We consider the compression of multidimensional signals on the aircraft board. We describe the data of such signals as a hypercube, which is "rotated" in a special way. To compress this hypercube, we use a hierarchical compression method. As one of the stages of this method, we use an adaptive interpolation algorithm. The adaptive algorithm automatically switches between different interpolating functions at each signal point. We perform computational experiments in real-world multidimensional signals. Computational experiments confirm that the use of proposed adaptive interpolator allows increasing (up to 31%) the compression ratio of the “rotated” hypercube corresponding to multidimensional hyperspectral signals.
The growing interest in applied problems in the field of processing multidimensional signals [1-4] determines the relevance of research in this area. One of the main issues is the data size corresponding to the multi-dimensional signals. For example, data of hyperspectral remote sensing [1, 3], as one of the types of multidimensional signal, contain up to several hundred large-sized components of high bit depth. With airborne compression, this information needs to be transmitted through a limited communication channel. In this situation, the use of compression [5-8] is the only possible solution. Currently, there are many methods of compression: wavelet [9], spectral [10], fractal [11], etc. However, all of them are very limited in airborne compression due to the high computational complexity and complexity of error control in the spatial domain. Besides, when a hyper-spectral data is received, a hypercube of a multidimensional signal is often “rotated” (see Figure 1-2) in a specific way: the first component (band) contains the first lines of the source bands; the second band includes the second lines, etc. We can see that this leads to the appearance of anisotropic correlation peculiarities in the signal cross-sections.
In the described conditions, the method of multidimensional signal compression based on hierarchical grid interpolation [12–13] has proven itself well. The most difficult stage of this compression method, which largely determines the efficiency of the method as a whole, is an interpolator of a multidimensional signal. In this paper, an adaptive interpolator based on hybrid technology is proposed for interpolation of specifically “rotated” multidimensional signals corresponding to hyperspectral remote sensing data. This technology combines the autoregressive prediction of the signal cross-sections and adaptive parameterized interpolation within the crosssections, which makes it possible to use the above-described correlation anisotropy of the multidimensional signal effectively.
To study the proposed multidimensional interpolator, computational experiments are performed on natural multidimensional signals.
Consider a multidimensional (D-dimensional) signal
X = {x (d0 , d1,.., dD−1 )} .
This signal can also be considered as a set of (D–1) dimensional cross-sections:
X t =(d1,.., {xt dD−1 )} =(t,d1,.., {x dD−1 )}, 0 ≤ t < T .
Often these cross-sections are highly correlated (see Figures 1-2), for example, when processing hyperspectral data of remote sensing (for a hyperspectral signal of this type D = 3, and T coincides with the number of spectral bands). To account for the interdependency of the components, we use the linear autoregressive model:
Pt =k1X t −1 + k2 X t −2 + ... + kN X t − N , (
Rt (d1,.., dD−1 ) =(d1,.., X t dD−1 ) − Pt (d1,.., dD−1 ) , 1 ≤ t < T . (
The variance of differential cross-section Rt is usually much smaller than the variance of the
original cross-section Xt, and the probability distribution is much less uniform. Therefore, the
differential cross-sections (
The difference cross-sections (
L−1 X = Xl , Xl
l=0 where Il is the set of sample indices for scale level Xl : ={x(d ) : d ∈ Il } , I L−1 = { 2L−1d} , Il =}\ { 2l d { 2l +1d}, 0 ≤ l < L .
The scale level number (L – 1) is the resampled “grid” of signal samples with the step of 2(L – 1) along each of the coordinates, and any other level (number l) is the grid of samples with the step of 2l, from which all samples with doubled step are excluded.
During compression, the scale levels are compressed sequentially in order of decreasing their
numbers. At the same time, more resampled levels are interpolated based on samples of less resampled
levels. Since peculiarities of post-interpolation residues are similar to peculiarities of differences (
The most critical step of hierarchical compression is the interpolator. In this work, differential
crosssections (
To ensure low computational complexity of hierarchical compression, we often use trivial interpolation procedures, based on averaging over the nearest samples of more resampled scale levels. Here is an example of an interpolation procedure:
1
xl(
This scheme is straightforward, but does not take into account local signal characteristics, for
example, the oblong structures of onboard remote sensing data [14-16] (see Figure 2). Therefore, in
this work, an adaptive interpolator is used to compress such remote sensing data. Besides function (
α ≤ µl (m, n ) ≤ β;
2, µl (m, n ) > β;
where α, β are parameters of the decision rule (thresholds for switching between interpolating
functions). These parameters are calculated automatically based on minimization of interpolation error
or entropy of quantized post-interpolation residues; µl (m, n ) is a feature describing the magnitude and
direction of the elongated structure in the local neighborhood of the current signal sample:
µl (2m + 1, 2n + 1) = xl+1 (m, n ) − xl+1 (m + 1, n + 1) − xl+1 (m, n + 1) − xl+1 (m + 1, n ) .
(
The optimization of the adaptive interpolator according to the parameters α, β is performed separately [17] for each signal realization (more precisely, separately for each hierarchical signal level).
The proposed hybrid interpolator, including an intercomponent approximator and an intracomponent adaptive interpolator, was implemented by software and built into the hierarchical compression method. Based on this software, we performed an experimental study of the proposed interpolator effectiveness in nature multidimensional signal sets, “rotated” in a specific way to airborne compression (see the example in Figure 2). We used the following signal sets: • signals [14] of hyper-spectrometer AVIRIS (224 bands); • signals [15] of «TokyoTech» hyperspectral dataset (31 bands); • signals [16] of hyper-spectrometer SpecTIR (360 bands, see Figure 1).
As a measure of the proposed interpolator efficiency, we used the gain G in the archive size (in %),
which was provided by replacing the averaging interpolator (
G = 100% ⋅ ( Snew − Sbase ) Sbase ,
(
ε2 G,%
We considered the compression of multidimensional signals, which are described by the hypercube, «rotated» in a manner specific for onboard processing. To compress this hypercube, we used the hierarchical compression method. As one of the stages of this method, we used the adaptive interpolation algorithm based on automatic switching between different interpolating functions at each point of the signal. We confirmed computational experiments in natural multidimensional signals, which confirmed that the use of proposed adaptive interpolator allows increasing (up to 31%) the compression ratio of the «rotated» hypercube corresponding to multidimensional hyperspectral signals.
The reported study was funded by: RFBR according to the research projects 18-01-00667, 18-0701312; the RF Ministry of Science and Higher Education within the state project of FSRC “Crystallography and Photonics” RAS under agreement 007-GZ/Ch3363/26.