7 Experimental evaluation of off-the-shelf compressors

This section describes the evaluation protocol and the experimental results of the compression of simulated data streams for PLANCK-LFI.

  
7.1 Evaluation protocol

First tests were performed on a HP-UX workstation on four compressors (Maris et al. 1998) but given the limited number of off-the-shelf compression codes for such platform, we migrated the compression pipeline on a Pentium III based Windows/NT workstation.

As described in Sect. 2 the signal composition is defined by many components, both astrophysical and instrumental in origin. In particular, it is important to understand how each component or instrumental parameter, introducing deviations to the pure white noise statistics, affects the final compression rate.

To scan systematically all the relevant combinations of signal compositions and off-the-shelf compressors, a Compression Pipeline was created. The pipeline is based on five main components: the signal quantization pipeline, the signal database, the compression pipeline, the compression data base, the post-processing pipeline. The signal quantization pipeline performs the operations described in the upper part of Fig. 1. The simulated astrophysical signals are hold in a dedicated section of the signal archive, they are processed by the quantization pipeline and stored back in a reserved section of the signal archive. So quantized data streams are generated for each relevant combination of the quantization parameters, signal composition and sky pointing.

Each compressor is then applied by the compression pipeline to the full set of quantized signals in the signal archive. Results, in terms of compression efficiency as a function of quantization parameters are stored in the compression database. The statistical analysis of Sect. 5 are performed with a similar pipeline.

Finally the post-processing pipeline scans the compression data base in order to produce plots, statistics, tables and synthetic fits. Its results are usually stored into one of the two databases.

The pipeline is managed by PERL 5.004 script files which drive FORTRAN, C, IDL programs or on-the-shelf utilities gluing and coordinating their activities. Up to $\approx 75\,000$ lines of repetitive code are required per simulation run. They are generated by a specifically designed Automated Program Generator (APG) written in IDL (Maris et al. 1998). The APG takes as an input a table which specifies: the set of compressors to be tested, the set of quantization parameters to be used, the order in which to perform the scan of each parameter/compressor, the list of supporting programs to be used, other servicing parameters. The program linearizes the resulting parameter space and generates the PERL simulation code or, alternatively, performs other operations such as: to scan the results data base to produce statistics, plots, tables, and so on. The advantage of this method is that a large amount of repetitive code, may be quickly produced, maintained or replaced with a minor effort each time a new object (compressor, parameter or analysis method) is added to the system.

   
7.2 Experimental results

Purpose of these compression tests is to give an upper limit to the lossless compression efficiency for LFI data and to look for an optimal compressor to be proposed to the LFI consortium.

A decision about the final compression scheme for PLANCK-LFI has not been taken yet and only future studies will be able to decide if the best performing one will be compatible with on-board operations (constrained by: packet independence and DPU capabilities) and will be accepted by the PLANCK-LFI collaboration.

For this reason up to now only off-the-shelf compressors and hardware where considered. To test any reasonable compression scheme a wide selection of lossless compression algorithms, covering all the known methods, was applied to our simulated data. Lacking a comprehensive criteria to fix a final compressor, as memory and CPU constrains, we report in a compact form the results related to all the tested compressors. We are confident that the experience which will be gained inside the CMB community developing ground based experiments (i.e. Lasenby et al. 1998), long duration flight balloon experiments (i.e. De Bernardis & Masi 1998) such as BOOMERANG (De Bernardis et al. 2000; Lange et al. 2000) and  MAXIMA (Hanany et al. 2000; Balbi et al. 2000), and specialized space missions such as the Microwave Anisotropy Probe (MAP) (Bennet et al. 1996), joined to the experience which will be gained inside the PLANCK collaboration developing on-board electronics prototypes, will provide us with a more solid base to test and improve the final compression algorithms applied to real data.

Tables 4 and 5 list the selected compression programs.

 

 

Table 4: Tested compressors and related parameters. The Macro column contains the names of the macros running a given compression Code with a given combination of Parameters

Macro	Code	Parameters	Note
ahuff-c	ahuff-c		Adaptive Huffman (Nelson & Gailly 1996)
AR	ar
arc	arc
arha	arhangel		http://www.geocities.com/SiliconValley/Lab/6606
arhaASC	"	-1	ASC method
arhaHSC	"	-2	HSC method
arith-c	arith-c		Arithmetic coding (Nelson & Gailly 1996)
arith-n	arith-n		Adaptive Arithmetic Coding (AC) (Nelson & Gailly 1996)
arith-n0	"	$-{\rm o} 0$	Zeroth order Arithmetic coding
arith-n1	"	$-{\rm o} 1$	First order AC
arith-n2	"	$-{\rm o} 2$	Second order AC
arith-n3	"	$-{\rm o} 3$	Third order AC
arith-n4	"	$-{\rm o} 4$	Fourth order AC
arith-n5	"	$-{\rm o} 5$	Fifth order AC
arith-n6	"	$-{\rm o} 6$	Sixth order AC
arith-n7	"	$-{\rm o} 7$	Seventh order AC
arj	arj
arj0	"	$-{\rm m} 0$	method 0 (no compression)
arj1	"	$-{\rm m} 1$	method 1
arj2	"	$-{\rm m} 2$	method 2
arj3	"	$-{\rm m} 3$	method 3
arj4	"	$-{\rm m} 4$	method 4
boa	boa
bzip	bzip2090
bziprb	"	-repetitive-best	best compression of repetitive blocks
bziprf	"	-repetitive-fast	fast compression of repetitive blocks
gzip1	gzip	-1	fast compression
gzip9	"	-9	best compression
huff-c	huff-c		Hauffman (Nelson & Gailly 1996)
jar	jar32
jar1	"	$-{\rm m} 1$	method 1
jar2	"	$-{\rm m} 2$	method 2
jar3	"	$-{\rm m} 3$	method 3
jar4	"	$-{\rm m} 4$	method 4
lha	lha
lzss	lzss
lzw12	lzw12
lzw15v	lzw15v

 

 

Table 5: See Table 4

Macro	Code	Parameters	Note
pkzip	pkzip		from PKWARE
pkzip-ef	"	-ef	fast compression
pkzip-en	"	-en	normal compression
pkzip-es	"	-es	super fast compression
pkzip-ex	"	-ex	extra compression
rar-m0	rar	-m0	level 0 compression
rar-m1	"	-m1	level 1 compression
rar-m2	"	-m2	level 2 compression
rar-m3	"	-m3	level 3 compression
rar-m4	"	-m4	level 4 compression
rar-m5	"	-m5	level 5 compression
splint	splint
SZIP00	szip		Rice Algorithm and Rice compression chip simulator
szip0ec	"	-ec	entropy coding compression mode
szip0nu	"	-nn	nearest neighbor compression mode
szipc0	"	-chip	compress exactly as chip
SZIPCEC	"	-chip -ec	as szip0ec + chip compression
SZIPCNU	"	-chip -nn	as szip0nu + chip compression
uses	uses	-n 16 -s 64 -rr	Universal Source Encoder for Space
			16 bits per sample,
			64 samples for scanline,
			correlates near samples (CNS)
uses008	"	-n 16 -s 8 -j 8	8 samples, 8 samples per block
uses008rr	"	-n 16 -s 8 -rr -j 8	as uses008 + CNS
uses016	"	-n 16 -s 16	16 samples per block
uses016rr	"	-n 16 -s 16 -rr	16 samples per block + CNS
uses032	"	-n 16 -s 32	32 samples per block
uses032rr	"	-n 16 -s 32 -rr	32 samples per block + CNS
uses064	"	-n 16 -s 64	64 samples per block
uses064rr	"	-n 16 -s 64 -rr	64 samples per block + CNS
uses320	"	-n 16 -s 320	320 samples per block
uses320rr	"	-n 16 -s 320 -rr	320 samples per block + CNS
uses960	"	-n 16 -s 960	960 samples per block
uses960rr	"	-n 16 -s 960 -rr	960 samples per block + CNS
zoo	zoo

Since the behaviour (and efficiency) of each compressor is determined by a set of parameters, one or more macro file operating a given combination of compressor code plus parameters is defined. It has to be noted that uses is a space qualified algorithm, based on Rice compression method, for which space qualified dedicated hardware already exists.

To evaluate the performances of each compressor, figures of merit are drawn like the one in Fig. 5 which shows the results for the best performing compressor: arith-n1.

Figure 5: Figures of merit for the arithmetic compression of order 1 ( arith-n1) for 30 GHz and 100 GHz channels. Here $\mbox{${\rm AFO}$ } = 0$ V, $\mbox{${{\rm VOT}}$ } = 1.0$ V/K, $\mbox{$N_{{\rm c}}$ } = 1$. The compression efficiency is plotted as a function of the incremental complexity of the signal composition: wn means white noise only, +1/f plus 1/f, +cmb plus CMB, +dipo plus dipole, +gal plus galaxy, +ex plus extragalactic sources

Looking at such figures it is possible to note as the compression efficiency does not depend much on the signal composition. This is true even when large, impulsive signals, as planets, affecting few samples over thousands are introduced. Again, this is a consequence of the fact that white noise dominates the signal, being the most important component to affect the compression efficiency. In this regard it has been speculated that the 1/f component should improve the correlation between neighborhood samples affecting the compression efficiency (Maris et al. 1998) no relevant effect may be detected into our simulations. As an example from Fig. 5 for the 30 GHz signal the addition of the 1/f noise to the white noise data stream affects the final $C_{{\rm r}}$ for less than $0.5\%$.

The only noticeable (i.e. about $6\%$) effect due to an increase in the signal complexity, occurs when the cosmic dipole is added. In the present signal the dipole amplitude is comparable with the white noise amplitude ($\approx 3$ mK) so its effect is to distort the sample distribution, making it leptocurtic. As a consequence compressors, which usually work best for a normal distributed signal, becomes less effective. Since the dipole introduces correlations over one full scan circle, i.e. few 10³ samples, while compressors establish the proper coding table observing the data stream for a small set of consecutive samples (from some tens to some hundred samples), even a self adaptive compressor will likely loose the correlation introduced by the dipole. A proper solution to this problem is suggested in Sect. 9. The other signal components do not introduce any noticeable systematic effect. The small differences shown by the figures of merit may be due to the compression variance and depend strongly on the compressor of choice. As an example a given compressor may be more effective to compress the simulated data stream with the full signal than the associated simpler data stream containing only white noise, 1/fnoise, CMB and dipole. At the same time another compressor may show an opposite behaviour.

As shown by Fig. 6, and as expected from Eq. (9) increasing ${{\rm VOT}}$, i.e. increasing the quantization step, increases the compression rate.

Figure 6: Compression rates for arith-n1 as a function of the ${{\rm VOT}}$ and $N_{{\rm c}}$ for a full simulated signal ( wn + 1/f + dp + cmb + dipo + gal + ex) (see also Fig. 5 for details). From top to bottom: Squares: $\mbox{${{\rm VOT}}$ } = 1.5$ V/K, Diamonds: $\mbox{${{\rm VOT}}$ } = 1.0$ V/K, Triangles: $\mbox{${{\rm VOT}}$ } = 0.5$ V/K

In addition $C_{{\rm r}}$ increases increasing $N_{{\rm c}}$ up to a $\approx 20 \%$. The increase is noticeable for $\mbox{$N_{{\rm c}}$ } < 15$ and saturates after $\mbox{$N_{{\rm c}}$ } = 30$. On the contrary its dependence on the offset (${\rm AFO}$) is negligible (less than $1\%$). For these reasons in the subsequent analysis the AFO dependency is neglected and the corresponding simulations are averaged.

   
7.3 Synthetical description

The full data base of simulated compression results takes about 14 MBytes, for practical purposes it is possible to synthesize all this information using a phenomenological relation which connects $C_{{\rm r}}$ with $N_{{\rm c}}$ and ${{\rm VOT}}$ whose free parameters may be fitted using the data obtained from the simulations. In short:

$$% \mbox{$C_{{\rm r}}^{{\rm Fit}}$ }(\mbox{${{\rm VOT}}$ }, \m... ...\rm c}}$ }) \ln \left[ \frac{{\rm VOT}}{1\;{\rm V/K}}\right]}$$ (12)

where $C_{{\rm r},1}$ is the $C_{{\rm r}}$ for $\mbox{$N_{{\rm c}}$ } = 1$, $\mbox{${{\rm VOT}}$ } = 1.0$ V/K, while ${\mathcal{I}(\mbox{$N_{{\rm c}}$ })}$ and $\mathcal{S}(\mbox{$N_{{\rm c}}$ })$ describe the $C_{{\rm r}}$ dependence on $N_{{\rm c}}$. In particular the relation is calibrated for any compressor imposing that $\mbox{$C_{{\rm r}}$ }(\mbox{${{\rm VOT}}$ }=1 \;{\rm V/K}, \mbox{$N_{{\rm c}}$ }=1) = \mbox{$C_{{\rm r},1}$ }$.

The linear dependency of $1/\mbox{$C_{{\rm r}}^{{\rm Fit}}$ }$ over $\ln \mbox{${{\rm VOT}}$ }$ is a direct consequence of Eq. (9), and is confirmed by a set of tests performed over the full set of our numerical results for the compression efficiency, the rms residual between the best fit (12) and simulated data being less than $1.5\%$, in almost the $92\%$ of the cases and less than $1\%$ in $72\%$ of the cases. The dependencies of its parameters $\mathcal{I}$ and ${\mathcal{S}}$ over $N_{{\rm c}}$ are obtained by a test-and-error method performed on our data set and we did not investigate further on their nature. For all practical purposes our analysis shows that these functions are well approximated by a series expansion:

$$% {\mathcal{I}}(\mbox{$N_{{\rm c}}$ }) \approx \exp \left( \sum_{k=1}^{2} A_k (\ln \mbox{$N_{{\rm c}}$ })^k \right),$$ (13)

$$% {\mathcal{S}}(\mbox{$N_{{\rm c}}$ }) \approx \mbox{${\mathc... ...eft( \sum_{k=1}^{5} B_k (\ln \mbox{$N_{{\rm c}}$ })^k \right).$$ (14)

here ${\mathcal{S}}_{1}$, A_k and B_k are free parameters obtained by fitting the simulated data, in particular ${\mathcal{S}}_{1}$ is the slope for $\mbox{$N_{{\rm c}}$ } = 1$.

Since an accuracy of some percent in determining the free parameters of $\mbox{$C_{{\rm r}}^{{\rm Fit}}$ }(\mbox{${{\rm VOT}}$ }, \mbox{$N_{{\rm c}}$ })$ is enough, the fitting procedure was simplified as follow. For a given compressor, signal component, swap status, and $N_{{\rm c}}$ value $\mathcal{I}$ and ${\mathcal{S}}$ where determined by a $\chi^2$ fitting procedure. The list of $\mathcal{I}$ and ${\mathcal{S}}$ as a function of $N_{{\rm c}}$ have been fitted by using relations (13) and (14) respectively. The fitting algorithm tests different degrees of the polynomial in the aforementioned relations (up to 2 for ${\mathcal{I}(\mbox{$N_{{\rm c}}$ })}$, up to 5 for $\mathcal{S}(\mbox{$N_{{\rm c}}$ })$) stopping when the maximum deviation of the fitted relation respect to the data is smaller than $0.5\%$ for $\mathcal{I}$ or 0.0001 for ${\mathcal{S}}$, or when the maximum degree is reached.

Tables 6-9 report the results of the compression exercise ordered for decreasing $C_{{\rm r},1}$.

  

Table 6: Compression rates at 30 GHz, white noise only. A y in the Swap column means that the compression efficiency is some percent better using a big Endian bytes ordering than with a little Endian bytes ordering. The polynomial degree D, as coefficients A₁, A₂, B₁, $\dots$, B₅ refer to Eqs. (12-14)

  

Table 7: Compression rates at 30 GHz, full signal

  

Table 8: Compression rates at 100 GHz, white noise only

  

Table 9: Compression rates at 100 GHz, full signal

The first column is the name of compression macro (i.e. a given compression program with a defined selection of modificators and switches) as listed in Tables 4, 5. The third and fourth column are the fitted $C_{{\rm r},1}$ and ${\mathcal{S}}_{1}$ as defined in: (12), (13), (14). From the $5^{{\rm th}}$ to the $7^{{\rm th}}$ columns and from the $8^{{\rm th}}$ to the $13^{{\rm th}}$ columns the polynomial degree and the expansion parameters for Eqs. (13) and (14) are reported.

Many compressors are sensitive to the ordering of the Least and Most Significant Bytes of a 16 bits word in the computer memory and files. Two ordering conventions are assumed: little Endian memory layout  i.e. Least Significant Byte is stored First or big Endian memory layout i.e. Most Significant Byte is stored First; Intel processors, as PENTIUM III, work with little Endian memory layout. For this reason each test was repeated twice, one time with the original data stream file, i.e. with the little Endian configuration, and the other after swapping bytes, so to simulate big Endian configuration. If the gain in $\mbox{$C_{{\rm r},1}$ }$ after byte swapping is bigger than some percent, big Endian compression is reported, otherwise the little Endian is reported. These two cases are distinct by the second column of Tables 6-9 which is marked with a y if swapping is applied before compressing. It is interesting to note that not only 16 bits compressors, such as uses, are sensitive to swapping. Also many 8 bits compressors are sensitive to it, probably this is due to the fact that if the most probable 8 bits symbol is presented first at the compressor a slightly better balanced coding table is built.

It should be noted that the coefficients reported here are obtained compressing one or more full scan circles at a time, so their use to extrapolate $C_{{\rm r}}$ when each scan circle is divided in small chunks which are separately compressed has to be performed carefully, especially for $\mbox{${{\rm VOT}}$ } \approx 0.5$ V/K where some extrapolated $C_{{\rm r}}$ grows instead of to decrease for a decreasing $N_{{\rm c}}$ as in most of the cases. However we did not investigate further the problem because the time required to perform all the tests over all the compressors increases decreasing $N_{{\rm c}}$, and because up to now a final decision about the packet length has not been made yet. Moreover, short data chunks introduce other constrains which are not accounted for by Eq. (9) but which are discussed in Sect. 8.

Apart from the choice of the best compressor, Tables 6 to 9 allow interesting comparisons.

The performances of the arithmetic compression arith are very sensitive to changes in the coding order n = 0, $\dots$, 7. The computational weight grows with n, while $C_{{\rm r}}$ is minimal at n=0, maximal for n=1 and decreases increasing nfurther.

Both non-Adaptive Huffman (huff-c) and Adaptive Huffman (ahuff-c) are in the list of the worst compressors, considering both the pure white noise signal and the full signal.

We implemented the space-qualified uses compressor with a wide selection of combinations of its control parameters: the number of coding bits, the number of samples per block, the possibility to search for correlations between neighborhood samples. We report the tests for 16 bits coding only, changing the other parameters. Uses is very sensitive to byte unswapping, when not performed uses does not compress at all. On the other hand, opposite to arith the sensitivity of the final $C_{{\rm r}}$ to the various control parameters is small or negligible. In most cases $C_{{\rm r},1}$ differs of less than 0.01 for changing the combination of control parameters, such changes are not displayed by the two digits approximation in the tables, but they are accounted for by the sorting procedure which fixes the table ordering. At 30 GHz most of the tested compressors cluster around $\mbox{$C_{{\rm r},1}$ } = 2.67$ and at this level arith-n3 is as good as uses. At 100 GHz the best uses macros clusters around $\mbox{$C_{{\rm r},1}$ } = 2.43 - 2.44$, equivalent to arith-n2 performance. In our tests uses performs worst at 8 samples per block without correlation search, but apart from it, in our case the correlation search does not improve significantly the compression performances. Some commercial programs such as boa, bzip compress better than uses.