Blind two-dimensional-SIMO channel identification using helix transform and cross relation technique

A device, method, and non-transitory computer readable medium that for two-dimensional blind single-input multiple-output channel identification for image restoration. The method includes receiving, by a receiver having independent channels, a two-dimensional image data matrix then transforming the received two-dimensional image data matrix to a one-dimensional image vector. Channel parameters can then be estimated using the one-dimensional image vector. The method can then construct a restored image using the estimated channel parameters and the two-dimensional image data matrix.

STATEMENT OF PRIOR DISCLOSURE BY AN INVENTOR

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of priority to U.S. Prov. App. No. 63/330,014, entitled “Blind two-dimensional-Simo Channel Identification Using Helix Transform and Cross Relation Technique”, filed on Apr. 12, 2022, and incorporated herein by reference in its entirety.

STATEMENT OF ACKNOWLEDGEMENT

The inventors acknowledge the financial support provided by provided by the Deanship of Scientific Research of King Fahd University of Petroleum and Minerals (KFUPM), Riyadh, Saudi Arabia under Research Grant SB181001.

BACKGROUND

Technical Field

The present disclosure is directed to a system and method for blind two-dimensional single-input multiple-output (SIMO) channel identification.

Description of Related Art

Multichannel blind image restoration is a technique used to recover an original image given several blurred or noisy image observations. Three general approaches are often used to recover the image. One approach involves direct restoration of the original image and is often referred to as equalization or deconvolution. The second approach initially identifies a channel and then restores the desired image. The last approach is similar to the second, but instead jointly identifies the channel and restores the image. Methods of the first approach are of special interest and can be further classified into stochastic methods (e.g., maximum likelihood, minimum mean square error solution, maximum a posteriori, etc.) and deterministic methods. In stochastic methods observed images are assumed to be random and the original image is estimated to be the most probable realization of a random process. However, stochastic methods are highly sensitive to perturbation and modeling errors because they depend on strong statistical hypotheses.

Deterministic methods however do not depend on such hypotheses and instead estimate the original image by minimizing a norm of a certain residuum. Most deterministic techniques have been applied to one-dimensional application, with only some being extended to two-dimensional applications. Minimum noise subspace (MNS) and symmetric minimum noise subspace (SMNS) are two such examples, with the latter being developed from the former. In SMNS, a certain amount of images are used more than others which leads to poor estimation of the original image. Another example includes least squared smoothing (LSS) which is robust against channel order overestimation but can only be applied to two-dimensional applications under restrictive conditions. Cross Relation (CR) is yet another method and is known for its simplicity, efficiency, and low computational costs.

Each of the aforementioned methods suffers from one or more drawbacks hindering their adoption. Accordingly, it is one object of the present disclosure to provide methods and systems for more efficient systems and methods to provide for blind two-dimensional-SIMO channel identification.

SUMMARY

In an exemplary embodiment, a method for two-dimensional blind single-input multiple-output channel identification for image restoration performed by a receiver. The method comprises receiving, by two or more receiver antennas of the receiver having independent channels, a two-dimensional image data matrix. The method then includes transforming the received two-dimensional image data matrix to a one-dimensional image vector. The method can then estimate channel parameters using the one-dimensional image vector. Then method can then construct a restored image using the estimated channel parameters and the two-dimensional image data matrix.

In another exemplary embodiment, a system for two-dimensional blind single-input multiple-output channel identification for image restoration is provided. The system comprises a transmitter comprising a transmitter antenna and a receiver. The receiver comprises two or more transmitter antennas having independent channels and configured to receive a two-dimensional image data matrix and a processing module. The processing module is configured to perform a method including: transforming the received two-dimensional image data matrix to a one-dimensional image vector; estimating channel parameters using the one-dimensional image vector; and constructing a restored image using the estimated channel parameters and the two-dimensional image data matrix.

In yet another exemplary embodiment, a non-transitory computer readable medium having instructions stored therein that, when executed by one or more processors, cause the one or more processors to perform a method of receiving, by two or more receiver antennas of the receiver having independent channels, a two-dimensional image data matrix; transforming the received two-dimensional image data matrix to a one-dimensional image vector; estimating channel parameters using the one-dimensional image vector; and constructing a restored image using the estimated channel parameters and the two-dimensional image data matrix.

DETAILED DESCRIPTION

Aspects of this disclosure are directed to a system, device, and method for blind 2-D single-input multiple-output channel identification. Embodiments use a helix transformation in conjunction with Cross Relation (CR) to recover an original image. Embodiments estimate a two-dimensional channel by using the helix transform to convert a two-dimensional convolution of an image and channel into one-dimensional convolutions. Embodiments then use Cross Relation to estimate unknown channel coefficients to identify the channel. One advantage achieved by embodiments is the ability to restore an image in circumstances when only two channels are available. More generally, embodiments allow for prior one-dimensional methods to be adapted for use in two-dimensional blind system identification. Embodiments can be used to enhance the quality of an image. For example, a camera or mobile phone can receive or image a noisy image and can use embodiments to deblur said image. Another example can include optical device, such as a microscope or telescope can employ methods to enhance the quality of imaging.

FIG.1shows a block diagram of a single-input multiple-output system according to certain embodiments. The single-input multiple-output system can comprise a transmitter100and a receiver110.

The transmitter100can comprise a transmitter antenna102. In some embodiments, the transmitter100may be included on a mobile phone, camera, or optical imaging devices such as a microscope, a telescope, an endoscope, etc. In such examples, the transmitter100can obtain an image using included imaging devices (e.g., through photography, downloading an image, tomography, etc.).

The receiver110can comprise a plurality of receiver antennas, shown as a first receiver antenna112A, a second receiver antenna112B, and an n-th receiver antenna112N, and a processing module114. Although the receiver110is shown to have three receiver antennas inFIG.1, the receiver100can comprise any suitable number of receiver antennas, such as 2, 4, 8, 16, 32, 64, 128, or 256. The processing module114can be used to perform computations and methods described herein. For example, the processing module114can enable the receiver110to perform the method described byFIG.3.

The receiver110can observe a single image F of size mf×nfthrough K independent noisy channels. For example, the receiver110can receive an image obtained and transmitted by the transmitter100. More specifically, the transmitter100can be a first mobile phone that took a photo of a dog and transmitted the photo to the receiver110that can be a second mobile phone. The image F can have a vectorized form of fi=[fi(1, 1), fi(1, 2), . . . ,], y1, . . . , yKcan represent the corresponding K vectorized blurred images, each of size my×ny, h1, . . . , hKcan represent the K vectorized channel's impulse response, each of size mh, nh×1 with hi=[hi(1, 1), hi(1,2), . . . , hi(mh, nh)]T, and w1, . . . , wKdenoting the additive noise term in each of the K channels. Adopting a causal notation, the system model of the i-th channel (e.g., the i-th image) can be written as:
yi(m,n)=xi(m,n)+wi(m,n)  (1)
where xl(m,n)=ΣΣl1=0mh−1Σl2=0nh−1hi(m,n)f(m−l1, n−l2). Based on the SIMO model presented, the blind estimation of the K unknown channels can be performed using the Cross Relation (CR) method.

Cross Relation is described Cross Relation is simple and has low computational complexity. This is achieved by exploiting the commutativity of the convolution according to the following equation.
hi*xj(m,n)=hj*xi(m,n)=hi*hj*f(m,n)  (2)
Equation (2) can be rewritten in a matrix form as follows:

[xi(m,n)T-xj(m,n)T][hjhi]=0,∀(i,j),∀(m,n)(3)
where hi, . . . , hKrepresent the column-wise vectorized versions of the channel matrices and xi(m,n)=[xi(m,n), xi(m−1, n), . . . , xi(m−mh, n−nh)]T. Considering all K(K−1)/2 equation pairs in a matrix leads to the following equation:
Ψ(m,n)h=0,∀(m,n)  (4)
where h=[h1T, . . . , hKT] is the channels parameter vector and Ψ(m,n) is defined by the following equations:

Ψ⁡(m,n)=[Ψ1(m,n)⋮ΨK-1(m,n)](5)
where the matrices are given by

h^=arg⁢minh⁢∑m,n⁢Ψ⁡(m,n)⁢h22=arg⁢minh⁢(hT⁢Qh)(6)
where Q represents the quadratic form of the Cross Relation criterion.

The helix transform, introduced and described further in J. Claerbout, “Multidimensional recursive filters via a helix,”Geophysics, vol. 63, no. 5, pp. 1532-1541, 1998 (incorporated herein by reference), requires two steps to be performed. The first step includes zero padding, while the second step includes the lexicographic ordering of elements. To illustrate, the convolution of two-dimensional signals FM×Nand HK×Lwhere M×N and K×L are the respective sizes of F and H is considered. The zero-padded matrices of the signals are as follows.

F(M+K-1)×(N+L-1)′=[F000]⁢H(M+K-1)×(N+L-1)′=[H000]
Letfandhbe the resulting vectors produced by the lexicographic ordering (either column-wise or row-wise) of F′ and H′ respectively. The last elements offandhcan be truncated to correspond to the last element of F and H respectively. The respective lengths offandhwill be [(N−1)×(M+K−1)+M] and [(L−1)×(M+K−1)+K]. It can be shown, such as in M. Naghizadeh and M. D. Sacchi, “Multidimensional convolution via a one-dimensional convolution algorithm,”The Leading Edge, vol. 28, no. 11, pp. 1336-1337, 2009 (incorporated herein by reference), that the one-dimensional convolution of f and h is equivalent to the two-dimensional convolution of F and H.

Embodiments apply the helix transformation to received images, and make use of the one-dimensional equivalent form of the above equation (1), shown as the following:
yi(n)=hi(n)*f(n)+wi(n)  (7)
whereyi(n),hi(n),f(n), andwi(n) are the one-dimensional counterparts (using the previously described helix transform) of the two-dimensional signals (e.g., yi(m,n), hi(m,n), f(m,n), and wi(m,n)).

An example of the helix transform transforming a two-dimensional channel and signal into one-dimensional equivalents is described. Consider a channel matrix H and a signal matrix F. The convolution of H and F yields a new matrix Y by equation (8) below.
Y=H*F(8)
As an illustration, let H be a 2×2 matrix and F be a 3×3 matrix as follows:

H=[h11h12h21h22]⁢F=[f11f12f13f21f22f23f31f32f33]
The two-dimensional convolution of H and F is equal to:

Y=[h11h12h21h22]*[f11f12f13f21f22f23f31f32f33]
To perform the helix transform, both H and F must be zero-padded and the zero-padded vectors should be vectorized.

In a first step, both matrices can be zero-padded based on their dimensions. In this case, two matrices of dimensions (2+3−1)×(2+3−1) are formed, that is:

In a second step, the zero-padded matricesHandFare vectorized, and the ending zeros are eliminated. This operation is described byh=Vec(H) andf=Vec(F) and are as follows:
h=[h11h210 0h12h22]
f=[f11f21f310f12f22f320f13f23f33]
In a third step, h and f are convolved to give a one-dimensional equivalent of a vectorized vectory=Vec(Y):
y=h*f(10)

In a fourth step, the vectorycan be reshaped into a (2+3−1)×(2+3−1) matrix that is equal to Y. Hence, Y=mat(2+3−1)×(2+3−1)[y]. For a general case of a matrix H of size a×b and a matrix F of size c×d, the length of each channelhis given as Lh=(a+c−1)(b−1)+a, while the length offis given as Lf=(a+c−1)(d−1)+c. It can be noted that the helix transform of the received two-dimensional signal is performed by vectorizing the two-dimensional signal and the estimated channel is accompanied by the padded zeros as shown in the vectorh. The knowledge of the channel size allows for the elimination of the padded zeros.

A fundamental assumption in two-dimensional channel estimation states that channels should be coprime and have no common factors is not violated by performing the helix transform. The helix transform does not destabilize the coprime status of the channel and does not introduce common factors into the channel matrix. The condition of no pairwise common factors for the channel matrices is satisfied with a probability of 1.

An approach similar to the above described can be followed to obtain equation (4) and can be followed to obtain the equivalent cross relationsΨ(n)h=0. MatricesΨ(n) andh=[h1T, . . . ,hKT]T, which are one-dimensional, can be considered as the helix transformed equivalent of the two-dimensional matrices Ψ(m,n) and h respectively.

FIG.2shows an exemplary illustration of a helix transform according to certain embodiments. Due to the inherent zero-padding of the helix transform, the parameter vector andhis of large dimensional size (i.e., its size is equal to K((nh−1)(mH+mf)+mh) and is proportional to the row size of the image) and hence a direct implementation of the one-dimensional CR method is prohibitive. To reduce the cost, the known zero-valued entries ofhare used to skip the unnecessary columns of the vectorΨ(n) and as such reduce the parameter vector size to Knhmh(i.e., the size of h). This is illustrated inFIG.2, where the darker shading (indicated by200inFIG.2) represents the portion that has the channel information, while the empty space (indicated by202inFIG.2) represents the zeros that are introduced intohas a result of the helix transform.

Once all zeros are eliminated (e.g., after obtaining the vector204), the h matrix can be obtained by solving a least squares minimization problem described by equation (11) below.

h^=arg⁢minh~⁢∑n⁢Ψ~(n)⁢h~22=arg⁢minh~⁢h~⁢Q~⁢h~(11)
To avoid the trivial solution {tilde over (h)}=0, the least squares criterion is optimized under the unit-norm constraint (i.e., ∥{tilde over (h)}∥=1). In this case, the desired solution is given by the least eigenvector fof the quadratic form Q.

Embodiments have been described in relation to the Cross Relation method, however, embodiments can be used with other methods similar to Cross Relation (CR) (e.g., Symmetric Cross Relation (SCR), Robust Cross Relation (RCR) or Robust Symmetric Cross Relation (R-SCR) which are described in F. Boudjenouia, K. Abed-Meraim, A. Chetouani, and R. Jennane, “Robust, blind multichannel image identification and restoration using stack decoder,”IET Image Processing, vol. 13, no. 3, pp. 475-482, 2018, (incorporated herein by reference)), or yet other identification methods (e.g., deterministic methods such as Subspace (SS), Minimum Noise Subspace (MNS) or Symmetric Minimum Noise Subspace (SMNS), Least Squared Smoothing (LSS), and Mutually Referenced Equalizers (MRE) that are described in W. Souidene, K. Abed-Meraim, and A. Beghdadi, “Deterministic techniques for multichannel blind image deconvolution,” inProceedings of the Eighth International Symposium on Signal Processing and Its Applications,2005., vol. 1, pp. 439-442, IEEE, 2005, (incorporated herein by reference)). Embodiments can employ helix transformation and Cross Relation to deal with the special case of two channels (i.e., K=2). This is because the diversity condition for the one-dimensional multichannel blind identification is that the channel's transfer functionhl(z) for i=1, . . . , K does not share common zeros (where z represents the z-transform). As mentioned above, this occurs with a probability of 1 if the coefficients of the channels hiare randomly distributed with non-degenerated probability density functions. In the case of two noiseless channels and considering f(m,n)=0, if (m,n) is not within a perfect reconstruction range defined as: [0, mf−1]×[0, nf−1], then then perfect reconstruction is possible.

FIG.3shows a method two-dimensional blind single-input multiple-output channel identification for image restoration according to certain embodiments. The method described byFIG.3can be performed between the transmitter100and the receiver110ofFIG.1. The processing module114can comprise circuitry that enables the receiver110to perform the method described below.

At step300, the receiver110can receive a two-dimensional image data matrix. The receiver110can receive the two-dimensional image data matrix using two or more receiver antennas112that have independent channels. As described by equation (1), the two-dimensional image data matrix can comprise a product of channel parameters and original image data, and additive noise. The two-dimensional image data matrix can be transmitted by the transmitter100using the transmitter antenna102to the receiver110. In some embodiments, the two-dimensional image data matrix can be obtained by the transmitter100using an accompanying imaging device. For example, the transmitter100can be on a first mobile phone that comprises a camera. The camera can be used to the two-dimensional image data matrix, which can then be sent to the receiver110, which can be a second mobile phone.

At step302, the receiver110can transform the received two-dimensional image data matrix to a one-dimensional image vector. The receiver110can perform the transformation using a helix transformation. More specifically, the helix transformation can include zero-padding the two-dimensional image data matrix and vectorizing the zero-padded two-dimensional image data matrix to form the one-dimensional image vector. The receiver110can further process the one-dimensional image vector to remove tail zeroes.

At step304, the receiver110can estimate channel parameters using the one-dimensional image vector. The receiver110can perform the estimation by using the least squares minimization seen in equation (11), which includes the quadratic form of the Cross Relation criterion. The least squares minimization can be optimized to the unit-norm constraint. In some embodiments, the estimation can be performed using an identification method similar to Cross Relation, or a deterministic identification method.

At step306, the receiver110can construct a restored image using the estimated channel parameters and the two-dimensional image data matrix. The restored image is constructed using a deconvolution filter computed from the pseudo-inverse of the estimated channel matrix (each row of the latter matrix represents a deconvolution filter with certain delays. Preferably, a delay(s) equal or close to the channel sizes should be considered.

The performance of the method above is compared to existing methods, such as SS, MNS, SMNS, CR, and LSS methods. In a first stage, the convolution of the image and channel is performed in two-dimensions, and is then helix transformed to rearrange the convolved signal and lexicographic ordering as if the convolution was performed in one-dimensional. The zeros introduced as a result of the helix transform are eliminated while forming the quadratic matrix Q, leading to a reduced computational complexity.

The performance of various methods are compared at different values of signal to noise ratios (SNR) using normalized mean squared error (NMSE). To reduce the scalar ambiguity of blind identification, NMSE is computed as:

NMSE=10⁢log10⁢minα∈ℝ(α⁢hest-h2h2)⁢NMSE=10⁢log10(1-(hestT⁢h)2h2⁢hest2)
where hestrefers to the estimated channel vector.

In the first experiment, the case with K=2 with a channel size varied in the range nh=mh∈[2, 4] is tested. The image is fixed, and the channels are randomly generated for each run of the Monte Carlo scheme for up to 100 runs.

FIG.4shows a first graph700according to certain embodiments.FIG.4depicts the results of the NMSE value vs SNR. It is observed that as the SNR increases, the NMSE improves dramatically.

In addition, the estimated channel coefficients are compared to exact values, the plot is shown byFIG.5.

FIG.5shows a second graph according to certain embodiments. The estimated channel coefficients are generated at a SNR of 30 dB. It is observed that the estimated channel values matches very well to the exact channel value.

As a second experiment, embodiments (denoted as H-CR inFIGS.6and7) are tested against existing methods for K=4 and nh=mh=3.

FIG.6shows a third graph600according to certain embodiments. The third graph600compares the NMSE vs SNR for various identification methods including SMNS, SS, MNS, two-dimensional-CR (two-dimensional Cross Relation), and LSS. Embodiments perform in a similar manner to the existing identification methods.

In a final experiment, embodiments are compared with existing identification methods for K=4 and nh=mh=3.

FIG.7shows a fourth graph700according to certain embodiments. The fourth graph700shows the NMSE vs SNR for various identification methods including SMNS, SS, MNS, two-dimensional-CR, and LSS. Embodiments perform similarly to other two-dimensional identification techniques. Embodiments use the helix transform to implement a computationally heavy two-dimensional algorithm in one-dimensional without loss of performance.

Next, further details of the hardware description of the computing environment according to exemplary embodiments is described with reference toFIG.8. The controller800can be representative of, or included as an additional component of, the transceiver100and/or receiver110ofFIG.1, in which the controller acts as a computing device which can perform processes described above/below. The process data and instructions may be stored in memory802. These processes and instructions may also be stored on a storage medium disk804such as a hard drive (HDD) or portable storage medium or may be stored remotely.

The hardware elements in order to achieve the computing device may be realized by various circuitry elements, known to those skilled in the art. For example, CPU801or CPU803may be a Xenon or Core processor from Intel of America or an Opteron processor from AMD of America, or may be other processor types that would be recognized by one of ordinary skill in the art. Alternatively, the CPU801,803may be implemented on an FPGA, ASIC, PLD or using discrete logic circuits, as one of ordinary skill in the art would recognize. Further, CPU801,803may be implemented as multiple processors cooperatively working in parallel to perform the instructions of the inventive processes described above.

The computing device inFIG.8also includes a network controller806, such as an Intel Ethernet PRO network interface card from Intel Corporation of America, for interfacing with network860. As can be appreciated, the network860can be a public network, such as the Internet, or a private network such as an LAN or WAN network, or any combination thereof and can also include PSTN or ISDN sub-networks. The network860can also be wired, such as an Ethernet network, or can be wireless such as a cellular network including EDGE, 3G, 4G and 5G wireless cellular systems. The wireless network can also be WiFi, Bluetooth, or any other wireless form of communication that is known.

The computing device further includes a display controller808, such as a NVIDIA GeForce GTX or Quadro graphics adaptor from NVIDIA Corporation of America for interfacing with display810, such as a Hewlett Packard HPL2445w LCD monitor. A general purpose I/O interface812interfaces with a keyboard and/or mouse814as well as a touch screen panel816on or separate from display810. General purpose I/O interface also connects to a variety of peripherals818including printers and scanners, such as an OfficeJet or DeskJet from Hewlett Packard.

A sound controller820is also provided in the computing device such as Sound Blaster X-Fi Titanium from Creative, to interface with speakers/microphone822thereby providing sounds and/or music.

The general purpose storage controller824connects the storage medium disk804with communication bus826, which may be an ISA, EISA, VESA, PCI, or similar, for interconnecting all of the components of the computing device. A description of the general features and functionality of the display810, keyboard and/or mouse814, as well as the display controller808, storage controller824, network controller806, sound controller820, and general purpose I/O interface812is omitted herein for brevity as these features are known.

FIG.9shows a schematic diagram of a data processing system, according to certain embodiments, for performing the functions of the exemplary embodiments. The data processing system is an example of a computer in which code or instructions implementing the processes of the illustrative embodiments may be located.

InFIG.9, data processing system900employs a hub architecture including a north bridge and memory controller hub (NB/MCH)925and a south bridge and input/output (I/O) controller hub (SB/ICH)920. The central processing unit (CPU)930is connected to NB/MCH925. The NB/MCH925also connects to the memory945via a memory bus, and connects to the graphics processor950via an accelerated graphics port (AGP). The NB/MCH925also connects to the SB/ICH920via an internal bus (e.g., a unified media interface or a direct media interface). The CPU Processing unit930may contain one or more processors and even may be implemented using one or more heterogeneous processor systems.

For example,FIG.10shows one implementation of CPU930. In one implementation, the instruction register1038retrieves instructions from the fast memory1040. At least part of these instructions are fetched from the instruction register1038by the control logic1036and interpreted according to the instruction set architecture of the CPU930. Part of the instructions can also be directed to the register1032. In one implementation the instructions are decoded according to a hardwired method, and in another implementation the instructions are decoded according a microprogram that translates instructions into sets of CPU configuration signals that are applied sequentially over multiple clock pulses. After fetching and decoding the instructions, the instructions are executed using the arithmetic logic unit (ALU)1034that loads values from the register1032and performs logical and mathematical operations on the loaded values according to the instructions. The results from these operations can be feedback into the register and/or stored in the fast memory1040. According to certain implementations, the instruction set architecture of the CPU930can use a reduced instruction set architecture, a complex instruction set architecture, a vector processor architecture, a very large instruction word architecture. Furthermore, the CPU930can be based on the Von Neuman model or the Harvard model. The CPU930can be a digital signal processor, an FPGA, an ASIC, a PLA, a PLD, or a CPLD. Further, the CPU930can be an x86 processor by Intel or by AMD; an ARM processor, a Power architecture processor by, e.g., IBM; a SPARC architecture processor by Sun Microsystems or by Oracle; or other known CPU architecture.

Referring again toFIG.9, the data processing system900can include that the SB/ICH920is coupled through a system bus to an I/O Bus, a read only memory (ROM)956, universal serial bus (USB) port964, a flash binary input/output system (BIOS)968, and a graphics controller958. PCI/PCIe devices can also be coupled to SB/ICH888through a PCI bus962.

The PCI devices may include, for example, Ethernet adapters, add-in cards, and PC cards for notebook computers. The Hard disk drive960and CD-ROM966can use, for example, an integrated drive electronics (IDE) or serial advanced technology attachment (SATA) interface. In one implementation the I/O bus can include a super I/O (SIO) device.

Further, the hard disk drive (HDD)960and optical drive966can also be coupled to the SB/ICH920through a system bus. In one implementation, a keyboard970, a mouse972, a parallel port978, and a serial port976can be connected to the system bus through the I/O bus. Other peripherals and devices that can be connected to the SB/ICH920using a mass storage controller such as SATA or PATA, an Ethernet port, an ISA bus, a LPC bridge, SMBus, a DMA controller, and an Audio Codec.