We have studied the $4d$ photoabsorption process of I, ${\mathrm{I}}^{+},$ ${\mathrm{I}}^{2+},$ and ${\mathrm{I}}^{3+}$ by using linear density-response theory together with density-functional theory to take into account the dynamic electron correlation, which is crucially important in reproducing the $4d$ giant resonance. The current approach, however, differs from previous approaches of this type in that we use an optimized effective potential and explicitly incorporate a self-interaction correction. We found that the $4d$ photoabsorption spectrum of the ${\mathrm{I}}^{+}$ ion is almost the same as that of the neutral I atom, and that in moving from I to ${\mathrm{I}}^{3+}$ the spectrum changes from one notably characterized by a giant resonance into one with multiple sharp peaks. Convoluting the spectra with the experimental energy resolution yielded spectra in excellent agreement with absolute experimental photoabsorption cross-section spectra. Moreover, our results showed better agreement with experiment than earlier theoretical studies. In addition, increasing the energy resolution revealed peaks not observed in the experimental spectra.