Most radio galaxies are hosted by giant gas-poor ellipticals, but some contain significant amounts of dust, which is likely to be of external origin. In order to characterize the mid-IR properties of two of the most nearby and brightest merger-remnant radio galaxies of the Southern hemisphere, NGC 1316 (Fornax A) and NGC 612 (PKS 0131-36), we used observations with the Wide-field Infrared Survey Explorer (WISE) at wavelengths of 3.4, 4.6, 12 and 22 micron and Spitzer mid-infrared spectra. By

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... lying a resolution-enhancement technique, new WISE images were produced at angular resolutions ranging from 2.6" to 5.5". Global measurements were performed in the four WISE bands, and stellar masses and star-formation rates were estimated using published scaling relations. Two methods were used to uncover the distribution of dust, one relying on two-dimensional fits to the 3.4 micron images to model the starlight, and the other one using a simple scaling and subtraction of the 3.4 micron images to estimate the stellar continuum contribution to the emission in the 12 and 22 micron bands. The two galaxies differ markedly in their mid-IR properties. The 3.4 micron brightness distribution can be well represented by the superposition of two Sersic models in NGC 1316 and by a Sersic model and an exponential disk in NGC 612. The WISE colors of NGC 1316 are typical of those of early-type galaxies; those of NGC 612 are in the range found for star-forming galaxies. From the 22 micron luminosity, we infer a star-formation rate of about 0.7 solar masses per year in NGC 1316 and about 7 solar masses per year in NGC 612. Spitzer spectroscopy shows that the 7.7-to-11.3 micron PAH line ratio is significantly lower in NGC 1316 than in NGC 612. The WISE images reveal resolved emission from dust in the central 1'-2' of the galaxies. (Abridged)