ECE 638: Principles of Digital Color Imaging Systems Lecture 7: Discrete Wavelength Models Sensors

Synopsis Discretization of model Sensor subspace Decomposition of stimulus into fundamental and nullspace components Graphical interpretation

Why discretize the model? So far we have assumed that is continuous Upside of this assumption matches real world Downside of this assumption for computation, we must discretize the model we can gain further intuition if we descretize the model

Discretization of wavelength It is common practice to make the following choices: This implies that we need samples to characterize the visible spectrum.

Notation This material is taken from the paper by Mark Wolski: “A review of linear color descriptor spaces and their applications,” which is available for download at the course website. Wolski’s notation differs somewhat from that used thus far to describe the continuous-wavelength trichromatic model. For ease of cross-reference between the lecture notes and Wolski’s paper, we will follow his notation here.

Discrete-wavelength trichomatic model Stimulus Sensor response Response of the i-th channel Define Span(S) defines HVS subspace Stack sensor outputs

Characterization of span(S) Can write any vector in span(S) as for some coefficient

Projection of stimulus onto HVS Suppose Task: Find that vector which is closest to in the sense of the Euclidean norm, i.e. is the projection of onto

Solution for projection To solve problem, write for some vector (3x1) Solution (see HW): satisfies We can do two things with (1):

1. Obtain expression for Invert to obtain provided is nonsingular. We can assume that has rank 3 is nonsingular.

2. Identify nullspace component of We can write (1) as since Now we rearrange (2) as Let Then we have

Application to trichromatic model for color Consider an arbitrary stimulus Can write where This implies that

Restatement of color matching conditions

Graphical interpretation Let number of channels = 1 (achromatic vision) Let number of samples in wavelength = 2 Assume Denote stimulus as Sensor response is a scalar

Metamerism Stimuli and are metameric, i.e. they look the same to the observer with response

Non-equivalence of sensors Stimuli and elicit same response to sensor but not to sensor , i.e.

Nullspace component For any stimulus, where is the fundamental component and the response to the nullspace component is zero Question 1: Are there real stimuli that are invisible? Clarification By stimulus, we mean signals in the visible wavelength region By real stimulus, we mean , and there exists some for which

Response to monochromatic stimulus Consider monochromatic stimulus What is response to ? i.e. transpose of the i-th row of sensor response matrix

What does this tell us about Question 1? Are there real stimuli that are invisible? For a stimulus to be invisible, we must have The human visual system doesn’t have this property So the answer to Question 1 is “no”. Question 2: Are there real stimuli for which ? i.e. there is no blackspace component Such stimuli might be regarded as being visually efficient We will answer this question later