresponding surface point.
• More precisely, the irradiance of the sensor surface is proportional to the radiance of the object surface in the direction of the lens.
• The irradiance of the object surface is the amount of light falling on the surface from a given direction. Light sources are usually modeled as a point (a ``point source'').
• The gray level value of a point (assuming a flawless imaging system) is therefore determined by
  • surface reflectance,
  • surface albedo,
  • surface orientation,
  • light source direction, and
  • light source intensity.
• Throughout the discussion we will assume orthographic projection.

Surface Brightness

• Introduce a coordinate system centered on a point in the scene and oriented with the Z axis coincident with the surface normal, $\eta$:

  
  
  

  
  
  

  • The direction of the ray, relative to the coordinate system, can be written in terms of the two spherical coordinate angles:

    
    $$\begin{align*} \phi &= \tan^{-1} Y/X \\ \theta &= \cos^{-1} Z/\sqrt{X^{2}+Y^{2}+Z^{2}}=\tan^{-1}\sqrt{X^{2}+Y^{2}}/Z\end{align*}$$
    

• In this coordinate system, the direction of the light source is described by $(\theta_{i},\phi_{i})$ and the direction of the camera is described by $(\theta_{e},\phi_{e})$.

  
  
  

  
  
  

• Surface reflectance can be described as $L(\theta_{e},\phi_{e})$,the amount of light (luminance or radiance) emitted from a point on the surface in direction $(\theta_{e},\phi_{e})$.
• It can then be written as a function of the amount of light emitted from the source, $E(\theta_{i},\phi_{i})$ :

  $$L(\theta_{e},\phi_{e}) = f(E(\theta_{i},\phi_{i}) )$$

• One example is a Lambertian or (matte) surface, which emits the same amount of light in all directions. In this case,

   

  $$L(\theta_{e},\phi_{e}) = ( c /\pi ) \cos{(\theta_{i})}E_0.$$ (1)

  The brightness (amount of light reflected off of the surface) is proportional to the angle between the surface normal and the light source, scaled by:

  • c, the ``albedo'' or fraction of light emitted (note that $0 \leq c \leq 1$), and
  • $E_0 = E(\theta_{i},\phi_{i})$, the intensity of the light source,
  $1/\pi$ is required to ensure that the total light emitted over the entire hemisphere of directions from which the surface is visible is c E₀, and
• As a second example, a purely specular surface (a mirror) has

  $$L(\theta_{e},\phi_{e})= E(\theta_{e},\phi_{e} - \pi).$$

• Often real surfaces are made up of a combination of Lambertian and specular components.