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Abstract: The radio-infrared correlation was explained as a direct and linear relationship between star formation and IR emission. However, one fact making the IR-star formation linkage less obvious is that the IR emission consists of at least two emission components, cold dust and warm dust. The cold dust emission may not be directly linked to the young stellar population. Furthermore, understanding the origin of the radio-IR correlation requires to discriminate between the two main components of the radio continuum emission, free-free and synchrotron emission. Here, we present a multi-scale study of the correlation of IR with both the thermal and non-thermal (synchrotron) components of the radio continuum emission from the nearby galaxies M33 and M31.