Quantimetric imaging

The quantity, denoted $q$, of light, to which an image sensor responds, is known as the photoquantity [7] which is neither radiometric nor photometric, but nevertheless provides quantifiable units of light. Quantimetric imaging, also known as (photo)quantigraphic imaging [7] measures the quantity of light in units that depend on the spectral response of the particular sensor used in a particular camera, rather than in units of flat spectral response (radiometry) or in units of the human eye's response (photometry).

Differently exposed images (e.g. individual frames of video) of the same subject matter are denoted as vectors: $f_0$, $f_1$, $\ldots$, $f_i$, $\ldots$, $f_{I-1}$, $\forall i, 0 \leq i < I$.

Each video frame is some unknown function, $f()$, of the actual quantity of light, $q({\bf x})$ falling on the image sensor: f_i = f(k_i q ( A_ix+b_i c_ix+d_i )), where ${\bf x} = (x,y)$ denotes the spatial coordinates of the image, $k_i$ is a single unknown scalar exposure constant, and parameters ${\bf A}_i$, ${\bf b_i}$, ${\bf c}_i$, and $d_i$ denote the projective coordinate transformation between successive pairs of images: $A\in \R^{2\times2}$ is the linear coordinate transformation (e.g. accounts for magnification in each of the $x$ and $y$ directions and shear in each of the $x$ and $y$ directions), $b$ is the translation in each of these two coordinate directions, and $c$ is the projective chirp rate in each of these two coordinate directions[3]. The additional constant $d$ makes the coordinate transformation into a group.

For simplicity, this coordinate transformation is assumed to be able to be independently recovered (e.g. using the methods of [3]). Therefore, without loss of generality, in this paper, it will be taken to be the identity coordinate transformation, which corresponds to the special case of images differing only in exposure.

Without loss of generality, $k_0$ will be called the reference exposure, and will be set to unity, and frame zero will be called the reference frame, so that $f_0=f(q)$. Thus we have: 1k_i f^-1 (f_i) = f^-1 (f_0), i, 0<i<I.

Taking the logarithm of both sides, F^-1 (f_i) - K_i = F^-1 (f_0), i, 0<i<I, where $K=log(k)$, and $F^{-1}$ is the logarithmic inverse camera response function (e.g. a LookUp Table converting pixel values into exposure values).

Re-arranging, we have: F^-1 (f_i) - F^-1 (f_0) = K_i, i, 0<i<I. This relation suggests a way to estimate the camera response function, $f$, from a pair of differently exposed images of the same subject matter. Before estimating the camera response function, we consider how the noise will affect the estimation.