An end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform

Abstract

The application discloses an end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform. The method decomposes an image through multi-stage wavelet transform to obtain sub-band coefficients of different sizes. The coefficients are input into an entropy coding model, and an arithmetic encoder driven by a context model is used to compress the coefficients into a bit stream. The application adopts two-dimensional non-separable wavelet transform, uses a lifting structure of wavelet transform, contains three prediction filters and an update filter, and the filter structure is a convolutional neural network structure. The entropy coding module also uses a neural network structure. All parameters are optimized as a whole through deep learning, the network is trained until convergence, and the trained model is used to realize image lossless compression. The application has excellent performance in lossless compression, is significantly improved compared with a traditional JPEG2000 lossless compression method, and is also superior to an end-to-end image lossless compression method based on a traditional separable wavelet transform.

Description

Technical Field

[0001] This invention belongs to the field of image compression coding and relates to an end-to-end lossless image compression method based on two-dimensional inseparable wavelet transform. Background Technology

[0002] Image compression is a key research area in modern digital signal processing and multimedia applications, aiming to improve the efficiency of information transmission and storage. Image compression algorithms include traditional image compression algorithms and end-to-end image compression algorithms based on deep learning. Traditional image compression standards such as JPEG and JPEG2000 are implemented through hand-designed modules, featuring low decoding complexity and strong versatility. However, these standards rely on fixed coding architectures, making it difficult to adapt to the diversity and complexity of image content. In contrast, end-to-end image compression algorithms utilize large amounts of image data for training, and through repeated iterations and optimization of neural networks, significantly improve coding efficiency and image quality.

[0003] As a classic image coding method, JPEG2000 uses wavelet transform to convert images into coefficients. In JPEG2000, the wavelet transform is implemented using a lifting form, which involves a series of prediction and update operations.

[0004] To further improve image coding performance, researchers applied wavelet transform to end-to-end image compression methods, proposing a learning-based wavelet transform-based end-to-end image coding method, iWave++. In this method, the wavelet transform employs prediction and update filters with convolutional neural network structures, and other modules of image coding also utilize neural network structures, thus forming a holistic end-to-end image compression framework. The iWave++ method exhibits superior performance in lossless compression, but there is still room for further improvement.

[0005] A primary reason for this optimization requirement is that lossless compression must be performed in the integer domain, thus requiring rounding operations after the filters constructed by convolutional neural networks. Traditional wavelet transform for processing two-dimensional images involves performing a one-dimensional transform first in the horizontal direction, then another in the vertical direction, resulting in four sub-bands. Each one-dimensional transform includes an update filter and a prediction filter, and rounding is required after processing the coefficients of each filter, resulting in a total of six rounding operations. Since rounding introduces rounding errors, excessive rounding can lead to error accumulation, thereby affecting the performance of lossless compression.

[0006] Therefore, in order to further improve the performance of this method, reducing the number of rounding operations by changing the structure of the two-dimensional wavelet transform is an important direction. Summary of the Invention

[0007] The purpose of this invention is to address the shortcomings of existing technologies by proposing an end-to-end lossless image compression method based on two-dimensional inseparable wavelet transform, comprising the following steps:

[0008] Step 1: Obtain the RGB format image and convert it to a grayscale image;

[0009] Step 2: Perform multi-scale decomposition of the grayscale image using multi-level wavelet transform to obtain multiple sub-bands;

[0010] Step 3: Input all subband information into the learning-based entropy coding module for compression coding to obtain the coded data of each subband coefficient;

[0011] Step 4: Decode the encoded bitstream and recover the corresponding sub-band coefficient values ​​according to the probability distribution of each sub-band coefficient;

[0012] Step 5: Perform multi-level wavelet inverse transform on each decoded subband coefficient, corresponding to Step 2, to reconstruct the pixel parameters of the image and reconstruct the original image;

[0013] Step 6: Convert the grayscale image obtained after decoding and reconstruction into the original RGB format image.

[0014] Furthermore, in step 2, the multi-scale decomposition of the grayscale image using multi-level wavelet transform is specifically as follows:

[0015] Perform a wavelet transform on the grayscale image to obtain four sub-band coefficients. , , , ;

[0016] For low-frequency subband coefficients Continue with wavelet transform to obtain higher frequency subband coefficients. , , , ;

[0017] And so on, performing wavelet transforms. Once, a total of Individual coefficients, , This indicates the wavelet transform order.

[0018] Furthermore, the wavelet transform employs a learning-based two-dimensional non-separable structure, including a decomposition step, a prediction step, and an update step.

[0019] Decomposition steps: Approximation coefficients of the grayscale image It is decomposed into four components according to the parity of the horizontal and vertical axes: , , , ,in Indicates the degree of wavelet decomposition. Indicates pixel position;

[0020] Prediction steps: Process each component using a prediction filter;

[0021] Update steps: Calculate the difference between the output of the prediction filter and each original component to obtain the processed subband component. , as well as Input the update filter, and obtain the result and Combined to generate updated low-frequency signals .

[0022] Furthermore, both the prediction filter and the update filter are convolutional neural networks. After filtering, rounding is performed, and wavelet transform is used to obtain the coefficients of the four output components. , , as well as ;

[0023] use , , , This represents four filters composed of convolutional neural networks. P Indicates a prediction filter. U This indicates the filter update; the formula for calculating the four coefficient values ​​of the output obtained from the four decomposed components is as follows:

[0024] ,

[0025] ,

[0026] ,

[0027] ,

[0028] In this context, [ ] represents rounding operation, and * represents convolution operation; the purpose of rounding operation is to ensure that the encoding is lossless.

[0029] Furthermore, both the prediction filter and the update filter are convolutional neural networks, including convolutional layers and nonlinear activation functions;

[0030] Feature extraction is performed on the input image through the first convolutional layer;

[0031] The features obtained from the first convolutional layer are further processed by two parallel tanh activation functions and a second convolutional layer to output a feature map;

[0032] The feature map is merged with the extracted features, and a third convolutional layer is used to fuse the features of the merged feature map.

[0033] The fused feature map is added to the input image to obtain the final output.

[0034] Furthermore, the learning-based entropy coding module includes:

[0035] The context model uses an autoregressive model to process all subband information sequentially, outputting a set of entropy parameters as coefficients for each subband. , , ..., Parameters of the probability distribution;

[0036] The probabilistic model predicts the probability distribution of each coefficient based on contextual information, and determines the probability of occurrence of the coefficient for arithmetic coding.

[0037] Arithmetic encoders compress and encode the coefficients of each subband based on the probabilities obtained from the probability model.

[0038] Furthermore, the context model includes a long-term context extraction model and a context fusion model;

[0039] Long-term context extraction model, for the encoded Extracting individual subbands to obtain long-term contextual information. ;

[0040] Context fusion model integrates long-term contextual information Information in and current subband The parameters of the current coefficient probability distribution are generated by combining the information of the already encoded coefficients.

[0041] Furthermore, in step 3, the probability model used in the arithmetic encoder is a Gaussian mixture model, which contains three Gaussian distributions.

[0042] Furthermore, in step 4, the encoded bitstream is decoded to recover the probability distribution of each subband coefficient and the corresponding subband coefficient value. , , ..., ;

[0043] Using the same context model as in the encoding stage, the parameters of the coefficient probability distribution are obtained, and the probability distribution of each coefficient is recovered.

[0044] Using the same probability model as in the encoding stage, the probability distribution of each coefficient is reconstructed based on the parameters generated by the context model for arithmetic decoding;

[0045] The arithmetic decoder decodes the bitstream step by step based on the output of the probability model, restoring the original values ​​of the coefficients of each subband.

[0046] Furthermore, in step 5, the multi-level wavelet inverse transform corresponding to step 2 is performed on the decoded sub-band coefficients, specifically starting from the highest frequency sub-band and performing the inverse transform step by step.

[0047] The four sub-bands generated by the last wavelet transform , , , Perform an inverse transform to obtain the lower frequency subband of the next higher level. ;

[0048] Merge the four sub-bands of the previous level , , , This yields the next higher level of low-frequency subband. ;

[0049] Repeat this process until the highest-level subband coefficient is obtained. , , , The values ​​are combined to obtain the complete pixel information of the image.

[0050] Furthermore, the two-dimensional inseparable wavelet inverse transform is the strict inverse process of the two-dimensional inseparable wavelet transform, including the following steps:

[0051] Inverse update, inverse prediction, and merging; the filter parameters used in inverse update and inverse prediction are exactly the same as those in the forward transform.

[0052] Similarly, rounding is required after each pass through the filter to ensure that the results of all operations are integers, thereby achieving lossless image reconstruction; the mathematical expressions for inverse update and inverse prediction are as follows:

[0053] ,

[0054] ,

[0055] ,

[0056] ,

[0057] in P , U The `[]` represents the prediction filter and the update filter, `[]` represents rounding, and `*` represents convolution. The resulting four component values... , , , The merging yields the inverse transformation result. .

[0058] Furthermore, it also includes step 7: constructing a loss function and using an optimizer to perform end-to-end training and optimization of the overall image compression model to obtain the optimal model.

[0059] The beneficial effects of this invention are:

[0060] Compared with the prior art, the significant advantages of this invention are:

[0061] (1) Traditional wavelet transform requires six rounding operations when performing two-dimensional decomposition, first horizontal and then vertical transformation. Two-dimensional inseparable wavelet transform only requires four rounding operations. The fewer rounding operations, the smaller the rounding error. Therefore, the image compression method using two-dimensional inseparable wavelet transform can achieve better lossless compression results compared with the existing method using classical two-dimensional wavelet transform.

[0062] (2) In the end-to-end image compression method, a two-dimensional inseparable wavelet transform is adopted, and its filter adopts a convolutional neural network structure. Together with the parameters used by other modules in the image compression method, the overall optimization is carried out based on deep learning. This process continuously trains and optimizes the model, thereby continuously improving its performance and achieving a more superior lossless compression effect.

[0063] Other features and advantages of the present invention will be described in detail in the following detailed description section. Attached Figure Description

[0064] The above and other objects, features and advantages of the present invention will become more apparent from the more detailed description of exemplary embodiments of the invention in conjunction with the accompanying drawings, wherein the same reference numerals generally represent the same components in the exemplary embodiments of the invention.

[0065] Figure 1 This is a diagram of the overall framework structure of the end-to-end lossless image compression based on two-dimensional inseparable wavelet transform of the present invention.

[0066] Figure 2 This is a diagram of the two-dimensional inseparable wavelet transform structure used in this invention.

[0067] Figure 3 This is a schematic diagram of the convolutional network structure used in the prediction filter and update filter of the two-dimensional inseparable wavelet transform structure.

[0068] Figure 4 This is a schematic diagram of the multi-scale decomposition used in this invention.

[0069] Figure 5 This is a schematic diagram of the context fusion model in the entropy coding module used in this invention. Detailed Implementation

[0070] Preferred embodiments of the invention will now be described in more detail with reference to the accompanying drawings. While preferred embodiments of the invention are shown in the drawings, it should be understood that the invention can be implemented in various forms and should not be limited to the embodiments set forth herein.

[0071] like Figure 1 As shown, the lossless image compression method based on learning-based two-dimensional inseparable wavelet transform includes the following steps:

[0072] Step 1: Input an RGB format image as a training image; First, convert the RGB format image to the YCbCr color space, then extract the Y channel information and convert the RGB color image to a grayscale image; Second, crop 128×128 pixels from the center of the processed grayscale image as training data.

[0073] Step 2; Perform the encoding step, and perform wavelet transform on each uncoded image; Use a learning-based two-dimensional inseparable wavelet structure, and adopt the wavelet transform lifting structure, which consists of three parts: decomposition, update and prediction;

[0074] like Figure 2 As shown, firstly, the image approximation coefficients are... Decomposed into four components based on parity in the horizontal and vertical directions: , , , ,in Indicates the degree of wavelet decomposition. This indicates the pixel position; each sub-band signal is processed by a corresponding prediction filter, and the difference between the output of the prediction filter and each original component is calculated to obtain the processed sub-band component. , as well as Input the update filter, and obtain the result and Combined to generate updated low-frequency signals After filtering, rounding is performed, and wavelet transform is used to obtain the coefficients of the four output components. , , as well as ;

[0075] Furthermore, using , , , This represents four filters constructed from convolutional neural networks, where P represents the prediction filter and U represents the update filter. The Z-transform calculation formula for the four output coefficient values ​​is as follows:

[0076] ,

[0077] ,

[0078] ,

[0079] ,

[0080] In this context, [ ] represents rounding operation, and * represents convolution operation; the purpose of rounding operation is to ensure that the encoding is lossless.

[0081] Furthermore, the prediction filter and update filter in the wavelet transform employ a convolutional neural network structure, as shown in the following figure. Figure 3 As shown, the neural network consists of multiple convolutional layers and non-linear activation functions. First, the input image layer has a size of 3×3×1, and then features are extracted through two 3×3×16 convolutional layers. Next, a combination of two parallel tanh activation functions and 3×3×16 convolutional layers outputs a feature map, which is then merged with the previous features. Next, the merged feature map is passed through a 3×3×1 convolutional layer for final feature fusion. Finally, this fused feature map is added to the input image to obtain the final output.

[0082] Step 3: Perform multi-scale decomposition on the image: First, perform a wavelet transform on an image to obtain four sub-bands: , , , Then to Continuing the transformation yields higher frequency subbands: , , , Then on Continuing the transformation, this method performs a total of four wavelet transform decompositions, yielding 13 sub-bands; The order in which the subbands are placed after the second wavelet transform is as follows: arrive , Figure 4 The diagram illustrates the placement order of each sub-band. During the arithmetic encoding / decoding stage, the generated sub-bands are denoted in reverse order of their generation. , , ..., Taking 13 sub-bands as an example, they are represented as follows: , , ..., All sub-bands will be in accordance with Figure 4The wavelet coefficients are compressed one by one in the order shown by the middle arrow; within each sub-band, the wavelet coefficients are compressed one by one in the order from left to right and from top to bottom.

[0083] Furthermore, the multi-level wavelet transform used in multi-scale decomposition shares the same set of convolutional neural network parameters; by sharing the same convolutional neural network parameters, the transformation process can maintain the same characteristics and effects for images of different resolutions, thereby achieving uniform encoding and decoding performance at different scales.

[0084] Step 4: Input the processed coefficients into the learning-based entropy coding module for further processing; The learning-based entropy coding module used in this invention includes: a context model that outputs a set of entropy parameters, a probability model that predicts the probability distribution of each coefficient, and an arithmetic codec. Figure 1 In the text, AE and AD represent arithmetic encoding and arithmetic decoding, respectively.

[0085] Furthermore, the context model in the entropy coding module is an autoregressive model, comprising a long-term context extraction model and a context extraction model: the long-term context extraction model processes previously encoded subbands. , , ..., Export long-term context information ; Context fusion model will Information in and current subband The parameters of the current coefficient probability distribution are generated by combining the information of the already encoded coefficients.

[0086] Furthermore, the long-term context extraction model uses a recurrent neural network structure, applying three convolutional-based long short-term memory layers; this is used to complete the entire subband. After the encoding process, the generated subbands Send it into a recurrent neural network structure to obtain long-term context information. In encoding the next subband hour, Remain unchanged; after completing the entire subband After the encoding process, then... The data is sent to the recurrent neural network to complete the process from... arrive The update will then As the next sub-band The long-term context information is then entropy encoded; this iterative process continues in each subsequent subband, enabling the recurrent neural network to capture and maintain relevant long-term context information between different subbands during the encoding process.

[0087] Furthermore, when the sub-band and When the resolution is different, appropriate adjustments need to be made to handle this situation; for example, in Figure 4 middle, and Different resolutions, when encoding At this time, it is necessary to output the long-term context. And upsampling is performed on the state variables of the recurrent neural network; the upsampling process ensures long-term contextual information and internal state and subband and The resolution is correctly aligned, thus enabling accurate and effective context modeling during the encoding process.

[0088] Furthermore, the context fusion model utilizes convolutional neural networks, such as... Figure 5 As shown; the model uses masked convolution to ensure that the network only utilizes the already encoded coefficients, i.e., information from the pixels above and to the left of the current pixel (as shown). Figure 5 middle (the upper half of the region); firstly, respectively... and Convolutional layers are applied, and a residual learning structure is incorporated. The intermediate results were integrated into The calculation process is as follows: the results of the two paths are then passed through a series of convolutional layers to finally generate entropy coding parameters.

[0089] Furthermore, the probabilistic model used in arithmetic coding is a Gaussian mixture model containing three Gaussian distribution components; each Gaussian distribution component has its corresponding mean, variance, and mixing coefficients, which are determined through training.

[0090] Step 5: Decode the encoded bitstream: Use the context model generated during the decoding process to generate parameters of the current coefficient probability distribution, and restore the probability distribution of each coefficient based on the parameters generated by the context model; use the arithmetic decoder to decode the bitstream step by step according to the output of the probability model, and gradually restore the original coefficient values.

[0091] Step 6: Perform the decoding step, applying four learning-based two-dimensional inseparable wavelet inverse transforms to the coefficients to reconstruct the pixel parameters of the image: First, starting from the highest frequency sub-band, perform four sub-bands generated by the last wavelet transform ( , , , Perform an inverse transform, merge these subbands, and reconstruct the lower-frequency subband of the next higher level. Then, the reconstructed low-frequency subband and its corresponding three high-frequency sub-bands ( , , Perform inverse transformation and merging operations; repeat this process until the original image is reconstructed.

[0092] Furthermore, the two-dimensional inseparable wavelet inverse transform is the strict inverse process of the two-dimensional inseparable wavelet transform, including the following steps: inverse update, inverse prediction, and merging. The filter parameters used for inverse update and inverse prediction are exactly the same as those for the forward transform. Similarly, rounding is required after each filter pass to ensure that the results of all operations are integers, thereby achieving lossless image reconstruction. The mathematical expressions corresponding to inverse update and inverse prediction are as follows:

[0093] ,

[0094] ,

[0095] ,

[0096] ,

[0097] in P , U The `[]` represents the prediction filter and the update filter, `[]` represents rounding, and `*` represents convolution. The resulting four component values... , , , The merging yields the inverse transformation result. .

[0098] Step 7: Convert the decoded grayscale image back to an RGB format color image;

[0099] Step 8: Construct the loss function as follows: ,in This represents the estimated bit rate. This represents the two-dimensional non-separable wavelet forward transform. Represents the training set, This represents the trainable parameters in the transformation module. This represents all trainable parameters in the entropy coding module; the Adam optimizer is used to perform end-to-end training and optimization of the overall image compression model to obtain the optimal model.

[0100] The training set used in this embodiment includes 800 images from the DIV2K dataset, 585 images from the CLIC2020 Professional dataset, and 1048 images from the CLIC2020 Mobile dataset. The training process was implemented using CompressAI. The Adam algorithm with its default settings was used during training, with a learning rate of 1e-4, and training lasted approximately 500 epochs.

[0101] The embodiments tested the present invention against the traditional compression method JPEG2000 and the end-to-end image compression method iwave++ based on learned wavelet classes. The test datasets included the Kodak and Tecnick datasets. The Kodak dataset contains 24 high-quality lossless color images, each with a resolution of 768x512 pixels. The Tecnick dataset contains 40 high-resolution images, each with a resolution of 1200x1200 pixels. In the testing phase, the RGB images were first converted to YCbCr format, and then the images were compressed on each of the three channels using the model trained on grayscale images during the training phase.

[0102] The embodiments use bits per pixel (bpp) as the evaluation metric. On the Kodak dataset, the average data size of JPEG2000 is 9.58 bpp, the average data size of iWave++ is 8.07 bpp, and the average data size of this invention is 7.99 bpp, a reduction of 1.59 bpp (approximately 16.6%) compared to JPEG2000, and a reduction of 0.08 bpp (approximately 0.99%) compared to iWave++. On the Tecnick dataset, the average data size of JPEG2000 is 7.23 bpp, the average data size of iWave++ is 6.44 bpp, and the average data size of this invention is 6.38 bpp, a reduction of 0.79 bpp (approximately 10.9%) compared to iWave++, and a reduction of 0.06 bpp (approximately 0.93%) compared to iWave++.

[0103] Experimental results show that, compared with the traditional image compression algorithm JPEG2000, this method significantly reduces storage requirements and data volume, demonstrating the substantial advantages of end-to-end image compression algorithms. Furthermore, this method outperforms the existing end-to-end image compression algorithm iWave++ in lossless compression on multiple datasets. This indicates that this method is more capable of lossless compression, achieving a higher compression ratio and smaller data volume.

[0104] The various embodiments of the present invention have been described above. These descriptions are exemplary and not exhaustive, nor are they limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments.

Claims

1. An end-to-end lossless image compression method based on two-dimensional non-separable wavelet transform, characterized in that, The method comprises the following steps: Step 1: Obtain an image in RGB format and convert it into a gray-scale image; Step 2: Perform multi-scale decomposition on the gray-scale image by using a multi-level wavelet transform to obtain a plurality of subbands; Step 3: Input all the subband information into a learning-based entropy coding module for compression coding processing to obtain coded data of the subband coefficients; Step 4: Decode the coded code stream, and restore the subband coefficient values according to the probability distribution of the subband coefficients; Step 5: Perform multi-level inverse wavelet transform on the decoded subband coefficients corresponding to step 2 to reconstruct the pixel parameters of the image and reconstruct the original image; Step 6: Convert the gray-scale image obtained after decoding and reconstruction into an original image in RGB format; In step 2, the multi-scale decomposition of the gray-scale image by using the multi-level wavelet transform is specifically as follows: Once wavelet transform is done on the gray image, four sub-band coefficients are obtained , , , ; Low frequency subband coefficients Continuing the wavelet transform, higher frequency subband coefficients are obtained , , , ; Similarly, wavelet transform is performed Total of subband coefficients, , denotes the number of wavelet transforms; The wavelet transform uses a learning-based two-dimensional non-separable structure, which comprises a decomposition step, a prediction step and an update step; Decomposition steps: Approximation coefficients of the grayscale image It is decomposed into four components according to the parity of the horizontal and vertical axes: , , , ,in This indicates the degree of wavelet decomposition. Indicates pixel position; The prediction step: each component is processed by using a prediction filter; Update step: compute the difference between the output of the prediction filter and each of the original components to obtain the processed subband components , and input the update filter and obtain the result and combine to generate the updated low frequency signal .

2. The end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform according to claim 1, characterized in that The prediction filter and the update filter are both convolutional neural networks, and after the filtering processing, a rounding operation is performed, and wavelet transform is performed to obtain four output component coefficients , , and ; with , , , represent four filters composed of a convolutional neural network, P represents a prediction filter, U represents an update filter; the calculation formula of the four coefficient values of the output obtained from the four decomposed components is as follows: ， ， ， ， Wherein, [ ] represents a rounding operation, and * represents a convolution operation; the purpose of the rounding operation is to ensure that the coding is lossless coding.

3. The end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform according to claim 1, characterized in that The prediction filter and the update filter are both convolutional neural networks, which comprise a convolution layer and a nonlinear activation function; The first convolution layer is used for feature extraction of the input image; The features obtained by the first convolution layer are further processed by two parallel tanh activation functions and the second convolution layer to output a feature map; The feature map and the extracted features are merged, and the third convolution layer is used for feature fusion of the merged feature map; The fused feature map and the input image are subjected to an addition operation to obtain a final output.

4. The end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform of claim 1, wherein, The learning-based entropy coding module comprises: Context model, using an auto-regressive model to process all sub-band information in turn, output a set of entropy parameters as each sub-band coefficient parameters of the probability distribution; A probability model is used to predict the probability distribution of each coefficient according to the context information, and the probability of occurrence of the coefficient is determined for arithmetic coding; An arithmetic coder is used to compress and encode each subband coefficient based on the probability obtained by the probability model.

5. The end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform according to claim 4, wherein, The context model comprises a long-term context extraction model and a context fusion model; A long-term context extraction model extracts long-term context information from the encoded subband. t ;​ Context fusion models combine information from long-term context information with information from already encoded coefficients in the current subband to generate parameters for the current coefficient probability distribution.

6. The end-to-end image lossless compression method based on two-dimensional non-separable wavelet transform according to claim 5, wherein, In step 3, the probability model used in the arithmetic coder is a Gaussian mixture model, which comprises three Gaussian distributions.

7. The end-to-end image lossless compression method based on two-dimensional nonseparable wavelet transform according to claim 5, wherein In step 4, the coded bitstream is decoded to recover the probability distribution of the subband coefficients and the corresponding subband coefficient values ; The same context model as that used in the coding stage is used to obtain the parameters of the coefficient probability distribution, and the probability distribution of each coefficient is restored; The same probability model as that used in the coding stage is used to restore the probability distribution of each coefficient according to the parameters generated by the context model for arithmetic decoding; An arithmetic decoder is used to decode the bit stream step by step according to the output of the probability model to restore the original values of the subband coefficients.

8. The end-to-end image lossless compression method based on two-dimensional nonseparable wavelet transform of claim 1, wherein In step 5, the multi-level inverse wavelet transform of the decoded subband coefficients corresponding to step 2 is specifically as follows: starting from the highest frequency subband, the inverse transform is performed step by step; The four sub-bands generated by the last wavelet transform , , , Perform an inverse transform to obtain the lower frequency subband of the next higher level. ; merge the four subbands of the previous stage , , , to obtain the low frequency subband of the next stage ; This process is repeated, eventually combining the highest-ranked subband coefficients , , , values to yield the complete pixel information for the image.

9. The end-to-end image lossless compression method based on two-dimensional nonseparable wavelet transform of claim 1, wherein The two-dimensional non-separable inverse wavelet transform is a strict inverse process of the two-dimensional non-separable wavelet transform, which comprises the following steps: Inverse update, inverse prediction and merging; the filter parameters used in the inverse update and the inverse prediction are completely consistent with those of the forward transform; Similarly, rounding operations need to be performed after each filter to ensure that the results of all operations are integers, thereby realizing lossless image reconstruction; the mathematical expressions corresponding to the inverse update and the inverse prediction are as follows: ， ， ， ， where P, U represent the prediction filter and the update filter, [ ] represents a rounding operation, * represents a convolution operation, and the obtained four component values , , , merge to obtain the inverse change result .

10. The end-to-end image lossless compression method based on two-dimensional nonseparable wavelet transform of claim 1, wherein Also includes step 7: build loss function, use optimizer to train and optimize the overall image compression model end to end, get the optimal model.

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