Visual Perception of Luminance

• Spectral energy distribution of light source:

  
  

  $$L(\lambda),\;\;\;\;350\;nm\;\leq\;\lambda\;\leq\;780\;nm$$
  
  

• Luminance (intensity) Light energy reflected by an object:

  
  

  $$I(\lambda)=\rho(\lambda) L(\lambda)$$
  
  

  where $\rho(\lambda)$ is the reflectivity of the object. $I$ represents the objective physics of the lighting of the object.

• Image signals:

  The light reflected by a 3D object is projected through the lens of the visual system (camera, eye) to become a 2D signal $I(x,y,\lambda)$, which is then detected by the sensors/receptors of the visual system:

  
  

  $$f(x,y)=\int_0^{\infty} I(x,y,\lambda) S(\lambda) d\lambda$$
  
  

  Here $S(\lambda)$ is the sensitivity (``luminous efficiency'') of the film, the CCD sensors, or the photoreceptors (rods and cones) in the retina. The function of human eye is a bell-shaped function of frequency.

• Apparent brightness (brightness):

  Brightness is the perception or sensation caused by the input light signal. It is a subjective and qualitative attribute of the object being observed, and it depends on the surroundings of an object as well as the luminance.

  Two objects with different surroundings could have the same luminance but different brightnesses. For example, the screen of a TV set may look gray, but when it is turned on, a black object in the scene may seem darker due to the comparison with the background, e.g., some white objects in the scene.

  More examples: White'sillusion and Wertheimer-Benary illusion

• Contrast

  Assuming the luminance of an object is $f$ and the luminance difference between the object and its surrounding is $df$, then according to Weber's law, the perceived contrast $dp$ (luminance difference) between the object and its surrounding is

  
  

  $$dp=\frac{df}{f}=d(ln f)$$
  
  

  which indicates that at higher level $f$, larger $df$ is needed to perceive the same contrast at lower level $f$ with a smaller $df$. In other wwords, equal increment in $ln f$, instead of in $f$, is perceived to be equally different (equal contrast).

  Integrating both sides, we get the perceived luminance

  
  

  $$p=\int d(ln f)=ln f + C$$
  
  

  The constant of integration $C$ can be obtained by assuming the perceived luminance is zero $p=0$:

  
  

  $$C=-ln f_0$$
  
  

  where $f_0$ is the threshold luminance not perceivable. Now we have

  
  

  $$p=ln \frac{f}{f_0}$$
  
  

  The relationship between stimulus $f$ and perception $p$ is logarithmic.

  Weber's law describes a general phenominon in human perception. Another example is the difference between different sound frequencies. The difference between $C_4$ (middle C, 261.63 Hz) and $C_5$ (523.25 Hz) is an octave, perceived the same as the difference between $C_5$ and $C_6$ (1046.5 Hz), although the frequency differences between the two pairs are quite different (261.63 Hz. vs. 523.25 Hz).

• Visual Illusions See examples.

Visual Perception of Colors

See here for more detailed discussion for color perception.

• Color Representation
  • What Determines the Color?

    Along the visible wave length (350 nm - 780 nm), there are only about 128 fully saturated colors that can be distinguished.

    It is the energy spectral distribution $P(\lambda)$ of the signal that determines the colors we perceive.

  • Three Components of Color
    • Hue: the dominant wavelength, the redness of red, greenness of green, etc.
    • Saturation: how pure the color is, or how much white is contained in the color. For example, red and royal blue are more saturated than pink and sky blue, respectively.
    • Luminance: the amount or intensity of light.

  • Tristimulus Theory

    There exist 3 types of cells (cones) in human retina of different response functions (luminous efficiency functions): $S_i(\lambda),\;\;(i=1,2,3)$. They overlap with each other and peak in the yellow-green, green and blue regions, respectively.

    The responses of these cells to a signal of intensity $C(\lambda)$ (a ``color'') are therefore
    

    $$r_i(C)=\int S_i(\lambda) C(\lambda) d\lambda\;\;\;(i=1,2,3)$$
    
    

    The perceived color is determined by the combination of these 3 responses $r_i,\;\;(i=1,2,3)$. In other words, if two colors $C_1(\lambda)$ and $C_2(\lambda)$ produce the same responses:
    

    $$r_i(C_1)=r_i(C_2)\;\;\;(i=1,2,3)$$
    
    
    then they are perceived as the same color.

  • Color Models

    There exist many different color models (all composed of three independent variables), for example:

    • RGB model: using Red, Green, and Blue as three primaries to represent a color.

    • HSV model: using Hue, Saturation, and Value (intensity) to represent a color

    • XYZ model (International Commission on Illumination, CIE)

• Color Matching and Reproduction
  • Color Matching

    It is possible for different colors, energy distributions, to produce exactly the same visual perception in the human visual system. These colors are said to be matched and are called metamers. Two matching colors $C_1(\lambda)$ and $C_2(\lambda)$ can be represented by
    

    $$C_1(\lambda) \equiv C_2(\lambda)$$
    
    
    Note that in general matching colors do not necessarily have identical energy distributions,
    
    $$C_1(\lambda) \neq C_2(\lambda)$$
    
    

  • Three-Color Theory

    Any color can be reproduced by mixing an appropriate set of three primary colors (e.g., CIE X, Y, Z, or red, green, and blue, not unique) with energy distributions $P_k(\lambda),\;\;(k=1,2,3)$.

  • Matching Colors with Primaries

    Suppose in order to match a given color $C(\lambda)$ the three primaries need to be mixed in proportions of $\beta_k,\;\;(k=1,2,3)$:

    
    

    $$C'(\lambda)=\sum_{k=1}^3 \beta_k P_k(\lambda)$$
    
    

    For the mixed color $C'(\lambda)$ to be perceived the same as the given color $C(\lambda)$, the responses of the three types of cone cells to $C'(\lambda)$ should be the same as those to $C(\lambda)$:
    

    $$r_i(C)=r_i(C') \;\;\;\;(i=1,2,3)$$
    
    

    The cone cells' responses to $C(\lambda)$ are
    

    $$r_i(C)=\int S_i(\lambda)C(\lambda) d\lambda\;\;\;\;\;(i=1,2,3)$$
    
    
    and their responses to the matching color $C'(\lambda)$ are
    

    $$r_i(C') = \int S_i(\lambda)C'(\lambda) d\lambda = \int [ \sum_{k=1}^3 \beta_k P_k(\lambda) ] S_i(\lambda) d\lambda$$

    $$= \sum_{k=1}^3 \beta_k \int S_i(\lambda)P_k(\lambda)d\lambda = \sum_{k=1}^3 \beta_k a_{ik}\;\;\;(i=1,2,3)$$

    
    
    where $a_{ik}$ is defined as the response of ith cells to the kth primary:
    
    $$a_{ik}\stackrel{\triangle}{=}r_i(P_k)= \int S_i(\lambda)P_k(\lambda) d\lambda \;\;(i,k=1,2,3)$$
    
    
    which can be found given the cone cells' sensitivities $S_i(\lambda)$ and the three primary colors $P_k(\lambda)$.

    For $C'(\lambda)$ to be perceived the same as $C(\lambda)$, we require
    

    $$r_i(C')=\sum_{k=1}^3 \beta_k a_{ik} = r_i(C)=\int S_i(\lambda)C(\lambda) d\lambda \;\;\;\;\;(i=1,2,3)$$
    
    
    These three equations are called the color matching equations. As both $a_{ik}$ and the right-hand side of the equations (available from the given $C(\lambda)$ and $S_i(\lambda)$) are known, the 3 coefficients $\beta_k$ $(i=1,2,3)$ can be obtained by solving the 3 color matching equations, and the matching color is produced by mixing the three primaries:
    
    $$C'(\lambda)=\sum_{k=1}^3 \beta_k P_k(\lambda) \equiv C(\lambda)$$
    
    

• CIE XYZ Primaries

  The Commission Internationale de l'Eclairage (CIE) defined three standard primaries called X, Y, and Z. Any color $C(\lambda)$ can be matched using these primaries with positive weights $X(C)$, $Y(C)$ and $Z(C)$.

  The chromaticity values of a color is defined by its weights for the three primaries normalized by the total energy $X+Y+Z$:

  
  

  $$x=\frac{X}{X+Y+Z},\;\;y=\frac{Y}{X+Y+Z},\;\;z=\frac{Z}{X+Y+Z}$$
  
  

  so that $x+y+z=1$. Chromaticity values depend on the hue and saturation of the color, but are independent of the intensity.

  All visible colors are represented by the points inside an enclosed area in the $X+Y+Z=1$ plane. And the chromaticity diagram is the projection of this enclosed area on $(X,Y)$ plane.