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Exposing Digital Forgeries Through Chromatic Aberration

Micah K. Johnson, Hany Farid, MM&Sec’06, September 26-27, 2006, Geneva, Switzerland

Information Technologies for IPR Protection

1 Chromatic Aberration

• Ideal imaging system – passes through the and is focused to a single point on the sensor.

• Longitudinal chromatic aberration – Longitudinal aberration manifests itself as differences in the focal planes for different of light.

• Lateral chromatic aberration – Lateral aberration manifests itself as a spatial shift in the locations when light of different wavelengths reach the sensor.

2 Lateral Chromatic Aberration (1/2)

• 1-D aberration

– Snell’s Law : θ = nn θ ff )sin()sin(

≈ xx br

r α b )( +−= xxxx 00

3 Lateral Chromatic Aberration (2/2)

• 2-D aberration – This model is simply an expansion/contraction about the center of the image. – It is common for lens designers to try to minimize chromatic aberration in .

Expansion center : yx 00 ),( r Each vector : = − − yyxxv brbr ),(

rr ≈ α yxyx bb ),(),(

r α b )( +−= xxxx 00

r α b )( +−= yyyy 00

Model parameters : yx 00 α),,(

4 Estimating Chromatic Aberration

• Assume that the lateral chromatic aberration is constant within each channel (RGB). • Using green channel as reference, estimating the aberration between

– the red and green channels, the model parameters yx α111 ),,(

– the blue and green channels, the model parameters yx α 222 ),,( • Seek the best model parameters to approximate following equations, which bring color channels back into alignment.

r = α1 g − )( + xxxx 11

r α1 g )( +−= yyyy 11

5 Alignment by Mutual Information

• A metric based on mutual information has been proven successful in such situation. • The mutual information between R and G, which are the random

variables from the intensities of rr yxGandyxR gg ),(),(

r = α1 g − )( + xxxx 11

r α1 g )( +−= yyyy 11

⎛ grP ),( ⎞ GRI = ∑∑ grP log),();( ⎜ ⎟ rG∈∈R g ⎝ gPrP )()( ⎠

• The model parameters are determined by maximizing the mutual information as follows (using brute-force iterative search): GRI );(maxarg yx ,, α111

6 Quantify Estimated Error

• Using average angular error to quantify the error between the estimated and known model parameters. • The angular error can be computed by the displacement vectors : r = − − yyxxyxv grgr ),(),( ⎛ α ))(( −+− xxxx ⎞ r yxv ),( = ⎜ 0 g 0 0 g ⎟ 0 ⎜ ⎟ ⎝ α 0 g 0 0 ))(( −+− yyyy g ⎠ ⎛ α ))(( −+− xxxx ⎞ r yxv ),( = ⎜ 1 g 11 g ⎟ 1 ⎜ ⎟ ⎝ α 1 g 1 1 ))(( −+− yyyy g ⎠ ⎛ ⋅ vv rr ⎞ −1 ⎜ 10 ⎟ θ yx = cos),( ⎜ rr ⎟ ⎝ vv 10 ⎠

• The average angular error over all P in the image is : 1 θ = ∑ θ yx ),( P , yx 7 Experiment of synthetic images

• Generate 512*512 color image with anti-aliased discs of various size and color. • Simulate aberration by warping blue channel to green channel. • center is image center. • Chosen 40 α between 1.0004 and 1.0078, 50 images for each. • Average angular error : 3.4 degrees • 93% error < 10 degrees • The result demonstrate the general efficacy of this algorithm.

8 Experiment of calibrated images

• The goal of this part is finding actual parameters. yx α000 ),,( • Calibration target : a board with ¼-inch diameter holes spaced 1-inch apart. The take 500 holes in each picture. • For each channel, computing the center of each hole. Compute displacements of (R, G) and (B, G). • Using brute-force search by minimizing r.m.s between the measured and modeled displacements to approximate the actual parameters.

9 Experiment of calibrated images cont.

• Test the efficacy of this approach on real images. • Use the same camera and calibrated lens. • Image size 3020*2008, TIFF format, 205 images. •Results : ¾ Average angular error is 20.3 degree with 96.6% < 60 degrees. Much error is due to other aberrations, that are not considered in this model. ¾ Quality 95% JPEG : error 26.1 degrees with 93.7% < 60 degrees ¾ Quality 85% JPEG : error 26.7 degrees with 93.4% < 60 degrees ¾ Quality 75% JPEG : error 28.9 degrees with 93.2% < 60 degrees

10 Experiment of forensics

• Based on inconsistent chromatic aberration. • Assume only small portion has been manipulated. • Then use global estimate compare against block estimates. • Judge : Any block that derivates significantly from the global estimate is suspected of having been manipulated. • Difficult to estimate aberration from block with little spatial frequency content. (e.g., sky) ¾ Only consider 50 blocks with the largest gradients.

2 2 ∇ = x + y yxIyxIyxI ),(),(),(

11 Experiment of forensics cont.

• Errors are estimated over 50 blocks per 205 images. •Result : ¾ Average angular error 14.8 degrees with 98% < 60 degrees ¾ That is, block error > 60 degrees is considered to be tampered.

12 13 Conclusion

• Digital introduce an inherent amount of noise uniformly spread across an image. When creating tampering, it is common to contain inconsistent pattern. Usually detect tamper image by those inconsistent pattern.

• The general digital forensics approach : – First, statistical changes associated with specific types of tampering. – Then, detection methods are designed to estimate these changes and differentiate.

• There is no general forensics detection method for all types of tampering. But the more detection method can provide the more confidence on tampering detection.

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