Multi-channel OAM hologram information transmission method based on double random phase coding encryption

Abstract

The invention discloses a multi-channel OAM hologram information transmission method based on double random phase coding encryption. The method comprises the following steps: carrying out preprocessing sampling on an original image by using a two-dimensional sampling array; carrying out OAM modulation on an OAM-preserved hologram generated by the sampling picture through a GS iterative algorithm, and then generating an OAM-selective hologram; random phase encryption of a space domain and a frequency domain is carried out on the OAM-selective hologram generated in the OAM encryption stage; carrying out OAM decryption on the hologram encrypted in the two encryption stages of OAM encryption and DRPE encryption; and a DRPE decryption process is carried out on the composite hologram to obtain a correctly decrypted image. According to the method, a double-level encryption scheme in which OAM physical encryption and DRPE mathematical encryption are organically combined is adopted, so that the security and attack resistance of optical image information are remarkably improved while the practicability of the system is kept.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of information transmission, and in particular to a multi-path OAM hologram transmission information method based on double random phase encoding. BACKGROUND

[0002] In recent years, with the rapid development of Internet technology, a large amount of sensitive information (i.e. financial data, medical records, personal privacy and military secrets, etc.) is transmitted and stored through the network, and the information security problem is increasingly prominent. In order to protect the security of information transmission, image encryption technology is widely used, which requires higher security strength and encryption efficiency. However, the current traditional encryption method mainly adopts various digital encoding (i.e. RSA, DES, AES, etc.) and algorithm optimization technology, which has disadvantages in computational complexity, key management and quantum attack resistance, and will not be able to meet the growing demand for encryption strength in the field of information security in the near future. In order to improve information security, two methods are usually adopted, i.e. new encryption mechanism and multi-encryption scheme. Compared with traditional digital encryption technology, optical encryption is considered as a new type of physical security technology, which can effectively improve the encryption strength. Since optical encryption has higher parallel processing capability and faster processing speed, it can be used for high-speed secure transmission of large-scale image data. However, due to the limitations of single optical encryption method (such as traditional DRPE) in key space and vulnerability to various attacks, it cannot achieve higher security level. Especially for critical information that needs to be kept secret for a long time, the security of single optical encryption technology cannot meet the requirements. Therefore, in order to achieve higher strength of security protection, the technical scheme of multi-dimensional composite optical encryption has become an inevitable trend, which will bring new development opportunities and technological innovation to the field of information security.

[0003] In order to overcome the above problems, the traditional optical encryption mainly adopts double random phase encoding (DRPE) technology, which realizes encryption by introducing random phase masks in the spatial domain and frequency domain, respectively, and is widely used in image security processing. However, due to the linear characteristics of DRPE and the development of phase recovery algorithm, the security of single encryption system based on DRPE will face severe challenges in the near future, and cannot be further enhanced by simple key expansion method. Allen et al. discovered a novel physical light property—orbital angular momentum (OAM) in 1992, which carries a vortex phase structure where l is the topological charge value, is the angular momentum. Due to the infinite-dimensional characteristics of OAM mode and the orthogonality between different OAM modes, the optical encryption technology based on OAM can significantly improve the security level and key space of the system.

[0004] To improve the security of optical encryption systems, a large number of studies have been conducted from both theoretical and experimental aspects. In 2004, Gibson et al. described an information transmission concept based on OAM, which used the orthogonal characteristics of OAM beams to realize data encoding, laying the foundation for optical encryption. In 2005, Carnicer et al. reported that the traditional DRPE system was vulnerable to chosen-plaintext attacks, revealing the security flaws of single DRPE methods. In 2015, Liu et al. designed a dual-image encryption system combining phase-shift digital holography and pixel scrambling, using DRPE combined with other technologies to improve security. In 2018, Singh et al. proposed an encryption scheme based on fractional Fourier transform combined with DRPE, and analyzed the effects of noise and attacks on encryption. In 2020, Li et al. demonstrated an optical encryption method based on OAM and chaotic systems, achieving high-security image encryption in the angular spectral domain. However, the above studies mainly focus on single encryption technology or simple combination methods, and the systematic study of deep integration of OAM and DRPE is limited. Therefore, the organic combination of the physical properties of OAM and the algorithmic advantages of DRPE to construct an optical encryption system with dual protection mechanism has great application prospects in the future, which can significantly improve the encryption strength and achieve multi-dimensional security protection, which is also conducive to the development of new generation of information security technology.

[0005] Optical image security protection schemes based on orbital angular momentum (OAM) single-layer encryption and double random phase encoding (DRPE) single-layer encryption have been widely studied and applied in the field of optical information security. The encryption technology based on OAM mainly uses the characteristics of light beams carrying orbital angular momentum to realize information encryption. This technology converts the original image information into a phase-type hologram through the GS algorithm, and then superimposes it with a spiral phase distribution of a specific topological charge value to form an OAM selective hologram. The core advantage of OAM encryption is to use the physical properties of orbital angular momentum of light as an encryption key, so that only OAM beams with opposite topological charge values can successfully decode the image information. On the other hand, DRPE as an encryption scheme based on pure mathematical algorithms, uses two independent random phase masks to modulate the image in the spatial and frequency domains respectively. The security of the DRPE scheme depends on the two random phase masks as the key, and the decryption process requires exactly the same two masks to accurately restore the original information

[0006] Although optical encryption based on orbital angular momentum (OAM) and dual random phase coding (DRPE) encryption have been widely used as independent schemes in the field of optical information security, these technologies still have significant limitations and security vulnerabilities in practical implementation. OAM single-layer encryption relies primarily on a single physical parameter (topological charge value) as the key, making it vulnerable to brute-force attacks within a limited range of values. While DRPE employs two independent random phase masks to provide high mathematical security, its linear nature makes it susceptible to chosen-plaintext and known-plaintext attacks, and it faces challenges in precise phase control during optical implementation, exhibiting high sensitivity to noise and system errors. Furthermore, these single encryption mechanisms generally carry the risk of information leakage; for example, the phase distribution of the hologram in OAM encryption may reveal features of the original image. Improving security often comes at the cost of system practicality, leading to increased system complexity and cost, and decreased decrypted image quality. In terms of key management, these schemes face practical difficulties in secure key distribution, especially in multi-user scenarios, and struggle to balance key sensitivity with system robustness. Insufficient key sensitivity leads to reduced security, while excessive sensitivity negatively impacts system usability. Summary of the Invention

[0007] Purpose of the invention: This invention provides a method for transmitting information in multi-channel OAM holograms using dual random phase coding encryption. This method is a two-level encryption scheme that organically combines OAM physical encryption with DRPE mathematical encryption, which significantly improves the security and anti-attack capability of optical image information while maintaining the practicality of the system.

[0008] Technical solution: The method for transmitting information through multiple OAM holograms with dual random phase coding encryption provided by this invention includes the following steps:

[0009] S1. Preprocess the original image using a two-dimensional sampling array;

[0010] S2. The OAM-preserved hologram generated by the GS iterative algorithm from the sampled image is modulated by OAM to generate an OAM-selective hologram.

[0011] S3. Perform random phase encryption in the spatial and frequency domains on the OAM-selective hologram generated during the OAM encryption stage;

[0012] S4. Decrypt the hologram after it has been encrypted through two encryption stages: OAM encryption and DRPE encryption.

[0013] S5

[0014] S1 includes:

[0015] S1-1. Obtain the original image: Read the target image to be processed into the system. This image can be a grayscale image or a color image. If the input image is a color image, convert it to a grayscale image and normalize the pixel values ​​to [0,1].

[0016] S1-2. Create a sampling grid: Based on the original image size, construct a regular sampling grid to determine the spatial distribution of sampling points;

[0017] S1-3. Perform sparse sampling: Perform spatial domain sparse sampling on the original image according to a preset sampling rate to extract key pixel information. The mathematical expression for the sampling array is:

[0018] Comb(x,y)=∑ m,n δ[(xx m ),(yy n )]

[0019] Where (x,y) is the Cartesian coordinate system, (x m ,y n ) represents the sampling network coordinates at the sampling interval "d", typically x m -x m-1 =d,y n -y n-1 =d, where δ is the two-dimensional Dirac function. The sampling interval "d" of the two-dimensional sampling array is determined by the spatial frequency domain distribution of the input OAM light, and the value of "d" is the same as the diameter of the OAM light.

[0020] S1-4, Brightness Adjustment: Apply an intensity factor to the sampled pixel values ​​to adjust the overall brightness level of the image;

[0021] S1-5. Denoising: The sampled image is denoised using a nonlocal mean filtering method to improve image quality.

[0022] S1-6. Generate preprocessing results: Save the processed sampled image as input for subsequent hologram generation steps.

[0023] Furthermore, S2 includes:

[0024] S2-1. Obtain sampled images from the image preprocessing stage;

[0025] S2-2, Generate the final OAM-preserved hologram from the sampled image using the GS iterative algorithm;

[0026] S2-3, Create an OAM mode with a specific topological load TC value as the encryption key;

[0027] S2-4. Perform a composite multiplication operation on the phase distribution of the OAM-preserved hologram and different OAM modes to achieve effective fusion of image information and OAM mode in the phase domain. The fused phase information is then converted into an OAM-selective hologram.

[0028] S2-5. Apply phase adjustment to the generated OAM-selective hologram and convert the generated selective hologram into a complex field form as input for subsequent DRPE encryption.

[0029] The algorithm steps in S2-2 are as follows:

[0030] The sampled image is transformed into the frequency domain using a Fourier transform to obtain its complex representation in the frequency domain;

[0031] The phase information is iteratively updated by multiplying the complex number in the frequency domain with the preset amplitude information.

[0032] The phase information in the time domain is obtained by performing an inverse Fourier transform on the complex result;

[0033] Repeat the above steps.

[0034] Furthermore, S3 includes:

[0035] S3-1, The OAM-selective hologram is used as input f(x,y) to input the random phase mask 1r1(x,y) to encrypt the OAM-selective hologram in the spatial domain;

[0036] r1(x,y)=exp[i2πφ1(x,y)]

[0037] φ1(x,y)=PRNG[k1,Size(x,y)]

[0038] g(x,y)=f(x,y)·r1(x,y)

[0039] Where φ1 is a uniformly distributed random function in the spatial domain, PRNG is a random generation function, k1 is the seed of the spatial domain controlled PRNG, Size(x,y) is the size of the spatial domain random phase mask, and g(x,y) is the output of the OAM-selective hologram after passing through the spatial domain random phase mask.

[0040] S3-2. After passing through a spatial domain random phase mask, the output g(x,y) is transformed from the spatial domain to the frequency domain by a Fourier lens.

[0041] G(u,v)=F[g(x,y)]

[0042] Where F represents Fourier transform;

[0043] S3-3. The complex field transformed to the Fourier domain is input into a random phase mask 2r2(u,v) for frequency domain encryption:

[0044] r2(u,v)=exp[i2πφ2(u,v)]

[0045] φ2(u,v)=PRNG[k2,Size(u,v)]

[0046] E(u,v)=G(u,v)·r2(u,v)

[0047] Where φ2 is a uniformly distributed random function in the frequency domain, k2 is the seed of the frequency domain controlled PRNG, Size(u,v) is the size of the frequency domain random phase mask, and E(u,v) is the output light field after frequency domain encryption by r2(u,v);

[0048] S3-4. After being encrypted using a random phase mask in the frequency domain, the output E(u,v) is transformed from the frequency domain to the spatial domain by an inverse Fourier lens:

[0049] ψ(x,y)=F -1 [E(u,v)]

[0050] Where ψ(x,y) is the complex field ultimately transformed into the spatial domain, which is the final output of DRPE encryption. -1 This represents the inverse Fourier transform.

[0051] Furthermore, S4 includes:

[0052] S4-1. Receive the complex light field after OAM encryption and DRPE encryption. This light field contains multiple image information and multiple encrypted information, which are superimposed to form a composite encrypted hologram H(x,y):

[0053]

[0054] Where s is the number of composite encrypted holograms;

[0055] S4-2. Based on the selected OAM mode, generate the opposite OAM mode and generate the corresponding OAM phase distribution;

[0056] S4-3. Combine the OAM phase distribution generated in S4-2 with the Gaussian amplitude distribution to form a complete reverse OAM optical field;

[0057] S4-4. Illuminate the encrypted composite hologram with reverse OAM light fields of different topological charges, and name the image after decryption by single OAM of the composite encrypted hologram after illumination by the reverse OAM light field as ψ'(x,y):

[0058]

[0059] S4-5. Perform Fourier transform on the illuminated light field to extract the amplitude information of the reconstructed image.

[0060] Furthermore, S4-5 includes:

[0061] Perform spatial frequency domain transformation; filter and denoise removal to eliminate noise introduced during decryption; amplitude normalization to obtain the final reconstructed image.

[0062] Furthermore, S5 includes:

[0063] S5-1. Receive the hologram after OAM illumination and obtain the hologram signal ψ'(x,y) after the reverse OAM light field illumination processing in S4.

[0064] S5-2. Generate random phase masks for decryption. Based on the pre-stored key seed value, regenerate two random phase masks that are exactly the same as those used in the encryption process: spatial domain random phase mask r1(x,y); frequency domain random phase mask r2(u,v).

[0065] Ensure that the random number generator uses the same seed value as the encryption process to guarantee mask consistency;

[0066] S5-3. Perform an inverse Fourier operation on ψ'(x,y) to transform the complex field in the spatial domain into a complex field ψ(u,v) in the frequency domain:

[0067] ψ'(u,v)=F -1 [ψ'(x,y)];

[0068] S5-4. Decrypting the complex field in the frequency domain, the decrypted complex field is G'(u,v):

[0069] G'(u,v)=ψ'(u,v) / r2(u,v)

[0070] S5-5. The decrypted complex field in the frequency domain is processed by Len1*, and a Fourier transform operation is performed to convert the complex field in the frequency domain into a complex field g'(x,y) in the spatial domain:

[0071] g'(x,y)=F -1 [G'(u,v)];

[0072] S5-6. The complex field g'(x,y) in the spatial domain is decrypted using RPM1 in the spatial domain, and the original image f'(x,y) is finally successfully decrypted:

[0073] f'(x,y)=g'(x,y) / r1(x,y).

[0074] Beneficial Effects: Compared with existing technologies, the significant advancements of this invention are as follows: By establishing a multi-dimensional encryption mechanism combining physical and mathematical layers, the vulnerability of the system caused by reliance on a single encryption layer is eliminated; simultaneously, the key security and diversity of the encryption system are improved by significantly increasing the key space through the introduction of multiple key parameters, effectively resisting brute-force attacks; the information leakage problem in existing technologies is solved by performing secondary encryption on the OAM hologram through DRPE, masking the feature information in the phase distribution of the original image; while improving security, the system's practicality is maintained by optimizing algorithm parameters and optical structure, giving the encryption-decryption process good operability and a certain degree of fault tolerance; the encryption system's ability to resist various optical noises and attacks is enhanced, improving the system's robustness in practical application environments; and the balance between key sensitivity and system robustness is resolved, achieving a differentiated mechanism that is friendly to legitimate users but hostile to attackers. By achieving the above objectives;

[0075] This invention provides a novel image encryption solution that is technically feasible, highly secure, and practically valuable in the field of optical information security. It effectively compensates for the security vulnerabilities of existing single-layer encryption mechanisms and provides more reliable protection for the transmission and storage of sensitive image information. Attached Figure Description

[0076] Figure 1 This is a flowchart of the present invention;

[0077] Figure 2 This is a schematic diagram of image preprocessing;

[0078] Figure 3 This is a schematic diagram of the OAM encryption and reuse phases;

[0079] Figure 4 This is a schematic diagram of the DRPE encryption phase;

[0080] Figure 5 This is a diagram illustrating the OAM decryption phase.

[0081] Figure 6 This is a schematic diagram of the DRPE decryption stage;

[0082] Figure 7 Decryption results of holograms encrypted for the OAM-DRPE system in three cases;

[0083] Figure 8 Decrypting the PSNR of images in different scenarios;

[0084] Figure 9 For the time performance analysis of the OAM-DRPE-based holographic system, (a) is the average time of each processing step, and (b) is the effect of image size on processing time.

[0085] Figure 10 To assess the performance under different noise attacks, (a) shows the reconstruction of the hologram system based on OAM-DRPE after three noise attacks, and (b) shows the reconstruction of the traditional hologram under three noise attacks.

[0086] Figure 11 The key sensitivity of the OAM-DRPE system is shown in (a) for symbol “2”, (b) for symbol “3”, and (c) for symbol “4”.

[0087] Figure 12 This study focuses on the reconstruction of complex images and their performance. Detailed Implementation

[0088] like Figure 1 As shown, this invention proposes a method for transmitting information from multiple OAM holograms using dual random phase coding encryption, which mainly includes the following steps:

[0089] S1. Image preprocessing.

[0090] like Figure 2 The image preprocessing stage diagram shown is composed of... Figure 1 The system consists of the original image and sampled images. The key is the use of a two-dimensional sampling array to sample the images, which is the foundation of the entire system. The steps are as follows:

[0091] S1-1. Obtain the original image: Read the target image to be processed into the system. The image can be a grayscale image or a color image. If the input image is a color image, convert it to a grayscale image and normalize the pixel values ​​to [0,1].

[0092] S1-2. Create a sampling grid: Based on the original image size, construct a regular sampling grid to determine the spatial distribution of sampling points.

[0093] S1-3. Perform sparse sampling: Perform spatial domain sparse sampling on the original image according to a preset sampling rate to extract key pixel information. The mathematical expression for the sampling array is:

[0094] Comb(x,y)=∑ m,n δ[(xx m ),(yy n )]

[0095] Where (x,y) is the Cartesian coordinate system, (x m ,y n ) represents the sampling network coordinates at the sampling interval "d", typically x m -x m-1 =d,y n -y n-1 =d, where δ is the two-dimensional Dirac function. The sampling interval "d" of the two-dimensional sampling array is usually determined by the spatial frequency domain distribution of the input OAM light. Usually, the value of "d" is the same as the diameter of the OAM light. If the vortex light carries multiple OAM modes, the diameter corresponding to the largest OAM mode is selected as "d".

[0096] S1-4, Brightness Adjustment: Apply an intensity factor to the sampled pixel values ​​to adjust the overall brightness level of the image.

[0097] S1-5. Denoising: The sampled image is denoised using a nonlocal mean filtering method to improve image quality.

[0098] S1-6. Generate preprocessing results: Save the processed sampled image as input for subsequent hologram generation steps.

[0099] S2 and OAM encryption.

[0100] The sampled images generated in the image preprocessing stage are processed by the GS iterative algorithm to produce OAM-preserved holograms, which are then modulated by OAM to generate OAM-selective holograms, such as... Figure 3 As shown, it includes the following steps:

[0101] S2-1. Obtain the sampled image from the image preprocessing stage. This image has undergone preprocessing such as format conversion, sampling, and denoising. It is suitable for subsequent generation of phase holograms.

[0102] S2-2. Generate the final OAM-preserved hologram from the sampled images using the GS iterative algorithm.

[0103] The algorithm's steps are as follows: First, the sampled image is transformed into the frequency domain using a Fourier transform to obtain its complex representation. Then, complex multiplication is performed—the complex number in the frequency domain is multiplied by preset amplitude information, iteratively updating the phase information. Next, the complex result is subjected to an inverse Fourier transform to obtain the phase information in the time domain. Finally, the above steps are repeated.

[0104] This hologram contains complete phase information of the image and preserves the spatial frequency characteristics of the image.

[0105] S2-3. Create an OAM mode with a specific topological charge (TC) value as the encryption key. Different values ​​can be selected for the OAM mode; here, we chose TC=3, TC=5, and TC=10 to form vortex phase structures with different rotational symmetries.

[0106] S2-4. Perform a composite multiplication operation on the OAM-preserved hologram and the phase distributions of different OAM modes to effectively fuse the image information and OAM modes in the phase domain. The fused phase information is then converted into an OAM-selective hologram. The generated selective hologram exhibits selective response characteristics to a specific OAM mode.

[0107] S2-5. Apply phase adjustment to the generated OAM-selective hologram to improve its spatial frequency characteristics and selective response. Then, convert the generated selective hologram into a complex field form as input for subsequent DRPE encryption.

[0108] S3 and DRPE encryption.

[0109] like Figure 4 The main function of the DRPE encryption stage shown is to perform random phase encryption in the spatial and frequency domains on the OAM-selective hologram generated in the OAM encryption stage, thereby further improving the security of the hologram. The steps are as follows:

[0110] S3-1, the OAM-selective hologram is input as f(x,y) to the random phase mask 1 (RPM1)r1(x,y). RPM1 encrypts the OAM-selective hologram in the spatial domain, adding an extra layer of security to the selective hologram in the spatial domain. RPM1 is generated by a seed-controlled random generation function.

[0111] The formulas mentioned above are:

[0112] r1(x,y)=exp[i2πφ1(x,y)]

[0113] φ1(x,y)=PRNG[k1,Size(x,y)]

[0114] g(x,y)=f(x,y)·r1(x,y)

[0115] Where φ1 is a uniformly distributed random function in the spatial domain, PRNG is a random generation function, k1 is the seed of the spatial domain controlled PRNG, Size(x,y) is the size of the spatial domain random phase mask, and g(x,y) is the output of the OAM-selective hologram after passing through the spatial domain random phase mask.

[0116] S3-2. After passing through the spatial domain random phase mask (RPM1), the output g(x,y) is further passed through lens 1 (Lens1), transforming the complex field from the spatial domain to the frequency domain. Lens1 acts as a Fourier lens. The formula is:

[0117] G(u,v)=F[g(x,y)]

[0118] Where F represents Fourier transform.

[0119] S3-3. The complex field, transformed to the Fourier domain by Lens1, is input into the random phase mask 2 (RPM2)r2(u,v) for frequency domain encryption, adding an extra layer of security in the frequency domain. Like RPM1, RPM2 is generated using a seed-controlled random generation function. The formulas involved in the above process are:

[0120] r2(u,v)=exp[i2πφ2(u,v)]

[0121] φ2(u,v)=PRNG[k2,Size(u,v)]

[0122] E(u,v)=G(u,v)·r2(u,v)

[0123] Where φ2 is a uniformly distributed random function in the frequency domain, k2 is the seed of the frequency domain controlled PRNG, Size(u,v) is the size of the frequency domain random phase mask, and E(u,v) is the output light field after frequency domain encryption by r2(u,v);

[0124] S3-4. After being encrypted using a random phase mask in the frequency domain, the output E(u,v) is transformed from the frequency domain to the spatial domain by lens 2 (Len2). Len2 acts as an inverse Fourier lens, and the relevant formulas are as follows:

[0125] ψ(x,y)=F -1 [E(u,v)]

[0126] Where ψ(x,y) is the complex field ultimately transformed into the spatial domain, which is the final output of DRPE encryption. -1 This represents the inverse Fourier transform.

[0127] S4, OAM decryption.

[0128] like Figure 5 The main function of the OAM decryption stage shown is to perform OAM decryption on the hologram after it has been encrypted through the OAM and DRPE encryption stages. The steps are as follows:

[0129] S4-1. Receive the complex light field after OAM encryption and DRPE encryption. The light field contains multiple image information and multiple encrypted information, namely multiple ψ(x,y) as mentioned above.

[0130] The formulas involved in superimposing multiple ψ(x,y) values ​​into a composite encrypted hologram H(x,y) are as follows:

[0131]

[0132] Where s is the number of composite encrypted holograms. In this invention, s is chosen to be 3, but theoretically s is infinite.

[0133] S4-2. Based on the selected OAM mode above, generate the opposite OAM mode respectively: For TC=3, 5, 10 in the OAM encryption process, we create TC=-3, -5, -10 respectively, and generate the corresponding OAM phase distribution respectively.

[0134] S4-3. Combine the OAM phase distribution generated in S4-2 with the Gaussian amplitude distribution to form a complete reverse OAM optical field.

[0135] S4-4. Irradiate the encrypted composite hologram with reverse OAM light fields of different topological charges:

[0136] (a) When using the reverse OAM light field with the correct topological charge, selective decryption and reconstruction of the corresponding original image are achieved;

[0137] (b) When using the reverse OAM light field with incorrect topological charge, it cannot be decrypted correctly, and the output result is still a noise distribution.

[0138] The image obtained by single OAM decryption of the composite encrypted hologram after illumination by a reverse OAM light field is named ψ'(x,y), and the relevant formula is:

[0139]

[0140] S4-5. Perform a Fourier transform on the illuminated light field to extract the amplitude information of the reconstructed image:

[0141] (a) Perform spatial frequency domain transformation;

[0142] (b) Filtering and noise reduction processing to eliminate noise introduced during the decryption process;

[0143] (c) Amplitude normalization to obtain the final reconstructed image.

[0144] S5 and DRPE decryption.

[0145] like Figure 6 The DRPE decryption stage diagram shown primarily illustrates the DRPE decryption process performed on the composite hologram after OAM decryption. This is the final stage of decryption; correct execution will yield a correctly decrypted image. The steps are as follows:

[0146] S5-1. Receive the hologram after OAM illumination and obtain the hologram signal ψ'(x,y) after the reverse OAM light field illumination processing in S4. This signal contains the image information to be decrypted.

[0147] S5-2. Generate random phase masks for decryption. Based on the pre-stored key seed value, regenerate two random phase masks that are exactly the same as those used in the encryption process:

[0148] Generate a spatial domain random phase mask r1(x,y);

[0149] Generate a frequency-domain random phase mask r2(u,v);

[0150] Ensure that the random number generator uses the same seed value as the encryption process to guarantee mask consistency;

[0151] S5-3. Perform an inverse Fourier operation on ψ'(x,y) to transform the complex field in the spatial domain into a complex field ψ(u,v) in the frequency domain, using Len2*:

[0152] ψ'(u,v)=F -1 [ψ'(x,y)]

[0153] S5-4. The complex field in the frequency domain is first decrypted in the frequency domain because encryption proceeds from the spatial domain to the frequency domain, and decryption must follow the same order. Decryption in the frequency domain is the same as encryption in the frequency domain; the complex field is then processed using RPM2. The complex field after RPM2 no longer has the encryption properties of the frequency domain. The decrypted complex field in the frequency domain is G'(u,v):

[0154] G'(u,v)=ψ'(u,v) / r2(u,v)

[0155] S5-5. The decrypted complex field in the frequency domain is processed by Len1*, and a Fourier transform operation is performed to convert the complex field in the frequency domain into a complex field g'(x,y) in the spatial domain:

[0156] g'(x,y)=F -1 [G'(u,v)]

[0157] S5-6. The complex field g'(x,y) in the spatial domain is decrypted using RPM1 in the spatial domain, and the original image f'(x,y) is finally successfully decrypted:

[0158] f'(x,y)=g'(x,y) / r1(x,y).

[0159] To comprehensively verify the effectiveness, security, and applicability of this method, a complete numerical simulation platform was built in the MATLAB R2022a environment. This platform can simulate the physical process of the entire optical encryption system, including key steps such as image preprocessing, OAM encryption, DRPE encryption, and the corresponding decryption process. The verification system is based on... Figure 1The illustrated dual-random phase-coded encrypted multiplexed hologram method organically combines an image preprocessing module, an OAM encryption module, a DRPE encryption module, and an OAM decryption module. The system first samples and preprocesses the original image to generate a phase hologram. Then, it generates an OAM-selective hologram using the OAM phase distribution. Next, it uses two random phase masks (mask 1 and mask 2) to perform DRPE encryption in the spatial and frequency domains, respectively. Finally, it achieves complete decryption through a reverse OAM optical field. To ensure comprehensive verification, this invention selects three grayscale test images with different characteristics—symbols "2," "3," and "4"—and assigns them different topological charge values ​​(l = 3, 5, 10), covering low, medium, and high ranges, enabling comprehensive testing of the system's performance under different OAM modes. For the DRPE key, this invention employs a random seed-based generation mechanism, assigning an independent key pair (seed1, seed2) to each test image, enhancing the system's flexibility and security. These carefully designed verification systems and parameters provide a solid foundation for verifying the method of this invention. The verification technology route of this invention mainly includes three aspects: basic function verification, security verification, and performance indicator verification. Basic function verification first comprehensively verifies the encryption process, including five key steps: original image sampling and processing, phase hologram generation, OAM phase modulation, DRPE encryption implementation, and composite hologram generation. Intermediate results are recorded at each step to ensure the transparency and verifiability of the encryption process. Decryption process verification includes three main steps: reverse OAM light field generation, DRPE decryption process, and image reconstruction and optimization. A complete decryption process is performed on each test image to verify the effectiveness and accuracy of the decryption. Regarding security verification, this invention designs comprehensive key sensitivity tests, including topological charge sensitivity tests, DRPE key sensitivity tests, and double error tests, to comprehensively evaluate the encryption system's sensitivity to key changes. Anti-attack tests evaluate the encryption system's performance under various attack conditions through noise attack tests, OAM-only decoding tests, and partial information leakage tests. Performance verification primarily involved quantitative analysis of the decrypted image quality through two dimensions: Peak Signal-to-Noise Ratio (PSNR) and visual quality assessment. The computational efficiency of the numerical simulation was also evaluated, including processing time statistics and parameter optimization assessment. This rigorous and comprehensive verification approach ensured a systematic evaluation of the performance of the method presented in this invention across all aspects.

[0160] Verification experiments show that the OAM-DRPE dual encryption method of this invention performs excellently in terms of functional effectiveness, security, and performance advantages. In terms of basic functional verification, the encrypted composite hologram exhibits a completely random phase distribution, making it impossible to visually identify any features of the original image. However, using the correct topological charge value and DRPE key, all three test images can be successfully decrypted, with PSNR values ​​reaching 31.8dB, 31.8dB, and 33.2dB, respectively. Security verification results show that this method has extremely high key sensitivity; a change of only 1 in the DRPE key seed value results in a decryption error rate exceeding 58%. When the topological charge value is incorrect, even with a correct DRPE key, the decrypted image remains unrecognizable. Furthermore, when both the topological charge and DRPE key are incorrect, the PSNR value drops to approximately 9.6dB, rendering the image completely unrecognizable. Anti-attack tests show that the method has good resistance to various noise attacks, maintaining a certain decryption quality even under high-intensity noise (0.2). Furthermore, using only the reverse OAM optical field without DRPE decryption, or with partial leakage of key information, fails to yield a meaningful image, demonstrating the necessity and effectiveness of the dual protection mechanism. In summary, through MATLAB numerical simulation verification, the OAM-DRPE dual encryption method of this invention innovatively constructs a dual protection mechanism by combining orbital angular momentum optical properties and double random phase coding technology, providing a new technical route for the field of optical image encryption. Comprehensive verification results demonstrate that this method has practical application value and broad development prospects. Future research can be conducted in areas such as verification in practical optical systems, broader application verification, and verification against quantum computing attacks, providing more comprehensive support for the improvement and application of this invention.

[0161] Based on the proposed experimental platform, three grayscale test images with different characteristics, namely symbols "2", "3", and "4", were selected for verification. Figure 1 The encryption process shown first involves using a sampling image module to perform two-dimensional sampling processing on the original image to generate a phase hologram. Then, in the OAM encryption stage, OAM phase distributions with topological charges of 3, 5, and 10 are superimposed onto the phase hologram to form an OAM-selective hologram. These three topological charge values ​​cover low, medium, and high ranges, enabling comprehensive testing of the system's performance under different OAM modes. Subsequently, the DRPE encryption stage is entered, using two independent random phase masks to perform double encryption on the OAM-selective hologram: first, encryption is performed in the spatial domain using random phase mask 1, and then a second encryption is performed in the frequency domain using random phase mask 2, ultimately generating a DRPE encrypted hologram. The entire encryption process strictly follows... Figure 1The system architecture shown ensures the effective combination of OAM modulation and DRPE coding, laying the foundation for subsequent security and effectiveness verification.

[0162] To verify the performance of the OAM-DRPE-based holographic system, such as Figure 7 As shown, the three numbers "2", "3", and "4" are clearly distinguishable in the image, indicating that the system can perfectly reconstruct the original information under ideal conditions. The second row represents the decryption attempt when only the correct inverse topological charge value is used and the DRPE key is missing. The result is that only a highlight appears in the center of the image, completely losing the structural information of the original image, verifying the necessity of the DRPE encryption layer. The third row shows the decryption effect when only the correct DRPE key is used but the wrong topological charge value is applied. This image presents a completely random noise distribution with no recognizable structural features, demonstrating the importance of OAM topological charge selection. The last row tests the effect of reversing the spatial and frequency domain order during DRPE decryption, also yielding an unrecognizable noise image, indicating that the decryption process must strictly follow the reverse order of encryption.

[0163] Comparative analysis of these four scenarios powerfully demonstrates the multiple security mechanisms of the OAM-DRPE composite encryption system: decryption requires both precise matching of topological charges and correct application of the DRPE key. Both are indispensable, and the order of operations is also crucial. This multi-layered, multi-parameter security dependency significantly enhances the system's ability to resist unauthorized access, providing a highly secure technical solution for the field of optical image encryption.

[0164] To verify the quality of the decrypted image, Peak Signal-to-Noise Ratio (PSNR) is used as an evaluation metric. The formula for calculating PSNR is:

[0165]

[0166] Where MAX is the maximum pixel value and MSE is the mean squared error. In the calculation of mean squared error, M and N are the horizontal and vertical dimensions of the holographic image, respectively, and f(i,j) and f'(i,j) are the pixel values ​​at points in the original and reconstructed images, respectively. The unit of PSNR is dB. Generally, the higher the calculated PSNR value, the better the quality of the reconstructed image.

[0167] Figure 8 The PSNR values ​​of the decrypted image are shown in five cases: correct key and correct opposite topology load, DRPE decryption order reversed, incorrect key and incorrect opposite topology load, incorrect key and correct opposite topology load, and correct key and incorrect opposite topology load. Figure 8A clear bar chart illustrates the PSNR of decryption quality under different key conditions. Specifically, the decryption quality performance is compared between completely incorrect keys and keys with only a single incorrect parameter. The experimental data clearly show that when only a single parameter is incorrect, the system's decryption quality is significantly improved compared to the case of completely incorrect keys, but its decryption effect is still significantly lower than that using completely correct keys. Notably, in the proposed OAM-DRPE-based holographic system, when using completely correct key combinations, the PSNR of the decrypted image consistently reaches over 30dB, and the PSNR value of the decrypted image with the DRPE decryption order reversed is very low. This fully demonstrates the system's excellent performance.

[0168] Further analysis reveals that, in terms of system security, the contributions of key parameters and topology payload parameters to overall security performance are roughly equal, exhibiting a balanced effect on cryptographic strength. The same erroneous key and topology payload were selected for reconstruction in both single-error scenarios, resulting in identical PSNR values ​​under single-error conditions. However, the relatively equal contribution of the key and topology payload to security is a pleasant surprise. This finding not only verifies the correctness of the encryption system designed in this application but also fully demonstrates its effectiveness and reliability in practical applications. Through this multi-parameter collaborative encryption mechanism, the system can provide more robust security guarantees and effectively resist various potential key attacks.

[0169] like Figure 9 As shown, this application verifies the real-time performance of the OAM-DRPE-based holographic system. Specifically, Figure 9 A comprehensive analysis and evaluation of the real-time performance of the OAM-DRPE-based holographic system was conducted. Figure 9 (a) presents the average time cost distribution for each processing stage of the system, while Figure 9 (b) reveals the quantitative impact of image resolution on processing efficiency. From Figure 9 As observed in data (a), the preprocessing stage has a relatively stable time consumption of approximately 0.5 seconds, demonstrating good algorithm robustness. It is noteworthy that the hologram generation stage accounts for the majority of the system's processing time, primarily due to the Gerchberg-Saxton (GS) iterative algorithm employed in this application. While this algorithm provides high-quality holograms, its computational complexity increases linearly with the number of iterations, leading to a corresponding increase in processing latency. Crucially, the system's core functional modules—OAM modulation, DRPE encryption, and decryption—all exhibit extremely low computational latency, fully demonstrating the computational efficiency and real-time processing capabilities of the proposed OAM-DRPE security scheme in core encryption and decryption operations. Figure 9(b) quantitatively demonstrates the functional relationship between image resolution and encryption / decryption time consumption. Experimental results show that as image resolution increases, the encryption time consumption exhibits a significant non-linear growth trend, while the decryption time consumption increases only slightly, almost negligible. It is important to note that the encryption time measured here is a comprehensive indicator encompassing multiple processing stages, covering not only… Figure 9 The two core steps shown in (a)—OAM modulation and DRPE encryption—also incorporate the time overhead of preprocessing and hologram generation. Figure 9 The experimental data in (a) and (b) lead to the conclusion that even in high-resolution image processing scenarios, the proposed OAM-DRPE-based holographic system maintains satisfactory computational efficiency, demonstrating that it has good scalability and real-time processing capabilities in practical applications, and provides an efficient technical solution for secure image transmission.

[0170] Furthermore, this application argues that robustness is indispensable for any hologram encryption scheme. To comprehensively evaluate the noise resistance and robustness of this encryption system, this application systematically selected three representative noise types to conduct quantitative attack tests on the encrypted holograms: Gaussian noise, salt-and-pepper noise, and speckle noise. The intensity of these three noise types was simultaneously set to 0.2 to ensure the consistency and comparability of the test conditions.

[0171] like Figure 10 As shown, Figure 10 (a) revealed that the decrypted images did not degrade significantly after being exposed to these noise attacks, and all reconstructed images retained well-preserved identifiable details and structural integrity. Figure 10 (b) shows the reconstruction results of the ordinary Gaussian hologram after a noise attack. It can be seen that the reconstruction results of the Gaussian hologram after the noise attack are very poor. These experimental results ultimately demonstrate that the proposed OAM-DRPE-based holographic system exhibits significant noise resistance performance, effectively resisting typical noise. Therefore, this function provides a reliable security guarantee for information protection in practical application environments.

[0172] Based on a comprehensive evaluation of the system's real-time performance and noise immunity, this application further analyzes the security performance of the OAM-DRPE system in depth, focusing particularly on key sensitivity, a key quantitative indicator of encryption system security. Key sensitivity is one of the core parameters in modern cryptography for evaluating the strength of encryption algorithms; it quantitatively describes the response function characteristics of an encryption system to small perturbations in the key.

[0173] From an information theory perspective, an ideal encryption system should exhibit a pronounced "avalanche effect"—that is, a small change in the key should lead to a drastic change in the decryption result, minimizing the statistical correlation between the ciphertext and plaintext. This characteristic is crucial for resisting advanced attacks based on key neighborhood search, while also significantly increasing the computational complexity of brute-force attacks. As Shannon's information entropy theory suggests, highly sensitive systems can maximize the uncertainty of the decryption result, thereby providing near-ideal information security.

[0174] Figure 11 The quantitative test results of the key sensitivity of the OAM-DRPE dual encryption system proposed in this study are presented. This test uses the decryption error rate as the evaluation index. By systematically introducing key perturbations and measuring the corresponding decryption error rate, a complete response curve of the system's key sensitivity is constructed. In the figure, the horizontal axis represents the amount of key difference, and the vertical axis represents the corresponding decryption error rate. Figure 11 (a), (b), and (c) represent the key sensitivity analysis results for test images with three different topological loads (l = 3, 5, 10), respectively. This diverse test sample ensures the universality and reliability of the experimental results. Experimental data analysis shows that even with only minor differences in the key, the decryption error rate exhibits significant nonlinear fluctuations, remaining at an extremely high level between 1.996 and 2.003 in most cases. Theoretically, when two independent random complex field signals are completely uncorrelated, the decryption error rate is close to 2.0, indicating that the result obtained by decrypting with an incorrect key has almost no exploitable correlation with the original image, yielding only an output similar to random noise. Particularly noteworthy is the absence of any obvious troughs in the error curve, proving that there are no vulnerabilities or "backdoor" structures in the key space that could be exploited by attackers.

[0175] This near-theoretical key sensitivity, combined with the real-time processing capabilities and noise immunity discussed earlier, constitutes the three major technical advantages of this system. From a systems engineering perspective, these three characteristics complement each other: real-time performance ensures the system's high usability in practical application environments; noise immunity enhances the system's robustness under non-ideal channel conditions; and key sensitivity provides solid security for the entire encryption framework. The organic combination of these three characteristics allows the OAM-DRPE system proposed in this study to achieve an excellent balance between security and practicality.

[0176] From a cryptographic perspective, the high key sensitivity of the OAM-DRPE system stems from the cascading amplification effect of its dual encryption mechanisms: OAM modulation introduces a complex phase structure related to topological charges in the spatial domain, while DRPE applies independent random phase masks in both the spatial and frequency domains. This multi-domain, multi-layered superimposed encryption strategy ensures that even a small change in any key parameter will result in a significant deviation during the entire decryption process.

[0177] In the preceding work, this application verified the basic functionality of the invention using simple images and analyzed in detail the system's real-time performance, noise immunity, and key sensitivity. Basic experiments demonstrated the feasibility and robustness of the proposed method. However, to more comprehensively evaluate the performance of this encryption system in practical applications, especially when processing complex images, we need to further expand the experimental scope. Compared to simple images, complex images have richer details, more complex texture structures, and wider spectral distributions, which places higher demands on the information preservation capability and reconstruction quality of the encryption system. Therefore, this application will use animated character images with rich details as test objects to further verify the invention's ability to process complex image content.

[0178] To verify the ability of this invention to process complex images, this application selected three animated character images as test objects: image nz1, image nz2 and image nz3. Figure 12 The encryption and decryption results and their values ​​for three complex images are presented. The experimental results show that all test images can be successfully decrypted and reconstructed to reproduce identifiable image content, with good preservation of the main contours and features. In terms of performance, image nz2 achieved the highest value of 31.66 dB, image nz1 30.8 dB, and image nz3 27.09 dB. The experimental results indicate that image complexity has a certain impact on reconstruction quality; images with clear contours and relatively simple structures achieve higher values, while images containing more detailed textures show a slight decrease in reconstruction quality. This phenomenon is consistent with the general rules of holographic reconstruction. The complex image experiments further confirm the robustness and practicality of the present invention, verifying that the system is not only suitable for simple geometric figures but can also effectively process real-world image content with rich details and complex structures.

Claims

1. A method for transmitting information from a multi-channel OAM hologram using dual random phase coding encryption, characterized in that, Includes the following steps: S1. Preprocess the original image using a two-dimensional sampling array; S2. The OAM-preserved hologram generated by the GS iterative algorithm from the sampled image is modulated by OAM to generate an OAM-selective hologram. S3. Perform random phase encryption in the spatial and frequency domains on the OAM-selective hologram generated during the OAM encryption stage; S4. Decrypt the hologram after it has been encrypted through two encryption stages: OAM encryption and DRPE encryption. S5. Then perform the DRPE decryption process on the composite hologram to obtain the correctly decrypted image.

2. The method for transmitting information through a multi-channel OAM hologram with dual random phase coding encryption according to claim 1, characterized in that, S1 includes: S1-1. Obtain the original image: Read the target image to be processed into the system. This image can be a grayscale image or a color image. If the input image is a color image, convert it to a grayscale image and normalize the pixel values ​​to [0,1]. S1-2. Create a sampling grid: Based on the original image size, construct a regular sampling grid to determine the spatial distribution of sampling points; S1-3. Perform sparse sampling: Perform spatial domain sparse sampling on the original image according to a preset sampling rate to extract key pixel information. The mathematical expression for the sampling array is: Comb(x,y)=∑ m,n δ[(xx m ),(yy n )] Where (x,y) is the Cartesian coordinate system, (x m ,y n ) represents the sampling network coordinates at the sampling interval "d", typically x m -x m-1 =d,y n -y n-1 =d, where δ is the two-dimensional Dirac function. The sampling interval "d" of the two-dimensional sampling array is determined by the spatial frequency domain distribution of the input OAM light, and the value of "d" is the same as the diameter of the OAM light. S1-4, Brightness Adjustment: Apply an intensity factor to the sampled pixel values ​​to adjust the overall brightness level of the image; S1-5. Denoising: The sampled image is denoised using a nonlocal mean filtering method to improve image quality. S1-6. Generate preprocessing results: Save the processed sampled image as input for subsequent hologram generation steps.

3. The method for transmitting information through a multi-channel OAM hologram with dual random phase coding encryption according to claim 1, characterized in that, S2 include: S2-1. Obtain sampled images from the image preprocessing stage; S2-2, Generate the final OAM-preserved hologram from the sampled image using the GS iterative algorithm; S2-3, Create an OAM mode with a specific topological load TC value as the encryption key; S2-4. Perform a composite multiplication operation on the phase distribution of the OAM-preserved hologram and different OAM modes to achieve effective fusion of image information and OAM mode in the phase domain. The fused phase information is then converted into an OAM-selective hologram. S2-5. Apply phase adjustment to the generated OAM-selective hologram and convert the generated selective hologram into a complex field form as input for subsequent DRPE encryption.

4. The method for transmitting information through a multi-channel OAM hologram with dual random phase coding encryption according to claim 3, characterized in that, The algorithm steps in S2-2 are as follows: The sampled image is transformed into the frequency domain using a Fourier transform to obtain its complex representation in the frequency domain; The phase information is iteratively updated by multiplying the complex number in the frequency domain with the preset amplitude information. The phase information in the time domain is obtained by performing an inverse Fourier transform on the complex result; Repeat the above steps.

5. The method for transmitting information through a multi-channel OAM hologram with dual random phase coding encryption according to claim 1, characterized in that, S3 include: S3-1, The OAM-selective hologram is used as input f(x,y) to input the random phase mask 1r1(x,y) to encrypt the OAM-selective hologram in the spatial domain; r1(x,y)=exp[i2πφ1(x,y)] φ1(x,y)=PRNG[k1,Size(x,y)] g(x,y)=f(x,y)·r1(x,y) Where φ1 is a uniformly distributed random function in the spatial domain, PRNG is a random generation function, k1 is the seed of the spatial domain controlled PRNG, Size(x,y) is the size of the spatial domain random phase mask, and g(x,y) is the output of the OAM-selective hologram after passing through the spatial domain random phase mask. S3-2. After passing through a spatial domain random phase mask, the output g(x,y) is transformed from the spatial domain to the frequency domain by a Fourier lens. G(u,v)=F[g(x,y)] Where F represents Fourier transform; S3-3. The complex field transformed to the Fourier domain is input into a random phase mask 2r2(u,v) for frequency domain encryption: r2(u,v)=exp[i2πφ2(u,v)] φ2(u,v)=PRNG[k2,Size(u,v)] E(u,v)=G(u,v)·r2(u,v) Where φ2 is a uniformly distributed random function in the frequency domain, k2 is the seed of the frequency domain controlled PRNG, Size(u,v) is the size of the frequency domain random phase mask, and E(u,v) is the output light field after frequency domain encryption by r2(u,v); S3-4. After being encrypted using a random phase mask in the frequency domain, the output E(u,v) is transformed from the frequency domain to the spatial domain by an inverse Fourier lens: ψ(x,y)=F -1 [E(u,v)] Where ψ(x,y) is the complex field ultimately transformed into the spatial domain, which is the final output of DRPE encryption. -1 This represents the inverse Fourier transform.

6. The method for transmitting information through a multi-channel OAM hologram with dual random phase coding encryption according to claim 1, characterized in that, S4 includes: S4-1. Receive the complex light field after OAM encryption and DRPE encryption. This light field contains multiple image information and multiple encrypted information, which are superimposed to form a composite encrypted hologram H(x,y): Where s is the number of composite encrypted holograms; S4-2. Based on the selected OAM mode, generate the opposite OAM mode and generate the corresponding OAM phase distribution; S4-3. Combine the OAM phase distribution generated in S4-2 with the Gaussian amplitude distribution to form a complete reverse OAM optical field; S4-4. Illuminate the encrypted composite hologram with reverse OAM light fields of different topological charges, and name the image after decryption by single OAM of the composite encrypted hologram after illumination by the reverse OAM light field as ψ'(x,y): S4-5. Perform Fourier transform on the illuminated light field to extract the amplitude information of the reconstructed image.

7. The method for transmitting information via a multi-channel OAM hologram with dual random phase coding encryption according to claim 6, characterized in that, S4-5 includes: Perform spatial frequency domain transformation; filter and denoise removal to eliminate noise introduced during decryption; amplitude normalization to obtain the final reconstructed image.

8. The method for transmitting information through a multi-channel OAM hologram with dual random phase coding encryption according to claim 6, characterized in that, S5 include: S5-1. Receive the hologram after OAM illumination and obtain the hologram signal ψ'(x,y) after the reverse OAM light field illumination processing in S4. S5-2. Generate random phase masks for decryption. Based on the pre-stored key seed value, regenerate two random phase masks that are exactly the same as those used in the encryption process: spatial domain random phase mask r1(x,y); frequency domain random phase mask r2(u,v). Ensure that the random number generator uses the same seed value as the encryption process to guarantee mask consistency; S5-3. Perform an inverse Fourier operation on ψ'(x,y) to transform the complex field in the spatial domain into a complex field ψ(u,v) in the frequency domain: ψ'(u,v)=F -1 [ψ'(x,y)]; S5-4. Decrypting the complex field in the frequency domain, the decrypted complex field is G'(u,v): G'(u,v)=ψ'(u,v) / r2(u,v) S5-5. The decrypted complex field in the frequency domain is processed by Len1*, and a Fourier transform operation is performed to convert the complex field in the frequency domain into a complex field g'(x,y) in the spatial domain: g'(x,y)=F -1 [G'(u,v)]; S5-6. The complex field g'(x,y) in the spatial domain is decrypted using RPM1 in the spatial domain, and the original image f'(x,y) is finally successfully decrypted: f'(x,y)=g'(x,y) / r1(x,y).

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