First-principle study of bismuth-related oxygen-deficient centers ($=$Bi$cdots$Ge$equiv$, $=$Bi$cdots$Si$equiv$, and $=$Bi$cdots$Bi$=$ oxygen vacancies) in Bi$_2$O$_3$-GeO$_2$, Bi$_2$O$_3$-SiO$_2$, Bi$_2$O$_3$-Al$_2$O$_3$-GeO$_2$, and Bi$_2$O$_3$-Al$_2$O$_3$-SiO$_2$ hosts is performed. A comparison of calculated spectral properties of the centers with the experimental data on luminescence emission and excitation spectra suggests that luminescence in the 1.2-1.3 $mu$m and 1.8-3.0 $mu$m ranges in Bi$_2$O$_3$-GeO$_2$ glasses and crystals is likely caused by $=$Bi$cdots$Ge$equiv$ and $=$Bi$cdots$Bi$=$ centers, respectively, and the luminescence near 1.1 $mu$m in Bi$_2$O$_3$-Al$_2$O$_3$-GeO$_2$ glasses and crystals may be caused by $=$Bi$cdots$Ge$equiv$ center with (AlO$_4$)$^-$ center in the second coordination shell of Ge atom.
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