Compression of Multispectral Images with Comparatively Few Bands Using Posttransform Tucker Decomposition

Up to now, data compression for the multispectral charge-coupled device (CCD) images with comparatively few bands (MSCFBs) is done independently on each multispectral channel. This compression codec is called a “monospectral compressor.” The monospectral compressor does not have a removing spectral redundancy stage. To fill this gap, we propose an efficient compression approach for MSCFBs. In our approach, the one dimensional discrete cosine transform (1D-DCT) is performed on spectral dimension to exploit the spectral information, and the posttransform (PT) in 2D-DWT domain is performed on each spectral band to exploit the spatial information. A deep coupling approach between the PT and Tucker decomposition (TD) is proposed to remove residual spectral redundancy between bands and residual spatial redundancy of each band. Experimental results on multispectral CCD camera data set show that the proposed compression algorithm can obtain a better compression performance and significantly outperforms the traditional compression algorithm-based TD in 2D-DWT and 3D-DCT domain.


Introduction
Multispectral CCD images can be used to identify the targets using their geometric feature and spectrum information simultaneously.Multispectral CCD imaging technology has extensive application in ground target detection, geology monitoring, climate monitoring, military reconnaissance, scientific research, and so on [1,2].With the increase of application requirements, the performance indexes of multispectral CCD camera, such as field of view, coverage width, spatial resolution, spectral resolution, time resolution, and radiation resolution, continue improving.These lead to rapid increase of amount of multispectral image data.However, the amounts of memory and the channel bandwidth used on the satellite are limited.Therefore, the suitable compression approaches play an important role in the successful applications of multispectral CCD camera used on the satellite [3][4][5].
Multispectral CCD camera has two types of instruments: (1) comparatively many spectral channels and (2) comparatively few spectral channels [6,7].Therefore, multispectral CCD images also have types: (1) multispectral CCD images with comparatively few bands (MSCFBs) and (2) multispectral CCD images with comparatively large number of bands (MSCLBs).MSCFBs and MSCLBs both have spectral redundancy and spatial redundancy.For MSCLBs, the correlation between the different bands is usually stronger than spatial correlation among pixels in the same band [8].There have existed several well-known MSCLBs compression algorithms, like principal components analysis (PCA) [9].The PCA approach combines all bands into one single spectral matrix.The covariance matrix of the spectral matrix can be obtained.The inverse of eigenvector matrix of covariance matrix multiplies by the spectral matrix to obtain new bands called principal components.The number of principal components is equal to the number of bands.There is no correlation between different components.Therefore, the PCA is mainly used to remove spectral redundancy.In order to remove the spectral and spatial correlation simultaneously, the MSCLBs compression algorithms use three dimensions (3D) transform approach, like 3D-DCT [10], The monospectral compressor does not remove spectral redundancy stage.In order to reach the excellent compression performance, MSCFBs compression algorithm must remove both spatial and spectral redundancies.In this paper, we propose an efficient compression approach for MSCFBs.In our approach, the 1D-DCT is performed on spectral dimension to exploit the spectral information, and the posttransform (PT) in 2D-DWT domain is performed on each spectral band to remove the spatial redundancies.MSCFBs are decomposed into several subtensors by PT.Then, these subtensors are decomposed by multilevel Tucker decomposition (TD) to reach compression results.

Proposed Algorithm
2.1.The Imaging Principle of MSCFBs.Figure 1 shows the imaging principle of MSCFBs.The long linear multispectral CCD has four parallel CCD linear arrays.Each array is one imaging spectral band.Four imaging spectral bands are red, blue, green, and near-infrared, respectively.The light radiated or reflected by the hundreds of kilometers of linear arrays of ground pixels is concentrated onto optical thin film of CCD detector through an optical system.The space and spectrum distributing of ground targets radiation, acquired by CCD detector, can be expressed as  (, , , ) = 1  [ 0 (, )  (, , )   () * ℎ  (, , )] +  (, ) , where  is the solar height angle,  is wavelength,  0 is ground luminometer,  is spectral reflectance,   is atmospheric transmittances, ℎ  is the point spread function, and  is radiant flux.The optical thin film of each linear array CCD allows the light of corresponding wavelength through. (, , , ) of each band is captured by a linear CCD array, and one dimensional spatial and spectral information is gained.When the CCD camera scans the ground target, spatial dimension information is gained.Therefore, MSCFBs are considered as 3D images,  ∈   1 × 2 × 3 , which is shown in Figure 2.
Based on the imaging principle of MSCFBs, MSCFBs have the spectral redundancies between the adjacent spectral bands and spatial redundancies between the adjacent pixels in each spectral band.Therefore, the compression algorithm must remove the spectral redundancies and spatial redundancies simultaneously.

Spectral Decorrelation.
MSCFBs have two types of correlations: spatial and spectral correlations.The spatial correlation is defined as the correlation between the adjacent pixels in each spectral band.The spatial line correlation of MSCFBs can be expressed as ( Mathematical Problems in Engineering  The spatial column correlation of MSCFBs can be expressed as where  is the line number of each band,  is the pixel number of each line, (, ) is the function of the band gray, and  is the average value of the band gray.The spectral correlation is defined as the correlation between the pixels located in the same spatial position in the different spectral band, which can be expressed as where (, ) and (, ) are the value of pixel located in (, ) in two adjacent bands and ℎ(0, 0) is the crosscorrelation function of (, ) and (, ).Based on (2)∼(4), the Landsat-7 Thematic Mapper (TM) images are chosen to perform an experiment of the spatial and spectral correlation.Figure 3 shows the test results.
The correlations between the adjacent bands for an MSCFB are in the range [0.318-0.926],and the line and column correlations between the adjacent pixels for an MSCFB are in the ranges [0.685-0.955]and [0.609-0.963],respectively.Therefore, the MSCFBs have higher spatial correlation than spectral correlation.Because the spectral correlations are comparatively weak, the traditional 3D-DWT-based compression algorithms are not suitable for MSCFBs.When the 3D-DWT-based approaches compress Landsat-7 images, the good compression results are not obtained.Simultaneously removing the spectral redundancies and spatial redundancies is primary target for the successful implementation of a highperformance compression method.
Because the MSCFBs have a smaller number of bands, 1D-DCT is performed on the spectral dimension to decorrelate the spectral bands.The number of bands is denoted as ; the size of each band is denoted as  × .An MSCFB is composed of  ×  1D spectral vector (Length ).The definition of the 1D-DCT for an input 1D spectral vector   ( = 1, 2, . . ., ×) and output image   ( = 1, 2, . . ., ×) is The    values are called the DCT coefficients of   .Equation ( 5) is performed  ×  times to complete the spectral transform.In order to speed up calculation, we proposed an iterative method.The iterative equation can be expressed as The DCT coefficients of the current spectral vector can be computed by the intermediate and output data obtained from the previous spectral vector.

Spatial Decorrelation.
The 2D-DWT can decompose the image into multilevel wavelet to obtain lower resolution subband and detail subbands.The multilevel decomposition can be regarded as the process of low pass and high pass filtering.
At each level, the low subband produced by previous level performs the high pass filtering to obtain detail information called wavelet coefficients, while performing the low pass filtering linked with scaling function to obtain the approximate information called scaling coefficients.The DWT has been widely employed exploiting the spatial correlations for remote sensing image, such as JPEG2000 and CCSDS-IDC.In this paper, we apply a 2D-DWT coupled with a posttransform to each band of MSCFBs.In our approach, the 2D-DWT is performed on each image band to reduce spatial correlations, and then the transformed wavelet coefficients are transformed by the posttransform approach to reduce remaining correlations in wavelet subbands.
After remote sensing MSCFBs are transformed by 2D-DWT, there exits residual correlation between coefficients in a small neighborhood (see Figure 4).Statistical dependencies between DWT coefficients have been studied for many years.In [25], correlations between nearby wavelet coefficients are reported to be in the range [0.01-0.54]at a distance less than or equal to 3 pixels.We found even wider range, and here we provide a more detailed discussion regarding this topic.We use a Pearson correlation coefficient  to analyze statistical dependency of DWT coefficients, which can be expressed as where  , denotes the covariance between the variables  and  and   and   denote the standard deviation of  and , respectively.According to project experience, threelevel 2D-DWT is appropriate for on-board compressor and we used three-level 2D-DWT in this paper.Three-level 2D-DWT is performed on each image band to produce one low-frequency subband (denoted as LL) and nine high-frequency subbands (denoted as LH  , HL  , HH  ,  = 1, 2, 3).The test is performed from ten MSCFBs.The residual correlations between wavelet coefficients in a small neighborhood of wavelet subbands are shown as in Figure 5.In Level  ( = 1), the residual directional correlation within 16-connected region is  (HL1, HH1, LH1, |Δ| ≤ 2, To obtain the optimum compression performance for MSCFBs, the compression algorithms must fully consider the abovementioned statistical properties.In [26,27], EBCOT has been reported to be very efficient method to remove these residual redundancy values.However, its implementation complexity is too high.In [28], Delaunay has proposed a posttransform to remove remaining redundancies between the adjacent wavelet coefficients within a small region.After the wavelet transform of an image, wavelet coefficients or scaling coefficients subbands are divided into several coefficients blocks.Each block has 4 × 4 wavelet coefficients.Posttransform performs on each coefficients block to remove the residual redundancies.From testing a large number of remote sensing images, the selected size of wavelet coefficients block is the best to a low complexity and efficient compression method.There are two major reasons: (1) the correlation between the adjacent wavelet coefficients is very weak when a distance is greater than or equal to 5 pixels; (2) the larger size of wavelet coefficients block leads to the higher computational complexity, and the smaller size of block results in increasing the number of coefficients blocks and the size information.In addition, since the posttransform is performed on coefficients blocks in wavelet transform domain, not on image domain, the compression algorithm has no additional overhead time to perform deblocking filter.
The core idea behind posttransform compression is that wavelet coefficients blocks further are transformed using one group particular direction basis (such as Bandelet, DWT, DCT, and PCA) in a dictionary.First, a 2D-DWT is applied to an image.Next, blocks of 4 × 4 DWT coefficients are projected on   orthonormal bases   of the dictionary .Then, the minimizing of the Lagrangian cost function is calculated to select the best posttransformed block.Finally, the posttransformed coefficients are encoded by entropy coding method.Each 4 × 4 DWT coefficients block  is Since a dictionary has   bases,   + 1 (including one original block) posttransformed blocks   can be obtained.Among all the posttransformed blocks   , the best posttransformed block   * is selected according to calculating the minimizing of the Lagrangian rate-distortion cost.The minimizing of the Lagrangian rate-distortion cost can be expressed as where    denotes the quantized posttransformed coefficients,  is the quantization step, and (   ) denotes the square error between the posttransformed coefficients   and the quantized posttransformed coefficients    . is a Lagrangian multiplier and (   ) denotes the required bit rates for encoding    and the associated side information  * .In the posttransform, the dictionary has multiple bases.The better the compression performance is, the more the number of bases is and the higher the computational complexity is.However, on-board compression requires a low computational complexity.The space MSCFBs compressor allows the dictionary  of posttransform to have only one basis.In [25], Delaunay et al. proposed a simple posttransform-based compression method.They only perform one-level wavelet transform and only use a Hadamard basis.Then, the posttransform only performs on the coefficients subbands.Compared with the compression method only using the DWT, the PSNR can only improve 0.4 dB∼ 0.6 dB.In this paper, to obtain a posttransform with the low computational complexity and still have high compression performance, we thus consider a very simple dictionary containing only one dynamic base, which is Hadamard basis at the low bit rates, and DCT basis at the high bit rates.
In [29], Delaunay has shown that the Lagrangian approach for selecting the best posttransformed block has two main drawbacks.First, the estimation of the required bit rate for (   ) is computationally intensive and not always accurate.Second, calculating the Lagrangian function depends on the quantization step .However, the coder does not define the quantization step when coding.Therefore, Delaunay proposed an  1 -norm minimization approach to select the best posttransformed block.In this paper, we propose a new method based on   -norm minimization approach to evaluate best posttransform, which replaces an  1 -norm minimization.It is very suitable for the space MSCFBs compressor which has the low complexity constraints.
At the low compression bit rate, the bit rate  can be expressed as where  is the number of nonzero posttransformed coefficients.We propose an  0 -norm ( = 0) minimization approach to select the best posttransformed block.
The selected best posttransformed block has the minimal sum of coefficients magnitude, which can be expressed as where   [] is a posttransformed coefficient.At the high compression bit rate, the bit rate  can be expressed as We use the  1 -norm ( = 1) minimization approach to select the best posttransformed block.The selected best posttransformed block has the minimal sum of coefficients magnitude, which can be expressed as Figure 6 shows the proposed posttransform architecture.Each high-frequency subband is performed by posttransform.The bit-rate comparator decides the type of coding bit rate.We define the coding bit rate greater than or equal to 0.5 bpp as the type of high bit rate (type 1) and that less than 0.5 bpp as the type of low bit rate (type 0).The posttransform is performed using DCT basis and  −1 -norm minimization when coding type 1 and Hadamard basis and  −0 -norm minimization when coding type 0.

Tucker Decomposition in Posttransform Domain
In [30], the posttransformed blocked coefficients are encoded by an adaptive arithmetic coding method.In [29], the posttransformed blocked coefficients are encoded by the bitplane encoding (BPE) method.However, the posttransforms destroy the zero-tree structure.The PSNR using BPE is less than 0.2 dB when compared with the method using DWT and even worse at high bit rates.A basis vector ordering approach has been used.This ordering is obtained by training or learning with thousands of wavelet coefficients blocks from a large number of remote sensing images.The ordering-based strategy has two major shortcomings for space MSCFBs compression.For the first weakness, the ordering of basis vector has huge calculations and is not always accurate.
For the second weakness, the ordering highly depends on a training set with thousands of wavelet coefficients blocks of remote sensing images.However, the training set is not captured when coding.Indeed, after 1D-DCT, 2D-DWT, and posttransform, the MSCFBs still have residual spatial and spectral correlation.In order to obtain farther excellent compression performance, we consider the posttransform coefficients of MSCFBs as a three-order 3D tensor.The tensor performs Tucker decomposition (TD) to remove residual spatial and spectral redundancies simultaneously.
A third-order tensor is denoted as  ∈   1 × 2 × 3 .The third-order tensor is decomposed into a low dimension core tensor multiplied by three factor matrices using TD (see Figure 7).The core tensor is denoted as  ∈   1 × 2 × 3 .The factor matrices are denoted as The third-order TD can be expressed as follows [31]: where  1 ,  2 ,  3 are dimensions of the core tensor , tensor Ŷ is an approximation of tensor , and tensor  is an error tensor.
Since the spatial dimension of each tensor is  1 /2  ×  2 /2  , the tensor decomposition computing cost is very high.To reduce the computational complexity, in this paper, we propose a multilevel TD approach.First, each tensor is divided into several subtensors (see Figures 8 and 9).In each tensor, each subband is one posttransformed block.Therefore, the size of each subtensor is 4 × 4 ×  3 .Each tensor contains  1 /2 +2 ×  2 /2 +2 subtensors, and the 9 tensors have ∑ 3 =1 ( 1 /2 +2 ×  2 /2 +2 ) three-order subtensors.In the first level, all subtensors for each tensor are performed by TD to produce the  1 /2 +2 ×  2 /2 +2 core tensor and 3 ×  1 /2 +2 ×  2 /2 +2 factor matrices.The produced core tensors are organized into a new tensor, which is performed by TD in the next level.In the proposed codec we are using one level of TD.Since the spatial dimension of each subtensor is 4 × 4, TD of each subtensor has low complexity.
To find optimal component matrices  and , we apply HALS-NTD algorithm [31] to each subtensor in this paper.The HALS-NTD algorithm can be shown as Algorithm 1. ( = 1, 2, 3).The CR can be defined as a ratio value between the required bit numbers for representing original image data and the required bit numbers for representing compressed bit-streams, which can be expressed as

Deep Coupling between
In order to efficiently determine the size of all core tensors of subtensor when each tensor performs TD, we proposed a deep coupling between posttransform and TD not only to determine the size of the core tensor and reduce the side information of posttransform but also to code these core tensors and complete the bit-rate control.Figure 10 shows the proposed deep coupling between the posttransform and TD.
The bit-rate allocation module allocates the bit rates for each tensor.The information evaluation module evaluates the information of each tensor.Now the target bit rates for different tensors can be allocated based on their information contents.Then, the allocated bit rates of each tensor are allocated to each subtensor, and the dimensions of each subtensor can be determined according to its allocated bit rates.
In the first place, the information content in LL subband  LL is calculated through an  −1 -norm approach.Let (, ) denote the coefficients in LL subband.The information content in LL subband can be calculated as In the second place, the information content of 9 tensors (LH   , HL   , HH   ,  = 1, 2, 3) is calculated.Let  LH   ,  HL   , and  HH   ,  = 1, 2, 3, denote the information content of 9 tensors, respectively.Since the selected representation   * reflects the image information, the information content of 9 tensors can be evaluated through   * .The dimensions of each tensor are denote the selected representation of th posttransform block.The information content of the tensor LH   can be calculated as Other tensors can be likewise calculated.
In the third place, the weight of bit-rate allocation for each tensor is acquired through Finally, the allocated bit rates of each tensor are allocated to each subtensor.The bit rate assigned to the th subtensor can be calculated as The side information  * denotes the posttransform used on this block.Since the posttransform has only one basis at low and high bit rates, the side information is one bit per block and is not entropy coded.Table 1 shows the side information cost.The side information always has 1 bpb of bit-rate overhead in whole coding process.The side information occupies 12.5% of the total bit rate at 0.5 bpp, 4.16% at 1.5 bpp, and only 2.08% at 3 bpp, respectively.Therefore, the side information has more effect on compression performance at low bit rates than at high bit rates.Indeed, these nonsignificant blocks do not require the side information because their coefficients are all zeros.Coincidentally, there are more nonsignificant blocks at low bit rate than high bit rate.Therefore, the allocated bit rate for side information can be reduced.As the number of nonsignificant blocks decreases, the required bit rate of side information will certainly increase.
In this paper, we use an improved BPE method to encode the core tensor.And then the side information is integrated into the bit-streams.We modify the SPITH algorithm [32] to complete the bit-plane coding.The bit-plane coding processes each core tensor of each subtensor at a time.After one core tensor is processed, the next core tensor is processed.For each time coding, the improved BPE defines two encoding paths: (1) significant path (SP) and ( 2) the refining path (RP).We define a significant state threshold .The element of core tensor is denoted as   1  2  3 at the location ( 1 ,  2 ,  3 ).We use   () to denote an index of significant state when the threshold is  = 2  in the th bit plane.The   () can be expressed as In the th bit plane, when   () = 1, the   1  2  3 is a significant element, which passes into the SP.When   () = 0, the   1  2  3 is insignificant and passes into the RP.The elements in RP perform BPE in the next bit plane.In the next bit plane,  =  − 1,  = 2 −1 .The coding method is the same with the previous bit plane.When traversing all the bit planes, the elements in SP are moved from the core tensor and encoded by entropy coding approach.If the SP for a core tensor has any elements, the side information is embedded into compressed bit-streams.Otherwise, the side information is not embedded into compressed bit-streams.
3.2.Our Proposed Architecture.Figure 11 shows the whole architecture of the proposed posttransform Tucker decomposition-(PTTD-) based compression method in this paper.
The whole process of PTTD-based approach can be divided into four steps as follows.
In the first step, MSCFBs perform 1D-DCT in the spectral dimension to remove the spectral correlation.
In the second step, each spectral band of the MSCFBs performs 3-level 2D-DWT to remove the spatial correlation.And 10 wavelet subbands (one low-frequency and nine highfrequency subbands) for each band can be obtained.
In the third step, all high-frequency subbands are divided into several blocks.Each block has 4 × 4 DWT coefficients.Each block performs the proposed posttransform.
In the fourth step, each posttransformed subtensor performs TD to remove the residual spectral and spatial redundancies.
In the fifth step, the posttransformed coefficients in low-frequency subband are encoded by DPCM.The factor matrices perform quantization.The core tensors perform the bit-plane coding and entropy coding using adaptive arithmetic coding.
In the sixth step, the bit streams are packed and transmitted to the decoder via signal channels.

Experimental Results
4.1.Experimental Scheme.In order to test the performance of the proposed algorithm in this paper, we use the research and development ground test equipment to build experimental system.Figure 12 shows the structure of experimental system to test the performance of the proposed algorithm.The experimental system is composed of image simulation resource, multispectral compression system, ground test equipment, DVI monitor, and server.The server injects the remote sensing image into the image simulation resource.The image simulation resource adjusts the size of image and line frequency of output to simulate the output of CCD.The multispectral compression system receives the test remote images from the image simulation resource and then compresses them to verify the proposed compression algorithm.The ground test equipment completes image decompression to gain the reconstructed images.The reconstructed images are transferred to the server through camera link bus.Finally, the compression performance is analyzed in the server.
The server is a high performance computer using 4.00 GHz and eight-core AMD processor and 28.0 GB of RAM.The image simulation resource has 64 GB NAND flash arrays to store remote sensing images.It outputs image as line-to-line way.The maximum number of pixels in each line is 4096, and the line frequency is 0∼8 KHz.In the multispectral compression system, the bit rate can be set to 2.0∼0.25 bpp.The working frequency of system is 88 MHz.

Multispectral Image Compression Validation and Analysis.
In order to verify the validation of the proposed algorithm, the server injects a series of the AVIRIS multiband remote sensing images into the image simulation resource.The group of multispectral images have four bands; the size of each band is 512 × 512.The bit depth of pixels is 8 bpp (bits per pixel).The bit rate is set to 1 bpp.Figure 13 demonstrated the four original and reconstructed bands.From the displayed images, the original image and reconstructed image are almost not different because the proposed compression algorithm has a high signal-to-noise ratio.The PSNR reaches 48.61 dB, 47.19 dB, 47.07 dB, and 46.38 dB, respectively, in Figures 13(e

)-13(h).
After each band is encoded by the proposed algorithm, the dynamic range of pixels is 0∼2 bits.And most pixels are about 1 bit.So, the proposed compression algorithm is feasible for multispectral CCD image.
where  is equal to 2  −1,  is the quantization bits of original images,  , denotes the value of the pixel of original images at (, ),   , denotes the value of the pixel of reconstructed images at (, ), and  and  denote the number of rows and columns of multispectral images, respectively.
In the first experimental part, in order to test the compressed performance of the proposed approach, we use three groups of SPOT-1 multispectral images having different texture characteristics.Each group is composed of red, green, blue, and near-infrared spectral bands.The size of each band is 512 × 512.The bit depth of pixels is 8 bpp (bits per pixel).Table 2 demonstrates the tested PSNR results of our approach at different bit rates.According to  In the second experimental, in order to compare compressed performance of our approach, we use several proposed compression algorithms, AT-3DSPIHT [33], 3D-DWT [34], 3D-PCA [35], 3D-SPECK [36], and SA-DCT [37], to compress remote sensing MSCFBs.We use the AVIRIS multiband remote sensing images in JPL laboratory.Each group is 256 pixels × 288 pixels × 4; the depth of every pixel is 8 bits.The images are Low Altitude, Lunar Lake, Jasper Ridge, and Cuprite.The compression bit rate is set to 2.0∼0.25 bpp. Figure 14 shows the comparison results of average PSNR for different compression methods.From Figure 14, due to the full usage of posttransform and TD, the proposed PTTD-based compression algorithm has the best compression performance.The PTTD-based method can improve 0.3∼1.3dB of average PSNR compared with other compression methods at 2.0∼0.25 bpp.
Overall, the PTTD-based compression method has excellent compression performance.It can fully meet the requirement of space multispectral CCD camera.

The Spectral Distortion Analysis.
We use the max similarity of spectrum (MSS) to evaluate the distortion of multispectral images for our algorithm.Let  be original remote sensing MSCFBs and let  be reconstructed remote sensing MSCFBs.The MSS can be expressed as where , , and  are line, column, and band number of multispectral image, respectively,   is the total number of bands, and  , and  , are the average value and mean square of the corresponding pixel in the spectral dimension, respectively.
The greater the value of MSS is, the greater the distortion of spectra is.The MSS between original and reconstructed multispectral images for the image group using different approach is shown as in Figure 15.
From Figure 15, our approach has the lowest MSS, so the proposed algorithm can protect spectral information well.
Figure 16 demonstrates the reconstructed third band of two remote sensing images, which is1024 × 1024; the depth of every pixel is 8 bits.The bit rate is set to 0.5 bpp.
In order to view the detailed information of reconstructed image, the part of image is 100 × 100.From Figure 14, under the low bit rate, 3DSPIHT-based approach causes the visible ringing effects and the blurred edge.However, our approach does not cause the blurred edge.So, the proposed algorithm can protect spectral information well.

Proposed Algorithm Complexity Analysis and Compression
Time.In our method, 1D-DCT is performed on spectral dimension to exploit the spectral information.It is reported that DCT has the lower amount of computations than DWT.When one of the most efficient methods is used to perform DCT, 8 × 8 points of 2D-DCT use only 462 time adding operations and 54 time multiplication operators.This means that each point of 2D-DCT needs only 0.84 time multiplication operators and 7.22 adding operations.Therefore,  bands of MSCFBs performing 1D-DCT need  × 0.84 multiplications and  × 7.22 additions.
After MSCFBs performing 2D-DWT, we use the Hadamard posttransform in low bit rate.Let each DWT coefficients block have  coefficients.The Hadamard-based posttransform needs the  ( log ) operations.In high bit  [38] 0.102 us/sample 3DSPIHT [3] 0.062 us/sample JPEG2000 [39] 0.181 us/sample Therefore, the calculation complexity of TD in the PTTDbased method is (    3 ) × ∑ 3 =1 ( 1 /2 +2 ×  2 /2 +2 ).In the following, we test and analyze the calculation times of the proposed PTTD-based compression method.Note that, in this test experiment, they only evaluate the calculation time of the PTTD-based compression because it is not the best optimized implementation on FPGAbased compression system.The evaluations testing performs the lossy compression for an image of size 3072 × 128 at 1.0 bpp on FPGA EVM board with system clock frequency at 88 MHz.Table 3 demonstrates the comparison of complexity between the proposed multispectral compression algorithm and others.
From Table 3, the processing time of our algorithm reaches 0.042 us/sample; the data throughput is 23.81MSPS, which is higher than JPEG2000, KLT, and 3DSPIHT, so our approach has the lowest complexity.In our project, the multispectral CCD camera works at an orbits altitude of 500 km, the scroll angle is −40 ∘ ∼ +40 ∘ , the latitudes are −70 ∘ ∼ +70 ∘ , and the line working frequency is 1.8094 kHz∼ 0.85946 kHz.In the line working frequency, capturing the image of 128 × 3072 required 70.74 ms.Using our approach, compressing the image of 128×3072 requires 16.51 ms.So, our approach can process the four bands images simultaneously.This meets the requirement of the project.
In addition, we use the XC2V6000-6FF1152 FPGA to implement the proposed algorithm.The design language  is VerilogHDL, the development platform is ISE8.2, and synthesis tool is XST.Table 4 demonstrates the occupancy of resources of our approach.
From Table 4, the LUTs occupy 67%, slices occupy 70%, and BRAM occupy 80%.Various indicators are lower than 95%, which meet the requirement of our project.

Conclusion
In this paper, we propose an efficient compression approach for MSCFBs.In our approach, the 1D-DCT is performed on spectral dimension to exploit the spectral information, and the PT in 2D-DWT domain is performed on each spectral band to exploit the spatial information.A deep coupling approach between the PT and TD is proposed to remove residual spectral redundancy between bands and residual spatial redundancy of each band.Experimental results show that the PTTD-based method has the average PSNR reaching upwards of 40 dB.And the PTTD-based method can improve 0.3∼1.3dB of average PSNR compared to other compression methods at 2.0∼0.25 bpp.The processing time of our algorithm reaches 0.042 us/sample; the data throughput is 23.81MSPS, which is higher than JPEG2000, KLT, and 3DSPIHT.Therefore, the proposed PTTD-based compression algorithm has the excellent compression performance and the low calculation complexity.It is very suitable for the space MSCFBs compression.

Figure 3 :
Figure 3: The spatial and spectral correlation test result.

Figure 4 :
Figure 4: The wavelet transform of high-resolution image with a zoom on some coefficients.

Figure 12 :
Figure 12: The structure of experiment system.
(b) Zoom of the part of original image (d) Reconstruction by our method (c) Reconstruction by 3DSPIHT (a) Image in the third spectral band

Table 1 :
Side information cost associated to the lp norm approach.

Table 2 :
Tested PSNR results of our compression.

Table 2 ,
the average PSNR reaches upwards of 40 dB.Therefore, the proposed algorithm based on deep coupling approach has the good compression performance and can satisfy the requirements of design index.Figure 11: Compression algorithm architecture for MSCFBs using the proposed PTTD method.

Table 3 :
The results of complexity comparison.

Table 4 :
The occupancy of resources., we use the DCT posttransform.2D 4 × 4 DCT requires only 27 multiplications and 231 additions.In addition, the best posttransform selection uses only 2 × ( − 1) time adding operations and one comparison operator.Therefore, the complexity of posttransform stage is (log 2  + 3) + 2 operations for each block with  coefficients.After 2D-DWT coefficients performing posttransform, the dimension of each subtensor is 4 × 4 ×  3 .Let   be the average dimensions of core tensor .The complexity of the calculation of tensor decomposition is (   3), where rate1 +  2 +  3 ) .