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##
Fault Tolerance Design for Discrete Cosine Transforms

### Introduction

Discrete cosine transform (DCT) is one of the basic building blocks for
JPEG. The discrete cosine transform was first applied to image
compression in Ahmed, Natarajan and Rao’s pioneering work, in which they
showed that this particular transform was very close to the
Karhunen-Loeve transform (KLT) , a transform that produces un-correlated
coefficients. Another important aspect of the DCT is the ability to
quantize the DCT coefficients using usually-weighted quantization values,
as will be discussed in this report. To detect a fault in DCT networks,
concurrent error detection (CED) design have been proposed. In this
report, the error due to the finite bit-precision is for the DCT is
investigated. In order to simplify the problem, the 32 bits integer is
used to implement the 2D-DCT. Reconstruction error is measured in terms
of total gray-level error. This evaluation is completed in the case no computer
failure errors injected into the transform system. When the fault tolerance
design, an efficient concurrent error detection scheme for DCT networks
is applied. By employing a check-sum scheme with effective checksum
weighting factors, the design allows high throughput as well as high
fault coverage. The overflow probability is also minimized.
### A Fast Algorithm for DCT Based on the DFT

There are a number of relationships between the DFT on real inputs and
the DCT. Vetterli and Nussbaumer showed that the N-point DCT can be
expressed in terms of the real and imaginary parts of an N-point DFT and
rotation of these DFT outputs. Harelick showed that the first N
coefficients of a 2N-point DFT with appropriate symmetry of input values
can be used to compute an N-point DCT. This property is used to build an
efficient scaled DCT structure as shown in Figure 1.

Figure 1. Flowgraph for 8-point DCT adapted from Arai,
Agui, and Nakajima.

a1 = 0:707; a2 = 0:541; a3 = 0:707; a4 = 1:307; and a5 = 0:383
The DCT maps integers from the intensity domain to the frequency domain.
The DCT theoretically is an exactly invertible mapping between the
spatial and frequency domains. Unfortunately, as commonly used the
DCT/IDCT does introduce error. The question is how much error can be
tolerated for a particular application, and how can the fault tolerance
system distinguish between finite resolution errors and the computer
failure errors. To get some intuition about error introduced by DCT, the
equation for the one-dimensional DCT can be written as:
**c**=**Ax**
From the above equation, the relationship between the spatial domain
pixels x and the frequency-domain values c is linear. Since A is
orthonormal, then *inv*(**A**) = **A'**. In many image
processing system, the output of the DCT is a frequency domain image with
integer pixels. This requires that the values of **c** be rounded to
the nearest integer and introduces error into the construction, even if
infinite precision was used to calculate Ax. In particular, if a 1D
signal is converted into the frequency domain by the DC, has its values
rounded to the nearest integer, and then converted back to into the
spatial domain using the inverse DCT. There is worst-case error in term
*x*[*i*] of
For *N*=8, this implies an error of up to 1.32 gray levels for one
dimensional data
In two-dimensions, the situation is similar. Once again, the
frequency-space values are rounded to the nearest integer. Since rounding
introduces an error of up to +0.5 or -0.5 in each frequency term, the
maximum error *E*[*m,n*] for a pixel in the reconstructed image
is.

### Proposed Error Detection Scheme for 8-point DCT

The upper limit of the round-off errors as described in the previous
section allows the fault tolerance system to know if a computer failure
error occurred. However, by using this error detection scheme, the
inverse discrete cosine transform is needed to compute the input data
from the coefficients output. Doing so could cost a double of time and
hardware resources.
An error detection scheme so called concurrent error detection for the
DCT is proposed. The protection design methodology uses integer number
parity values, similar to previous integer FFT fault tolerance
techniques. Figure 2 describes the structure of the concurrent error
detection scheme for 8-point DCT.

Figure 2. Concurrent Error Detection Structure for 8-point DCT
Two comparable parity values are calculated, one from the input data
vector **f** and the other from the output transform coefficients in vector
**F**. The parity values are based on input and output weighting vectors
**b** and **w**. The input parity value is computed by taking the
inner product between **f** and **b**. Similar for the output parity
value. The parity checking operation involves a totally self-checking
checker adapted for real number comparisons.The decision threshold is selceted based
on the gap between the round off errors and the computer failure errors.
Assume the output weighted factor vector **w** is known, then the input
weighted factor vector **b** can be computed. For example, the weight
factor at the output coefficient F(*k*) is 1/(*k*+1),
then the input weight factors are found as shown below.
### Experiment Results

The following figures show the results of a computer program implementing the proposed
concurrent error detection scheme.

Figure 3. Performance curves of the concurrent error detection design
for 8-point DCT. The mean

square error versus the noise level (*left*),
and detection performance vs. the desision threshold (*right*)
The fault coverage is evaluated as shown in Figure 3. In this simulation, a functional error
is an additive white noise occurring at the link of the transform stages. Only one fault
is injected at random in one of the links of a module. The error threshold was decided
based on the machine dependent constant, the word length and the input checksum. If the
difference of the two checksums is greater than the error threshold, it is assumed that
a fault is detected.
### Fault Coverage

From the detection performance curves, given a noise level, the smaller the
threshold to be selected, the higher fault detection coverage. However, when the
threshold is in the round off error region, false alarm may occur. Therefore, the
decision threshold is selected based on the level of corrupted output due to the system
failure and the fault detection coverage requirement.
Fault coverage, by intuitive definition, is a measure of system's ability to perform fault
detection, fault location, fault containment, and fault recovery. Fault detection coverage
is a measure of a system's ability to detect faults. A system requirement may be at a
certain fraction of all faults be detected. Fault coverage can be calculated as the ration
of the number of faults detected to the number of faults that occur. In this design, if
the output noise grater than a selected threshold *T*, it is assumed that fault exists
in the network.

### Throughput

The thoughput of a DCT network is th number of transforms performed per cycle.
In this design, like some other chechsum designs, a high throughput can be achieved
since the second try is required only when the total noise value (the noise due to the
round off error at the DCT values) of the network is over a certain round-off error bound.
### Report papers

More information on DCT, particular for JPEG can be found in the following
reports

Apr. 12, 2003 Fault Detection Implementation for
the Discrete Cosine Transform(1.6M)
### Error Detection/Correction Package

Apr. 12, 2003 Fault Tolerance Implementation for DCT(358k)

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