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Exposing Digital Forgeries Through Chromatic Aberration

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

Multimedia Security

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Chromatic Aberration

• Ideal imaging system– Light passes through the lens 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 wavelengths of light.

• Lateral chromatic aberration– Lateral aberration manifests itself as a spatial shift in the

locations when light of different wavelengths reach the sensor.

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Lateral Chromatic Aberration (1/2)

• 1-D aberration– Snell’s Law : )sin()sin( ffnn

00 )( xxxxxx

br

br

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Lateral Chromatic Aberration (2/2)

• 2-D aberration– This model is simply an expansion/contraction about the center of t

he image.– It is common for lens designers to try to minimize chromatic aberrat

ion in lenses.

),( brbr yyxxv

),( 00 yx

00

00

)()(),(),(

yyyyxxxxyxyx

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bbrr

Expansion center :Each vector :

Model parameters : ),,( 00 yx

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Estimating Chromatic Aberration

• Assume that the lateral chromatic aberration is constant within each color channel (RGB).

• Using green channel as reference, estimating the aberration between– the red and green channels, the model parameters– the blue and green channels, the model parameters

• Seek the best model parameters to approximate following equations, which bring color channels back into alignment.

),,( 111 yx),,( 222 yx

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)(

)(

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xxxx

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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 pixel intensities of

• The model parameters are determined by maximizing the mutual information as follows (using brute-force iterative search):

Rr Gg gPrPgrPgrPGRI)()(),(log),();(

);(maxarg111 ,, GRIyx

),(),( ggrr yxGandyxR

111

111

)(

)(

yyyy

xxxx

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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 :

• The average angular error over all P pixels in the image is :

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cos),(

))(())((

),(

))(())((

),(

),(),(

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yyyyxxxx

yxv

yyyyxxxx

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),(1

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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.• Distortion 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.

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Experiment of calibrated images

• The goal of this part is finding actual parameters.• Calibration target : a board with ¼-inch diameter holes spaced

1-inch apart. The camera 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 mea

sured and modeled displacements to approximate the actual parameters.

),,( 000 yx

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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 :

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.

Quality 95% JPEG : error 26.1 degrees with 93.7% < 60 degrees

Quality 85% JPEG : error 26.7 degrees with 93.4% < 60 degrees

Quality 75% JPEG : error 28.9 degrees with 93.2% < 60 degrees

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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) Only consider 50 blocks with the largest gradients.

),(),(),( 22 yxIyxIyxI yx

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Experiment of forensics cont.

• Errors are estimated over 50 blocks per 205 images.• Result :

Average angular error 14.8 degrees with 98% < 60 degrees That is, block error > 60 degrees is considered to be tampered.

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Conclusion

• Digital cameras 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.