Accuracy of the Reconstruction

maintaining sufficient energy density at the focal point [61]. Their resolution is typically defined as the full width at half maximum (FWHM) of the system’s optical transfer func-tion or, respectively, point spread funcfunc-tion (PSF) in the case of a point-like object. The system PSF is a convolution of the illumination PSF, modeled in Fig. 2.2C, and the PSF resulting from the detection light path. Necessarily, the system PSF’s FWHM is smaller than or equal to the FWHM of the illumination PSF. Wilhelm et al. [61] provide simple equations to estimate the FWHM of an ideal confocal microscope. It is assumed that illu-mination and detection PSF are identical, which is only strictly true for the diffraction-limited case of an infinitely small pinhole diameter. The FWHM in the optical plane (FWHMlateral) and along the optical axis (FWHMaxial) are then estimated by

(Eq. 2.1)

(Eq. 2.2)

(Eq. 2.3)

with a = 0.37 and b = 0.64; λ denotes wavelength, and NA is the numerical aperture of the applied objective.

For high-resolution images it is advisable to minimize the pinhole diameter, select a dye with a small Stokes shift and an excitation wavelength as small as realizable with the available equipment. To estimate the resolution of the system under consideration the finite size of the pinhole (0.5 AU) needs to be considered by increasing the factors a and b in Eqs. (2.2) and (2.3) to 0.44 and 0.76, respectively; NA = 1.3, and λexcitation equals 405 nm, whereas λemission is the emission maximum of the V450 dye at 448 nm. The HCX PL APO 63×/1.3 GLYC CORR CS (21°) is designed to work with glycerol/water mixtures of a

re-fractive index of 1.451 at the mercury e-line (ne). Assuming a dispersion to 1.457 at a mean wavelength of 424 nm, the above wave-optical estimate of the FWHM yields 0.14 µm in the optical plane and 0.40 µm along the optical axis. Practically, the axial FWHM can be determined with a standard mirror test at 488 nm to characterize the quality of the applied objective [62]. For the utilized confocal system the axial FWHM was measured at 0.36 µm with this test.

The Kinetex particles examined exemplarily in this study have a nominal particle di-ameter of 2.6 µm. On the basis of the small ratio of particle didi-ameter and assumed optical resolution power of the microscope system, the elimination of all residual aberrations orig-inating from the sample setup will be of utmost importance for a reliable reconstruction of the packing, since spherical aberrations are the major source of deviation from ideal behav-ior and signal degeneration in optical microscopy. We therefore extend our previous dis-cussion on sample preparation [50] by additionally considering the dispersive behavior of the matching liquids.

To design a sample setup with minimized spherical aberrations it is necessary to elimi-nate the capillary wall to function as a lens and avoid refractive index mismatch between sample and embedding medium. Fortunately, this is unproblematic for silica-based packings since refractive indices of capillary wall and packing medium are almost identi-cal. They can be matched with a solvent of nD = 1.458 that mimics the dispersive behavior of fused silica (cf. Fig. 2.2B).This will leave a slight mismatch with respect to the applied objective which works optimal for a refractive index of ne =1.451 (cf. Fig. 2.2A). As a con-sequence, aberrations increase with sampling depth unless the immersion medium for the lens is also replaced by a medium with nD = 1.458. In that case, aberrations due to the re-fractive index mismatch are constant and independent from sampling depth. Since glycerol objectives are designed to be used with coverslips of nD = 1.523 an aberration correction for a 170 µm coverslip is incorporated into the lens system. Thus, it is possible to consider the resulting PSF of the system to be a function of the cov-erslip thickness t (illustrated in Fig. 2.2A) which describes a minimum when the aberrations introduced by matching liq-uids and coverslip are equivalent to a t = 170 µm coverslip.

Numerical solutions for the system PSF help to identify suitable values for t. We used

“PSF Lab” [59] to calculate the axial FWHM in dependence from t for illumination PSFs considering a focal length of 310 µm (280 µm of free working distance + 30 µm lens guard). The refractive indices of the matching liquid and the fused-silica wall at 405 nm

were assumed to be 1.470, whereas coverslip and objective were expected to disperse to n = 1.534 and 1.458, respectively. Fig. 2.2C presents the results of these calculations and illustrates that measurements with a coverslip of about t = 108 µm instead of a 170 µm coverslip are, in principle, free of aberrations. A practical implementation of these results is to select a “type 0” coverslip and fine-tune the correction collar of the objective accord-ingly. This will yield optimally resolved images from any part of the capillary.

Knowing the system PSF also enables the application of deconvolution techniques to the recorded image data. Schrader et al. [63] suggested that improvements in resolution by a factor of two to four can be achieved depending on the accuracy of the determined PSF.

Still, the accurate reconstruction and extraction of quantitative morphological data require the determination of the particles centers and blurred particle–void interface. The former can be achieved with high confidence by looking at the 2D-Fourier trans- form of the im-ages. In the Fourier transform of an image all features are characterized by sets of cies. The highest frequencies describe the object boundaries, whereas the lowest frequen-cies resemble slow intensity variations in the image background. Particles for HPLC are optimized for a narrow size distribution and good sphericity. The frequencies that charac-terize their individual size and their intensity distribution are thus located in a narrow band, whereas information on the particles positions is contained in higher frequency domains.

The application of a highpass filter with a cut-off frequency below the first harmonic of the size-characterizing frequencies or an optimized bandpass filter results in spots of maximal intensity at the particles centers in the inverse transformed image, which are dark spots in the original image. Therefore, the subtraction of the original image leaves a set of pixel-clusters whose geometrical centers define a list of all particle centers in the image. Merged particles of two or more cores are not problematic since they are typically detected by mul-tiple particle centers close together. A manual examination of the output is still required because, e.g., circular pores of comparable size will also be detected as a particle.

The knowledge of the particle centers significantly reduces the amount of calculations required to estimate the core–shell and shell–void interfaces. Though the optical transfer function of a complex object differs from the theoretical PSF of a confocal microscope, surfaces found in the image are still readily described by a three-dimensional Airy pattern which is the typical shape of a PSF in confocal microscopy (Fig. 2.6A). The location of the exact interface of the object itself is unknown, resulting in an increased relative error in

Figure 2.6: Deconvolved intensity signal along a core–shell particle (A) and its first derivative (B). The dye only reacts with the accessible silica surface, which is the porous shell, whereas the particle core remains dark resulting in a ring-like signal (A). However, the signal is not binary, but its intensity changes gradually. The intensity decay at the materials surface is described by an Airy function which is exemplarily displayed for a core–shell interface (purple). Since the ratio of the shell thickness to the resolution is small, the complete signal of a shell can be approximated by a Gaussian distribution (blue), or as a superposition of two Gaussian distribution functions when another particle is touched (orange). The intensity gradient becomes maximal close to the edges of the shell. Thus (B), which is the first derivative of (A), displays two maxima. Segmen-tation is performed at these maxima of the gradient image allocating voxels to a specific parti-cle’s core or shell (B).

final quantifications for objects that are merely resolved. Thus, placement of the object’s interface to generate a quantifiable image with defined regions that are segmented into different features is crucial and heavily depends on the applied segmentation algorithm.

However, in the case of particles additional information, like sphericity and approximate particle size, can be used to overcome the limiting depth resolution. Segmentation in the plane around the already known particle center has the highest accuracy, not only because of the better lateral microscopic resolution, but also because the particle diameter is maxi-mal and least stray light from adjacent particles is guaranteed.

With 0.35 µm the nominal thickness of the particle shell itself is only two to three times the lateral resolution. As a consequence, the recorded intensity signal contains a sin-gle maximum and is well approximated by a Gaussian distribution (see Fig. 2.6A). The maximum of this distribution may already vary for a single particle, but segmentation based on the signal’s maximum gradient can provide reproducible results. We decided to calculate the first derivative of each optical slice and draw a set of linear intensity profiles from the center of each particle to the surface of a virtually surrounding circle. In this way, each profile is recorded orthogonal to the particle surface so that any directional bias is eliminated from the calculations. Every pixel from the origin to the first maximum is iden-tified as belonging to the core and given a label with respect to the corresponding particle center. Pixels from the first to the second maximum are assigned to the particle’s shell. In the worst case, two particles of differing signal intensity touch and the interface between these particles will be shifted towards the brighter particle. Still, the difference to modeling each distribution function is only a few nanometers and does not effect the overall recon-struction or arithmetic mean of the shell thickness (Fig. 2.6). Finding the three-dimensional surface of the particle is the more error-prone task. From the segmented central particle plane we calculate a provisional core and particle diameter. We then assume the particle to be approximately spherical and extrapolate the surface to the other recorded layers. Imper-fections of the particle are taken into account by shifting the assumed surface to the nearest gradient maximum which results in a three-dimensional reconstruction of the rugged parti-cle surface (Fig. 2.5).