We have measured the center-to-limb variation (CLV) of parameters describing geometric and photometric statistical properties of the solar granulation at 6708 Å. This work is based on an excellent series of white-light images obtained with the Swedish Vacuum Solar Telescope at Roque de los Muchachos Observatory, La Palma, during the partial solar eclipse of 1994 May 10. The lunar limb profile, which is visible in each frame, was used as a calibration tool for estimating the point-spread function of the combined optical system formed by the atmosphere and the telescope. Before restoration, noise was removed from the images by a novel application of the so-called optimum filter for two-dimensional objects. The latter was optimized in terms of rms error and was constructed from very precise smoothed models of the specific power spectrum of the granulation at each position on the solar disk. The determination of the positions on the solar disk was achieved with high accuracy by matching the position of the Moon's limb in our images to a numerical simulation of the eclipse geometry. The CLV curve of the ΔIrms granular contrast shows one of the steepest gradients among those reported in the literature and quite a high value (9.6%) at the disk center considering that our working wavelength is in the far-red range of the solar spectrum. The elliptical shape of the restored power spectra with ellipticities equal to those expected just from foreshortening proves that radiative transfer effects do not alter the isotropy of the horizontal intensity pattern of the solar granulation, at least up to μ=0.4. The mean wavenumber, k, derived from the two-dimensional power spectra azimuthally integrated along the ellipses amounts to a value of 6.15 Mm-1 at the center of the solar disk and then shows a decrease toward the limb. Apart from the power spectra analysis, a direct statistical study of the granulation size and brightness, based on the image segmentation for defining granular contours, has also been performed. A general increase in both granular and intergranular areas is found as we move toward the solar limb. The mean granular cell area varies from 1.36 Mm2 at μ=1 up to 2.06 Mm2 at μ=0.6, and in parallel, the granular filling factor (the percentage of area of the image covered by granules) decreases from 44.2% to 42.8%. In the small area range, the granular brightness increases linearly with the granular cell size and is preserved constant, on average, for granular cells larger than ~2.0". No slope variation is found for the intergranular intensities versus granular cell areas. Observations close to the solar limb detect granular structures as small as 0.53" or even smaller up to a distance of at least ~0.5" from the limb, showing that the ΔT associated with the granulation persist at least until z~200 km. However, this penetration could be different for small and large granules because we find several hints indicating the progressive disappearance of small structures toward the limb.