Introduction

Lossless or near-lossless image compression techniques are especially applicable to medical or remote-sensing images where too much information loss is not allowed. These images carry a lot of information and hence actual images can’t be recovered in lossy compression processes. Due to this problem, lossy compression processes are not used in medical or remote-sensing images as detailed analysis is not possible for further processing. So, this is a challenging problem for lossless compression techniques where storage space consumption is also a major issue. We studied a few of the very recent image compression methods (last 5 years) and applied their output to our method.

The application of vector quantization with principal component analysis is one of the latest techniques used for lossless compression of JPEG images [1]. The neural network system is also used for lossless compression of remote-sensing images which improves the PSNR of the output [2, 3]. A combination of DWT and residual network provides a 2.5 compression ratio for JPEG images [4, 5]. Deep learning is applied to video surveillance in underground mines [6] which perform lossless compression as well as object detection. Lifting wavelet lossless image compression improves storage space optimization using a high compression ratio (sometimes more than 7) [7]. DICOM system helps to maintain a very high PSNR value for medical images (> 80) [8]. DCT is also used for near-lossless compression, especially for dental images [9]. A 3D version of it is effectively used in endoscopic images [10].

The researchers applied the grating system for multiplexing and compression of a series of images as well as video [11, 12]. Using either amplitude or phase grating, multiple images are modulated at different angles using variable spatial frequencies. A part of the final spectrum is transmitted for storage space optimization. The main drawback of the system is that the number of images is limited. In the modified version of it, this problem is overcome. Here, the selected spectra are placed sequentially in the canvas which is to be transmitted to a long distance [13, 14].

In this article, using amplitude grating, selected images are modulated one by one along the x-axis. A high value of spatial frequency is maintained to overcome the aliasing problem. As a result of this modulation, three spectra are generated. One is the center spectrum which is the direct Fourier Transform of the image. Other spectra are the side lobes generated due to the modulation. Both of them contain the same information hence only one is considered for further processing. With the application of the Inverse Fourier Transform in addition to a low pass filter is utilized for the extraction of the images. During this process, a pixel intensity graph is drawn along the x-axis, and the intensity of the central point and the periphery is measured which is useful for cut-off frequency detection. Lastly, the PSNR value of the extracted images is measured.

Methodology

Let us assume that the selected image is marked as \({k}_{1 }(x,y)\)

Image \({k}_{1 }(x,y)\) is modulated using the amplitude grating Eq. 

$$g\left(x\right)= \frac{m}{2}\left(1+cos2\pi {b}_{0}x\right)$$
(1)

where \({b}_{0}\) is treated as the spatial frequency (lines/cm) used for modulation

As we consider the 1st order only, hence the value of \(m=1\)

So, the Eq. (1) is written as

$$g\left(x\right)= \frac{1}{2}\left(1+cos2\pi {b}_{0}x\right)$$
(2)

Due to the modulation in the image \({k}_{1 }(x,y)\) with the grating system \(g\left(x\right)\), the resultant object is

$$r\left(x,y\right)= {k}_{1 }(x,y)*g\left(x\right)$$
(3)
$$r\left(x,y\right)= {k}_{1 }(x,y)*\frac{1}{2}\left(1+cos2\pi {b}_{0}x\right)$$
(4)

So. Its Fourier Transform is

$$R\left(u,v\right)= \frac{1}{2}{K}_{1 }\left(u,v\right)+\frac{1}{4}{K}_{1 }\left(u-{b}_{0},v\right)+\frac{1}{4}{K}_{1 }(u+{b}_{0},v)$$
(5)

\(\frac{1}{2}{K}_{1 }\left(u,v\right)\) denotes the centre spectrum and \(K\left(u\pm {b}_{0},v\right)\) denotes the side spectra. Both spectra contain the same information, hence only one of them is sufficient for the next process. We considered spectrum \({K}_{1 }(u+{b}_{0},v)\) in our research work. The main reason for selecting only the ±1st order is the very poor value of the diffraction efficiency for the ±2nd order and beyond. So, it is impossible to reconstruct the images from those side bands.

By applying the Inverse Fourier Transform and a low pass filter with a suitable cut-off frequency, we recovered the image information.

Result

In this research work, we have chosen five different images which are used by the researchers in the reference research works. As we used different pixel resolution images, we resized them by 512 × 512 for comparative study and analysis of our result. These images are displayed in Fig. 1(a-e).

Fig. 1
figure 1

(a-e) Selected images

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Fig. 2
figure 2

Grating system

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Fig. 3
figure 3

(a, b) Modulated spectra and selected spectra

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Figure 2 indicates the enlarged version of the amplitude grating system with a high spatial frequency. The high spatial frequency is used to avoid the aliasing effect of the system. In the case of the aliasing effect, the spectra overlap with each other, hence proper reconstruction would be impossible. We used the spatial frequency value as 1000 (lines/cm) which is sufficient to avoid overlap.

Figure 3(a) shows the modulated spectra (contains three spectra in the same canvas). The center spectrum is the direct Fourier Transform of the image whereas the other spectra represent \(K\left(u\pm {b}_{0},v\right)\). As both of them contain the same information, only one is enough for further processing. We selected \({K}_{1 }\left(u+{b}_{0},v\right)\) which is shown in Fig. 3(b).

Fig. 4
figure 4

(a, b) Extraction process from the selected spectrum

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Figure 4(a) and Fig. 4(b) show the diagram of the extraction process from the spectrum \({K}_{1 }\left(u+{b}_{0},v\right)\). During this process, initially, we calculate the intensity of the central point and the periphery of the spectrum. These two points are marked as A and B and their difference is denoted by R. It has been observed that the intensity at the periphery (point B) is just half of the centre value intensity (point A). It has been observed that with the increase of the diameter length, the size of the reconstructed images also increases hence more space is required to store the recovered images. So to optimize the storage space, this particular value is selected.

In the next process, the value of R (pixel distance) is calculated which is the cut-off value of the low pass filter. It is possible to extract almost 99% of the image information.

This intensity-based cut-off value selection is useful for the coding of the system. The cut-off frequency value varies from image to image. Hence for any image, where the intensity is half of the center point intensity, this value is treated as the maximum value of R. Now the pixel distance value of R indicates the cut-off frequency value.

In our research work, a value of R = 130 is observed. This is the final value for the selection of the cut-off frequency. The intensity graph is displayed in the Fig. 5.

Fig. 5
figure 5

(a, b) Filtered images

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Figure 5(a-e) displays the extracted images with a pixel resolution size of 260 × 260.

Using three tables, we highlighted the compression ratio (Table 1), the PSNR value of the output images (Table 2), and the improved compression ratio with the existing methods (Table 3)

Table 1 Compression ratio
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Table 2 PSNR value
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Table 3 Improved compression ratio of existing methods
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Special features of this method

  1. 1.

    The compression ratio is almost 4 for the image 512 × 512. In existing lossless or near-lossless methods, the value of the compression ratio is nearly 3. So, in addition to this method, researchers can optimize storage space.

  2. 2.

    The PSNR value is greater than 38 in all cases, which indicates a satisfactory result (> 30 value of PSNR is generally known as good quality output).

  3. 3.

    Especially applicable for remote-sensing and medical images, where too much information loss is not allowed and suitable for all types of images whereas few applications are suitable for JPEG images only.

Conclusion

In this communication, we propose a preprocessing or postprocessing near-lossless compression which can be additionally used with researchers’ compression methods. This near-lossless process provides a compression ratio of 3.9 which is practically more than 20 times (Table 3) for any lossless method. The PSNR value of the output image is greater than 38 which indicates an excellent result. This system is also applicable to colour images. In that case, the entire operation is performed in each plane of the color image and then they are reconstructed together.