3D measurement of both front and back surfaces of transparent objects by polarization imaging

We present a method to recover the 3D shape of both front and back surfaces of smooth transparent objects, such as glass windows or containers. We use a combination of two methods known for the 3D reconstruction of specular surfaces: shape from distortion and shape from polarization. As each transparent surface reflects and transmits incident light, one can see two shifted images by observing the reflection of a pattern on two surfaces nearby. Looking at the reflection of one known point source on the front surface with a calibrated camera, the depth and the orientation of this surface can be determined up to a one dimensional space of solution. This ambiguity is lifted by using the degree of polarization of the reflection, which depends on the incidence angle. Supposing that the front surface is locally flat, we show that there is the same ambiguity between position and orientation for the observed reflection coming from the back surface of the object. This ambiguity can again be lifted by using ray-tracing and Mueller calculus. Thus our method enables to measure both the position and the orientation of the two surfaces of a transparent object, with only one polarimetric image. We present an experiment on real objects to evaluate this method.