Handling multiple materials for exposure of digital forgeries using 2-D lighting environments

Abstract

The distribution of incident light is an important physics-based cue for exposing image manipulations. If an image has been composed from multiple sources, it is likely that the illumination environments of the spliced objects differ. Johnson and Farid introduced a proof-of-principle algorithm for a forensic comparison of lighting environments. However, this baseline approach suffers from relatively strict assumptions that limit its practical applicability. In this work, we address one of the biggest limitations, namely the need to compute a lighting environment from patches of homogeneous material. To compute a lighting environment from multiple-color surfaces, we propose a method that we call “intrinsic contour estimation” (ICE). ICE is able to integrate reflectances from multiple materials into one lighting environment, as long as surfaces of different materials share at least two similar normal vectors. We validate the proposed method in a controlled ground-truth experiment on two datasets, with light from three different directions. These experiments show that using ICE can improve the median estimation error by almost 50 %, and the mean error by almost 30 %.

Appendices

Appendix

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Intrinsic dataset error due to indoor illumination

We perform an error calculation to illustrate the effect of a light source at finite distance, as used for capturing both datasets. A light source at a finite distance effectively casts a cone of light. Assuming that object normals are selected from a light cone of 50cm diameter around the object center, only the central ray of the cone is with exactly the same angle incident to the object as if the rays were parallel. The maximum angular deviation of a ray incident to the object is

$$ \phi_{\text{max,incand}} \tan^{-1}(25\,\text{cm}/150\,\text{cm}) = 9.46^{\circ}\enspace, $$

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which occurs at the outer boundary of the cone. Under the method’s assumption of Lambertian reflectance, this error propagates into the cosine between the ray direction and the surface normal. The derivative (and hence variation) of the cosine is maximum at π/2, with sin(π/2)=1. Distributing the error symmetrically around π/2 leads to a upper bound of the intrinsic dataset error of

$$ e_{\text{orth,incand}} = 2\cdot \cos(\pi/2 - (9.46^{\circ}/2)) = 9.44^{\circ}. $$

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This worst case bound occurs if a normal is located at the cone-beam boundary and is directed orthogonally to the light source. If a normal lies at the cone beam boundary but points towards the light source, the intrinsic dataset error is negligible, i. e.,

$$ e_{\text{towards,incand}} = 1- (2\cdot \cos(9.46^{\circ}/2))=0.39^{\circ}. $$

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Similarly, the closer a normal is to the center of the cone-beam, the smaller is its estimation error. In other words, normals that are orthogonal to the light source and located at the cone boundary exhibit the same intensities as if the light source-contour angle were 90°+9.44°≈100° degrees under an infinitely distance light source. All normals that do not adhere to both of these conditions exhibit a lower error, with minima at the center of the cone-beam or whenever normals are parallel to the central ray of the light source.

Performing the same calculation for the flash dataset yields ϕ _max,flash = 9.04° for the closest (“ 0°”) light source, and ϕ _max,flash = 6.20° for the most distant (“ 60°”) light source. The maximum errors e _orth,flash are therefore 9.03° and 6.20°, respectively. The minimum errors e _{towards,flash} are 0.36° and 0.17°, respectively.