Digital Photomicrography

We wish to digitally record with a CCD or CMOS sensor the images obtained with a microscope, for example equipped with an objective \(10\times /0.25\) or \(40\times /0.70\) designed to work with an eyepiece of field number 22. We have to form a real image of the observed object at a finite distance, on the sensor. In that goal, we can either directly use the intermediate image formed by the objective (and the tube lens, if there is a tube lens), or use an additional optic called projective ocular which will make another image of this intermediate image. The first solution is rarely chosen on the current microscopes for various practical reasons: it can be mechanically difficult to access the intermediate image, it can be preferable to simultaneously perform standard visual observation and digital acquisition, the intermediate image might have some residual aberrations,...; therefore we will only consider the second solution, and we will look at the required characteristics for the projective ocular. The problem is to find a good compromise between resolution and observed field. Indeed, the image formed by the sensor must, rigorously, follow the Shannon sampling theorem to be correctly registered without artifacts ('aliasing'). This means that the image spatial cutoff frequency must be smaller than the sensor Nyquist frequency, i.e. to half its sampling frequency. Noting \(g_{y,~ocProj}\) the magnification of the projective ocular in the conjugation : intermediate image → sensor, one must have :

2. NA obj λ × 1 | g y | ⋅ | g y , ocProj | < ~ F NyquistSensor = F Sampl . Sensor 2 {2. NA_{nitalic obj}} over %lambda times {1 over { lline g_y rline cdot lline g_{y,nitalic ocProj} rline}} " " {"<"} csub {"~"}" " F_{nitalic NyquistSensor}" "=" "F_{nitalic Sampl.Sensor} over 2

For a standard 'video' sensor of format ½ inch (i.e. of dimensions \(6.4mm\; \times \;4.8mm\)), with \(752\times 582~pixels\), and of pixel pitch \(\sim 9~\mu m\), the sensor spatial sampling frequency is \(\sim 110~mm^{–1}\) and its Nyquist frequency is \(\sim 55~mm^{–1}\). Noting that the cutoff frequency of the intermediate image, \(2\cdot NA_{obj} / (\lambda \cdot |g_y | )\), is \(\sim 90~mm^{–1}\) for the objective \(10\times /0.25\) and \(65~mm^{–1}\) for the objective \(40\times /0.70\), we therefore deduce that \(|g_{y,~ocProj}|\) must be close to (or larger than) \(\sim 1.6\) in the first case, and \(\sim 1.2\) in the second case. In the first case, for the sensor to cover the same field of view than the one seen with the ocular of field number 22, it must have a width of \(1,6\times 22~mm = \sim 35~mm\)! Thus, with a video sensor \(1/2 ”\) of width \(6.4~mm\), the observed image linear dimension cannot exceed, respectively, a fifth or a fourth of the image observed with the eye (\(1/25\) or \(1/16\) if we speak about surfaces) in order to not undersample the image and to benefit from the objective full resolution.

For a digital camera sensor of \(8~Mpixels\) (e.g. \(2448\times 3264\)) of dimensions \(\sim 10.8~mm× \sim 8.1~mm\) (\(\sim 13.5mm\) on the diagonal) and of pixel pitch \(\sim 3.3~\mu m\), the spatial sampling frequency is \(\sim 300~mm^{–1}\) and the Nyquist frequency \(\sim 150~mm^{–1}\). The projective ocular magnification, \(|g_{y,~ocProj}|\), must therefore be on the order of (or larger than) \(\sim 0.61\) for the objective \(10\times /0.25\), and \(\sim 0.43\) for the objective \(40\times /0.70\) to respect Shannon sampling theorem. The field-of-view of diameter \(22~mm\) in the intermediate image space would therefore be projected onto dimensions of \(13.4~mm\) and \(9.5~mm\) (respectively), which are smaller than the sensor dimensions. In that case, the magnification will be increased so that the circular image covers the rectangular sensor, which means that the intermediate image field will be matched with the diagonal of the sensor; a projective ocular magnification of \(13.5/22=\sim 0.62\) would therefore be well-adapted for the two objectives considered here.