Strong light bridges (SLBs) represent an abrupt change in the physical conditions of sunspot umbrae. They divide the umbra into separate units (fragments, umbral cores) and very often show a granular structure. A photometric and spectroscopic study of SLBs, and also of abnormal granulation (AG) in a region of pores, is presented. Slit-jaw images (bandpass 5425 +/- 50 A) and spectra (line Fe I 5434.5 A), with a spatial resolution of 0.3 sec, were acquired during the period 1991 July 1-10 at the Swedish Solar Telescope at the Observatorio del Roque de los Muchachos (La Palma). The power spectra of intensity fluctuations in the slit-jaw images, were used as the primary diagnostic tool, complemented by the analysis of line profiles. The following results were obtained. 1. The structures present in SLBs and AG are generally smaller than the granules in the quiet photosphere. The typical size of SLB granules is 1.2 sec (in quiet granulation, 1.5 sec). In AG, the distribution of power is more complex and is characterized by peaks corresponding to scales of 2.5 sec, 1.3 sec, and 0.64 sec. 2. SLB and AG power spectra show an excess of power (compared to quiet granulation) at scales of 0.5 sec. This power enhancement reflects the presence of small bright grains, clearly visible in SLBs and AG, with a mean nearest neighbor distance of 0.5 sec. 3. Two of these small bright grains, together with a dark lane between them, were resolved spectroscopically in a SLB. The line shifts and bisector shapes suggest a convective origin of these structures. 4. The SLB power spectra in the log P/log k scale indicate the presence of a Kolmogorov turbulent cascade in the structures between 1.7 sec and 0.64 sec. However, the small bright grains do not pertain to the turbulent cascade. They are particular phenomena, probably with a convective origin. The power spectrum of AG does not give any indication of the presence of a turbulent cascade. 5. Small bright features (umbral dots or bright grains) can be found in unbral cores, light bridges, and AG regions as well.