Andrew C. Gallagher1

1
Detection of Linear and Cubic Interpolation in
JPEG Compressed Images

• Andrew C. Gallagher
• Eastman Kodak Company
• andrew.gallagher_at_kodak.com

2
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
The Concept
i(x0)
y(n0)
y(n1)
y(n2)

• A interpolated sample is a linear combination of
  neighboring original samples y(n0).
• 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.

x0
n0
n1
n2
Original samples and an interpolated sample
4
The Periodic Signal v(x)
i(x0 -D)
i(x0 -D)
i(x0)
y(n0)
y(n1)
y(n2)

• 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.

d
x0
x0-D
x0D
n0
n1
n2
Original samples and an interpolated sample
1 period
v(x)
n0
n1
n2
Distribution (standard deviation) of interpolated
samples
5
The Periodic Signal v(x)

• Linear Interpolation

• Cubic Interpolation

This property can be exploited by an algorithm
designed to detect linear interpolation.
6
The Algorithm
p(i,j)

• 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 derivativeof each row
average across rows
compute Discrete Fourier Transform
estimate interpolation rate N
(assuming no aliasing)
7
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 rateswill alias to N.
• Only two possible solutions for rates Ngt1
  (upsampling). An infinite number of solutions for
  Nlt1.
• The correct rate can often be determined through
  prior knowledge of the system.

(Q a positive integer)
8
Example Signals N 2

• After computing DFT

After summing across rows v(x)
9
Example DFT Signals
10
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.

interpolation N 2.8JPEG compression
no interpolationJPEG compression
o(i,j)
digital zoom by N
After computing DFT
JPEG compression
p(i,j)
11
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
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
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.