Camera Calibration Technical Report:
Camera Calibration

Camera Calibration

We wish to establish the dependence of the measured image value, g, on the light intensity, I, incident on the measuring system which can be taken to include the microscope from the objective to the camera, the camera and its internal electronics, the video transmission cable and finally the digitisation board in the host computer. The calibration is to be performed with respect to a reference density, ${\delta \rho}$, with transmission coefficient ${\delta t}= 10^{-{\delta \rho}}$. The experimental setup in highly schematic form is shown in figure 1 where the reference density can be inserted or removed from the light path as required.

  

Figure: Schematic form of the camera calibration experimental setup.

In figure 1 the camera is represented as a single detector and for convenience in the subsequent implementation of the theory we convert the measured value to a value which goes to zero as the intensity goes to zero. This makes no difference to the following derivation but in the application of the method to video cameras it is important to take proper account of the dark-field image. Therefore we define

G(I) = g(I) - g(0),

and clearly G(0) = 0.

For each value I of the incident light intensity we can measure two values from the detector, G and G' by removing and inserting the reference density respectively. From the definition of the transmission coefficient we know that Gand G' correspond to two values of intensity I and I' respectively, related by

$$I' = {\delta t}I = 10^{-{\delta \rho}} I.$$

By varying the incident intensity it is possible to generate the curve G' = F(G(I)) as a function of G(I) parameterised by the (unknown) I. This curve will range from zero to the maximum measurable value which will depend on the system but is commonly 255.

We now define a sequence of values $\{G(I_i) \mid i > 0\}$ by repeated application of the function F viz.

G(I_i) = F(G(I_i-1)),

G₀ is arbitrary and the sequence is terminated when $\vert G_i - G_{i-1}\vert < \Delta$. $\Delta$ should be determined by the noise characteristics, in this paper we used $\Delta = 1.0$. From above the sequence of values I_i are related by $I_i = {\delta t}I_{i-1}$ and by repeated application we establish

$$I_i = {{\delta t}}^i I_0,$$

which implies that we have calibrated the measured values G_i in terms of a single unknown intensity I₀ and the known transmission coefficient ${\delta t}$ i.e.

$$G_i = H( {{\delta t}}^i I_0 ).$$

We can equally regard this as a function of density,

$$G_i = \overline{H}( \rho_0 + i \delta\rho ).$$

This iteration can be understood in graphical terms as shown in figure 2. The curved line represents a typical measured set of (bright, dark) values of G. The diagonal line is simply G(I') = G(I). The iteration above corresponds to tracing the dashed horizontal and vertical lines to each new value in the sequence.

  

Figure: Diagramatic representation of the iterative scheme to establish the sequence of measured values that correspond to a geometric progression of intensity values or equivalently, equal increments of the reference density

To fill in values of the response function Gbetween those points determined from the curve at intervals of ${\delta \rho}$we must invoke some additional constraint. The constraint can only arise from the nature of the physical device, for example that the device responds smoothly and monotonically with respect to intensity or exhibits a particular asymtotic behavior. We establish density values for all possible measured values by interpolating with respect to log(G), which for most CCD cameras is approximately linear. We use cubic spline interpolation [5].

Once the function $G({\rho}_0 + i {\delta \rho})$ has been established then optical densities are determined by measuring the response with and without the object of interest and then to subtract the corresponding densities. In practice this implies the storage of the bright-field image because the illumination is almost never sufficiently even that shading effects can be ignored. Clearly the method can equally determine transmission coefficients, but now the bright-field correction involves division rather than subtraction.