1. May 9, 2005 Andrew C. Gallagher1 CRV2005 Detection of Linear and Cubic Interpolation in JPEG Compressed Images Andrew C. Gallagher Eastman Kodak Company andrew.gallagher@kodak.com

2. May 9, 2005 Andrew C. Gallagher2 CRV2005 The Problem An image consists of a number of discrete samples. Interpolation can be used to modify the number of and locations of the samples. Given an image, can interpolation be detected? If so, can the interpolation rate be determined?

3. May 9, 2005 Andrew C. Gallagher3 CRV2005 The Concept A interpolated sample is a linear combination of neighboring original samples y(n 0 ). The weights depend on the relative positions of the original and interpolated samples. Thus, the distribution (calculated from many lines) of interpolated samples depends on position. Original samples and an interpolated sample n0n0 n1n1 n2n2 y(n 0 )y(n 1 )y(n 2 ) x0x0 i(x 0 )

4. May 9, 2005 Andrew C. Gallagher4 CRV2005 The Periodic Signal v(x) The second derivative of interpolated samples is computed. The distribution of the resulting signal is periodic with period equal to the period of the original signal. The expected periodic signal can be calculated explicitly for specific interpolators, assuming the sample value distribution is known. n0n0 n1n1 n2n2 y(n 0 )y(n 1 )y(n 2 ) x0x0 i(x 0 ) x0-x0- i(x 0 -  ) x0+x0+ Original samples and an interpolated sample n0n0 n1n1 n2n2 Distribution (standard deviation) of interpolated samples 1 period  v(x)

5. May 9, 2005 Andrew C. Gallagher5 CRV2005 The Periodic Signal v(x) Linear Interpolation Cubic Interpolation This property can be exploited by an algorithm designed to detect linear interpolation.

6. May 9, 2005 Andrew C. Gallagher6 CRV2005 The Algorithm In Matlab, the first three boxes can be executed as: –bdd =diff(diff(double(b))); –bm = mean(abs(bdd),2); –bf = fft(bm); A peak in the DFT signal corresponds to interpolation with rate compute second derivative of each row p(i,j) average across rows compute Discrete Fourier Transform estimate interpolation rate N (assuming no aliasing)

7. May 9, 2005 Andrew C. Gallagher7 CRV2005 Resolving Aliasing The algorithm produces samples of the periodic signal v(x). Aliasing occurs when sampled below the Nyquist rate (2 samples per period). All interpolation rates will alias to N. Only two possible solutions for rates N>1 (upsampling). An infinite number of solutions for N<1. The correct rate can often be determined through prior knowledge of the system. (Q a positive integer)

8. May 9, 2005 Andrew C. Gallagher8 CRV2005 Example Signals: N = 2 After summing across rows v(x) After computing DFT

9. May 9, 2005 Andrew C. Gallagher9 CRV2005 Example DFT Signals

10. May 9, 2005 Andrew C. Gallagher10 CRV2005 Effect of JPEG Compression Heavy JPEG compression appears similar to an interpolation by 8. Therefore, the peaks in the DFT signal associated with JPEG compression must be ignored. digital zoom by N o(i,j) JPEG compression p(i,j) After computing DFT no interpolation JPEG compression interpolation N = 2.8 JPEG compression

11. May 9, 2005 Andrew C. Gallagher11 CRV2005 Experiment A Kodak CX7300 (3MP) captured 114 images 13 images were non-interpolated. The remainder were interpolated with rate between 1.1 and 3.0. N

12. May 9, 2005 Andrew C. Gallagher12 CRV2005 Results Algorithm correctly classified between interpolated (101) and non-interpolated (13) images. Interpolation rate was correctly estimated for 85 images. For the remaining 16 images, the algorithm identified the interpolation rate as either 1.5 or 3.0. This is correct but ambiguous.

13. May 9, 2005 Andrew C. Gallagher13 CRV2005 Conclusions Linear interpolation in images can be robustly detected. The interpolation detection algorithm performs well even when the interpolated image has undergone JPEG compression. The algorithm is computationally efficient. The algorithm has commercial applications in printing, image metrology and authentication.