  We investigate the energy sources of the infrared (IR) emission and their
relation to the radio continuum emission at various spatial scales within the
Scd galaxy M33. We use the wavelet transform to analyze IR data at the Spitzer
wavelengths of 24, 70, and 160$\mu$m, as well as recent radio continuum data at
3.6cm and 20cm. An H$\alpha$ map serves as a tracer of the star forming regions
and as an indicator of the thermal radio emission. We find that the dominant
scale of the 70$\mu$m emission is larger than that of the 24$\mu$m emission,
while the 160$\mu$m emission shows a smooth wavelet spectrum. The radio and
H$\alpha$ maps are well correlated with all 3 MIPS maps, although their
correlations with the 160$\mu$m map are weaker. After subtracting the bright
HII regions, the 24 and 70$\mu$m maps show weaker correlations with the 20cm
map than with the 3.6cm map at most scales. We also find a strong correlation
between the 3.6cm and H$\alpha$ emission at all scales. Comparing the results
with and without the bright HII regions, we conclude that the IR emission is
influenced by young, massive stars increasingly with decreasing wavelength from
160 to 24$\mu$m. The radio-IR correlations indicate that the warm dust-thermal
radio correlation is stronger than the cold dust-nonthermal radio correlation
at scales smaller than 4kpc. A perfect 3.6cm-H$\alpha$ correlation implies that
extinction has no significant effect on H$\alpha$ emitting structures.
