  Long-baseline interferometry is an important technique to spatially resolve
binary or multiple systems in close orbits. By combining several telescopes
together and spectrally dispersing the light, it is possible to detect faint
components around bright stars. Aims. We provide a rigorous and detailed method
to search for high-contrast companions around stars, determine the detection
level, and estimate the dynamic range from interferometric observations. We
developed the code CANDID (Companion Analysis and Non-Detection in
Interferometric Data), a set of Python tools that allows us to search
systematically for point-source, high-contrast companions and estimate the
detection limit. The search pro- cedure is made on a N x N grid of fit, whose
minimum needed resolution is estimated a posteriori. It includes a tool to
estimate the detection level of the companion in the number of sigmas. The code
CANDID also incorporates a robust method to set a 3{\sigma} detection limit on
the flux ratio, which is based on an analytical injection of a fake companion
at each point in the grid. We used CANDID to search for the companions around
the binary Cepheids V1334 Cyg, AX Cir, RT Aur, AW Per, SU Cas, and T Vul.
First, we showed that our previous discoveries of the components orbiting V1334
Cyg and AX Cir were detected at > 13 sigmas. The companion around AW Per is
detected at more than 15 sigmas with a flux ratio of f = 1.22 +/- 0.30 %. We
made a possible detection of the companion orbiting RT Aur with f = 0.22 +/-
0.11 %. It was detected at 3.8{\sigma} using the closure phases only, and so
more observations are needed to confirm the detection. We also set the
detection limit for possible undetected companions. We found that there is no
companion with a spectral type earlier than B7V, A5V, F0V, B9V, A0V, and B9V
orbiting V1334 Cyg, AX Cir, RT Aur, AW Per, SU Cas, and T Vul, respectively.
