Two-dimensional satellite image compression using compressive sensing

ABSTRACT


INTRODUCTION
Image processing applications plays a vital role in the present digital world.In that satellite imaging has many useful applications in agriculture, remote sensing, urban planning and mineral exploration.Despite many useful applications, it is critical to transmit the satellite image data into the ground station in the limited transmission bandwidth.The satellite will be in contact with the ground station only at certain intervals.The data need to be stored on-board until the next communication window of the satellite.For all the reasons, it is very important to compress the satellite image data to make the communication and storage feasible.Image compression algorithms are broadly classified into lossless and lossy image compression algorithms.The satellite image is not only used for viewing purposes but also used to make important exploratory analysis.Therefore, lossless image compression [1] is the modality used in most of the applications.But in certain cases, to improve the image acquisition rates and to improve the compression rate lossy satellite image compression algorithms are made use in missions like SPOT 4 [2].
Nyquist sampling theorem [3] is the well-known method for signal acquisition and faithful recovery of signal from the measurements.Advances in compressive sensing-based methodologies gives satisfactory proof on retrieving the signal from a smaller number of samples when the signal is compressible in nature.The modality of compressive sensing is very useful when it is not practical to acquire many samples and when it is very expensive to collect large number of samples.In both cases compressive sensing can contribute by making the image acquisition hardware simpler, reducing the image acquisition time, and thereby achieving image compression.Where, the images are transformed using discrete cosine transform (DCT) or discrete wavelet transform (DWT).The coefficients are then encoded using encoding techniques like run-length encoding, Golomb coding [4], [5], Arithmetic coding [6], and rice coding [7].Many conventional image compression algorithms have been proposed in the literature for the compression of satellite images.The consultative committee for space data systems (CCSDS) document [8] describes the lossless compression methods for satellite images.The image compression algorithm based on DWT called set partition in hierarchical trees (SPIHT), where the significant DWT coefficients are encoded first for the fast and efficient coding of images is presented in [9].Aranki et al. [10], the field programmable gate arrays (FPGA) implementation of the lifting scheme of DWT is proposed.The research work in [10] and [11] discuss the concept of finding the DWT coefficients of an image segment, in a pipeline fashion, instead of waiting for the entire image from the push-broom scanning, since the values of DWT coefficients depends on the pixels in the neighbourhood.JPEG-LS algorithm for the lossless compression of images is presented in [12].A low complexity context based lossless image compression algorithm is presented in [13].The JPEG-2000 algorithm, which is a DWT version of DCT based JPEG to achieve different degrees of image quality based on different applications is proposed by the joint photographic experts' group in [14]- [16].Compressive sensing [17]- [19] is based on the principle that if the signal is compressible in nature, it is possible to reconstruct the signal from very few measurements.The compressive samples are generated using the random sampling matrix which captures the signal in very few measurements.
Figure 1 shows the general block diagram of the conventional image compression algorithms like JPEG and CCSDS standard.As it is evident from the block diagram, the satellite image goes through many processing steps at the encoder stage.First the image is transformed using one of the transform coding techniques like DCT or DWT.The insignificant coefficients are discarded after the transform coding.The remaining coefficients are subjected to quantization in the next stage.The quantized coefficients are further encoded using entropy coding techniques like Huffman coding, and arithmetic coding.The encoder is very simple in case of CS based compression algorithms as shown in Figure 2. The only processing step at the encoder is generation of compressive sensing samples from the random compressive sensing matrix.The decrease in computational steps in the encoder is of great significance in case of remote sensing application because memory and power are the two important critical resources in case of any remote sensing application.

Figure 1. General block diagram of the conventional satellite image compression algorithm
The most significant aspect of using CS for the compression in remote sensing applications can be attributed to the property of making the encoder very simple by removing all the processing steps in the encoder stage of the conventional compression algorithm.The compressive sensing samples are communicated to the ground station in case of satellite imaging applications.The disadvantage of CS based algorithms for compression is that the reconstruction process from compressive sensing samples is quite computationally expensive.This aspect can be overlooked as the reconstruction process happens in the ground station where resources are available in abundance.The research work makes use of compressive sensing techniques to compress the satellite images.Compressive sensing samples are generated using a CS matrix containing random Fourier transform coefficients.The satellite image is compressed at various sampling rates to analyse the performance of the algorithm at different levels of compression.The satellite image is reconstructed from the compressive sensing samples by a class of interior point methods called log barrier method.The iterative technique minimizes the objective function with total variation regularization to solve the problem of image reconstruction from compressive sensing samples.The rest of the article is organized as follows; Section 2 describes the compressive sensing technique proposed to reconstruct the satellite image from compressive sensing samples.Section 3 presents the experimental results using different datasets.The performance of the algorithm is analysed using quantitative metrics like peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM).Section 4 concludes the research work presented in the article.

CS-CR ALGORITHM
Consider an image , which has dimensions  ×  The total number of pixels in  is denoted by .The image  is vectorized as  ≔ (), where the length of the vector  is .The results in papers [20]- [22] demonstrates the possibility of recovering signals from fewer number of measurements i.e.,  =  ∈   ,  ≪  with considerable accuracy.The basis pursuit problem to solve for optimal value of  from fewer compressive sensing samples  is given by (1): The optimization problem solves for  with minimum  1 norm, which explains the compressive sensing measurements .Another strategy to solve for  is given by (2): where, () represents the total variation [23] of .() is defined as (3), where, The problem in (2) can be generalized as (6), where,   represents the  linearity constraints.The optimization problem in ( 6) can be converted into a series of sub problems with linearity constraints using the Log-barrier method [24] as (7): Log-barrier method augments the inequality constraints into the objective by log penalty function.At each iteration , the value of  is increased.When  is sufficiently large, the solution   approaches the optimal value of the original problem in (6).The second order approximation of ( 7) at ∆ is given by ( 8): where,   represents the gradient, which is given by ( 9): Represents the Hessian matrix, which is given by ( 10): If  is in feasible set, ∆ which represents the minima of second order approximation in ( 8) is given by the following set of linear: is the Lagrange multiplier for the equality constraints.
The flowchart of the CS-CR algorithm based on Log-barrier method for the reconstruction of image from compressive sensing measurements is given in Figure 3.The algorithm is initialized with   , , , ,  in step 1.In step 2 (7) is solved using Newton method using  −1 as initial point.In step 3, if

RESULTS AND DISCUSSION
Experiments are conducted using MATLB R2016b in windows 10 laptop, 64-bit operating system with Intel(R) i7 processor, 16 GB RAM.For the experimentation two datasets [25] have been considered.A band from coastal_b1 data of Landsat NASA is considered as the first data.The image is 1024×1024 in size.Another data considered for experimentation is spot_la_b3 from SPOT 3 CNES.The image is 500×500 in size.Three algorithms [26] are considered to compare and validate the performance of the CS-based reconstruction algorithm.The algorithm proposed in CCSDS document, uses float DWT to achieve lossy compression of satellite images, where the parameters are tuned to achieve different rate distortion performance and full frame compression.The JPEG-2000 algorithm based on float DWT in both scan mode and frame mode is considered for the comparison where rate allocation optimization is done using the Lagrange multipliers.Another algorithm considered for comparison is the set partitioning in hierarchical trees (SPIHT), which is based on float DWT to down sample the image into different decomposition levels.The methods are compared in terms of quantitative metrics like PSNR and SSIM.PSNR provides the information about the average numerical similarity between the original and reconstructed image.SSIM provides the measure of average spatial structural similarity between original and reconstructed image.The measures PSNR and SSIM are defined as follows:  Peak signal to noise ratio (PSNR):

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ISSN: 2088-8708 Int J Elec & Comp Eng, Vol. 12, No. 1, February 2022: 311-319 312 Conventional image compression algorithms rely on the Nyquist sampling theorem where large numbers of samples are collected only to be discarded at the stage of compression.Conventional image compression algorithms in the literature are generally based on transform coding techniques.

Figure 2 .
Figure 2. General block diagram of compressive sensing-based algorithm
algorithm is terminated, else the algorithm goes back to step 2.

Figure 3 .
Figure 3. Flowchart of the CS-CR algorithm

Figure 6 .
Figure 6.These figures are; (a) PSNR using different compression techniques, (b) PSNR at various sampling rates using CS based compression technique, (c) SSIM at various sampling rates using CS based compression technique, (d) run time in seconds at various sampling rates using CS based compression technique, (e) original and compressed size of coastal_b1 image at different sampling rates, (f) original and compressed size of spot_la_b3 CNES image at different sampling rates