Reconstruction of missing image lines due to defect sensor elements and transmission losses
2. RECONSTRUCTION MODELS
The actual reconstructions in this paper are performed for each band individually. For highly
correlated bands the accuracy of the estimated missing data values could possibly be improved
if the additional information of the other bands is utilised. However, generally this approach is
only possible for the vertically oriented artefacts since the CCDs are independent from each
other. But in case of transmission losses the information of the different bands is often interlaced
and therefore a loss affects all bands.
2.1. Autoregressive (AR) Model
The name ‘autoregressive’ is derived from the concept of regressing an unknown value xj based
on previous known values xi with i<j. In terms of the problem on hand the xi represent pixels left
or right of the missing pixel while the xj corresponds to the defect CCD element. The estimation
follows a linear model, i.e.
where e is the uncorrelated prediction error with zero mean and constant variance. The variable
p determines the order of the model and hence the number of known values that are incorporated
in the reconstruction of xj while the ak are weight coefficients with ak Î[-1,1]. If the gap is wider
than one pixel, i.e. xj,…,xj+n are distorted, then an iterative approach is applied that estimates the
unknown sequentially while utilising previous estimates as actual available values. Before the
actual reconstruction can be performed, the order of the process as well as the coefficients have
to be determined. Various algorithms are available for this purpose, however, the most promi-nent
are the least square, Yule-Walker and Burg approach (Priestley, 1994; Rezek and Roberts,
1997). De Hoon et al. (1996) highly recommend the Burg method for AR parameter selection
since usually Yule-Walker leads to poor parameter estimates and the least square method to un-stable
models. Nevertheless all three techniques are used in this paper to provide a full compari-son
and to reflect on their popularity.
The least square (LS) technique considers a simple regression between the predictor x and the
dependent or response variable Y. Therefore the observation Y can be expressed by the random
variable x and the unknown regression coefficients can be computed using a set of available
observations while minimising the sum of squares of the residuals.
The Yule-Walker technique is also known as the autocorrelation method. The Levinson-Durbin
algorithm is used to solve the Yule-Walker equation, which explores the Toeplitz structure of
the matrix to be inverted. This solves for the AR parameters more efficiently. Note that the ap-proach
by Yule and Walker is expected to give the smallest prediction error of the three AR
prediction models in this investigation.
The main idea behind the argumentation for the Burg algorithm is that knowledge of the auto-correlation
coefficients is not sufficient to identify a process uniquely. Instead the coefficients
are determined by a maximum entropy method by estimating the reflection coefficients at suc-cessive
orders. The algorithm exhibits some bias in estimating the central frequencies of sine
components. Moreover, higher order fits are affected by "splitting", a phenomenon where multi-ple
peaks in the spectrum are generated instead of a single feature.
2.2. Radial Basis Functions
In this paper polyharmonic radial basis functions (RBF) are used for the reconstruction of the
missing data. Basically, the image data describe multi-dimensional points that are approximated
by the corresponding surface of the RBFs. For a given set of points xi with xi being a column
vector in the multi-dimensional space, the RBF is described by
where p is a low-order polynomial, the li coefficients and F the actual basis function. Unlike
other interpolation functions, e.g. linear, cubic and sinc, the RBF utilises all N available points.
This is of particular interest, since the spatial extend of F is not limited as for the other listed in-terpolation
kernels. The utilised function in this paper is the thin-plate spline F(r)=r 2 log(r).
The major problem with respect to Equation (2) is to determine the coefficients of the polynom
p and the li. Given the constraint
for all polynomial functions k with a degree smaller or equal to m, the problem can be solved by
solving the corresponding linear system (Carr et al., 2001). For the actual realisation of the
function fitting in this investigation, the special implementation of the RBF provided by the
company aranz was chosen (ARANZ, 2003).
