Rosette Trajectories for Fast MRI Based on an Adaptive Reconstruction Method

Non-Cartesian MRI k-space trajectories provide faster and more motion-robust data acquisitions than those of Cartesian trajectories. In this paper, we focus on the rosette trajectory and generalize the weighted rosette trajectories for fast undersampled k-space data acquisition. However, single-slice imaging using the rosette trajectory will be affected by the off-resonance effect. To reduce the artifacts from off-resonance slices, this paper introduced an adaptive iterative reconstruction method derived from the TV and nuclear norm regularization terms for raw MRI data reconstruction. This reconstruction problem was separated into two subproblems and solved with an adaptive regularization parameter. The results show that the weighted rosette trajectory that offers short acquisition times and good off-resonance behaviors with little blurring can be used for reconstruction with adaptive regularization parameters and achieve a superior performance.


I. INTRODUCTION
Magnetic resonance imaging (MRI) is widely used in medical diagnosis; however, the procedure of obtaining magnetic resonance imaging requires patients to hold their breath for up to several seconds, creating an uncomfortable situation for many patients and introducing artifacts and blurring. Reducing the time of data acquisition and improving the image quality has been a core objective of MRI development.
In recent years, the theory of compressed sensing imaging has provided a new way to shorten the time of data acquisition [1], [2]. According to the theory of CS-MRI [3], [4], if our target MR image is sparse or if the image can be mathematically transformed to be sparse, then it is possible to use that sparsity to recover a high-definition image from substantially less acquired data [5]. Moreover, if the undersampling is incoherent, using a suitable nonlinear reconstruction method can remove the aliasing artifacts and blurring. Tremendous progressive improvements have been achieved in CS-MRI. Lustig et al. applied compressed sensing for rapid MR images [3] and introduced a randomly perturbed The associate editor coordinating the review of this manuscript and approving it for publication was Chao Tan . spiral trajectory for fast imaging [6]. Li et al. [7] proposed a 3D spiral FLAIR technique to improve the image quality of conventional 3D Cartesian FLAIR. Chen et al. [8] exploited the wavelet tree structure to improve CS-MRI. Marko Panic et al. [9] presented a practical CS-MRI reconstruction algorithm that implemented a Markov random field prior model for spatial clustering of subband coefficients solved by proximal splitting.
In general, CS-MRI is related to two main factors: one is undersampling, which depends on the incoherence condition; the other is the reconstruction method, which meets the sparsity or compressibility condition.
For undersampling in k-space, shortening the acquisition time by sampling only a fraction of data inevitable leads to aliasing artifacts. To solve this problem, an irregular undersampling pattern has been studied. Compared with Cartesian sampling, non-Cartesian sampling patterns narrow down the scope of scanning k-space as to avoid signal attenuation, improve the sampling rate and reduce the scanning time [10], [11]. In 1992, Meyer et al. [12] successfully developed the k-space spiral sampling trajectory, owing to its insensitivity to flow and fast data acquisition it has since then been widely applied in MRI areas. Santos et al. [13] Margarita et al. [14] implemented variable-density spiral trajectories in data acquisition, however, it is difficult to avoid having the blurring artifacts that appear as an offresonance effect of spins. Another radial sampling [15] approach that could be considered a variable-density spiral technique also produced streak artifacts from undersampling, and the stochastic sampling pattern could generate noise-like artifacts.
Rosette trajectories, first proposed in [16], are robust to motion with great SNR efficiency, exhibit self-correction of inhomogeneities without the need for a separate acquisition and have good off-resonance behavior with little blurring [17]. According to the distribution of k-space data, to acquire sparser data with more image energy means the trajectory should sample more data close to the k-space central region. Considering the advantage of rosette trajectories and the need to solve the artifact problem associated with undersampling, this paper presents a variable-density rosette sampling approach to improve the effectiveness of acquiring data and achieve faster and more accurate reconstruction results. Although Rosette trajectory has not been clinically validated and used in our study, we only carry out the theoretical discussion and simulation, and meanwhile provide a more competitive exploration. Bush et al. [18] have carried out rosette T2* MRI for iron quantification with a model-based reconstruction offers a clinically useful motionrobust and minimal artifacts. With the developed MRI technology, we believe rosette data acquisition technique should be clinically validated and comprehensive clinical application takes long time as the same as the PROPELLER [19] and BLADE [20]acquisition technique. Clinical validation is in the best possible way.
For the reconstruction methods, the regularized iterative algorithm can effectively remove artifacts and noise. Lustig et al. [3] first proposed the conjugate gradient (CG) to solve slow and impractical reconstruction problems for real MR images. Ma et al. [21] proposed an efficient algorithm that jointly minimizes the L 1 norm and total variation in compressed MRI (TVCMRI) based upon an iterative operator-splitting framework. Yang et al. [22] presented an algorithm (RecPF) that requires a small number of iterations for fast MR image reconstruction. Huang et al. [8], [23] proposed the composites splitting algorithm (CSA) based on the effective acceleration scheme in the fast iterative shrinkage thresholding algorithm (FISTA) and proposed the fast composites splitting algorithm (FCSA) to achieve outperforming results in MR image reconstruction, additionally exploiting wavelet tree models (WatMRI) in MRI recovery. Yao et al. [24] proposed the FTVNNR model for dynamic magnetic resonance (MR) reconstruction. To extend the TV reconstruction problem, consider the special structure of undersampling matrix A in reconstruction, Chen et al. [25] proposed fast iterative reweighted least squares (FIRLS) method, could be easily applied to the TV and JTV sparsity patterns, outperformed in accuracy and computer speed. Xu et al. [26] exploit the JTV regularization to increase sampling speed for pMRI, JTV model demonstrate also superior performance in both accuracy and efficiency. Since practical MR data are complex-valued, some of these methods solely work for real-valued images. Therefore, methods to improve the effectiveness of the complex raw data need to be studied.
In this paper, we introduced weighted rosette trajectories that can acquire the real and complex incoherent sampling data and then further proposed an adaptively regularized iterative reconstruction algorithm. This algorithm minimizes the combination of the two terms, namely, the TV and nuclear norm. To solve this large-scale problem, we decomposed the original problem into two subproblems and solved each problem respectively with adaptively updated regularization parameters, finally obtaining the reconstructed MR image by iteratively averaging the solutions of the two subproblems. Experimental results demonstrate that the reconstruction quality is successfully improved by the proposed method.

II. THE GENERALIZED ROSETTE TRAJECTORY A. K-SPACE SAMPLING
Rosette trajectories were first studied and implemented for spectrally selective magnetic resonance imaging by Noll [27]. The formulation of this trajectory consists of a rapid oscillation with frequency ω 1 and a slower rotational frequency ω 2 , as described by where k max determines the coverage range radius, and ω 1 and ω 2 are chosen as the frequency of oscillation and rotation, respectively. Additionally, ω 1 is associated with the density of sampling that crosses the center of k-space, and ω 2 determines the sample shape of the rosette trajectory. To generate rosette trajectories with different petals, different values of ω 1 and ω 2 are chosen. The form of the gradient waveforms can be shown to be [17] and the constraints for the peak gradient amplitude and peak rate of change of the gradient amplitude (slew rate). The G and S (slew rate) are given by and for ω 1 ω 2 we conclude  Based on the above, the gradient and slew rate are limited by the ω 1 term. To generate a more acceptable acquisition time, a multishot version of the rosette trajectory was described in [17]. This version can be represented in the related polar coordinates [28].
where ω 1 t = nθ, n represents the leaves of the rosette, and θ is the angle of the leaves' rotation. In addition, if n is sufficiently large, the trajectory can densely swap the entire k-space. Figure 1 illustrates the n-rosette trajectory with 2 * 18 petals. There is destructive interference when the trajectory retraces the origin in k-space multiple times, and the offresonance can be calculated, which leads to a loss of image intensity, but with very little blurring. Self-correction of inhomogeneities is also preserved in the n-rosette. Due to these advantages, the n-rosette provides a shorter acquisition time for each oscillation in the multishot rosette trajectory with 2 passes across k-space. That is, in the one-shot scenario, the rosette only needs to cross k-space using one pass. For the multishot rosette, the readout length is correlated with the number of oscillations per shot n * 1 and the oscillation frequency ω 1 , which is represented as For the n-rosette, 1/2 of the oscillations can swap the entire k-space, which yields a2n * 1 times shorter readout length.
The application of this n-rosette trajectory exhibits a reduction in SNR due to the dispersion of off-resonance [17]. To solve this problem in undersampling conditions, we consider the original data distribution of k-space. For CS-MRI, the ideal non-Cartesian sampling trajectory can provide higher incoherence and meet several requirements such as short traversal time, smoothness, variable density and incoherence [28]. Among them, regarding variable density, it is known that the MR image energy is concentrated in the central low-frequency region, and while n-rosette trajectories cross through the region many times, it is not completely covered, as shown in Figure 1.
Variable density characteristics are therefore not particularly obvious based on the rosette trajectory. We proposed a new pattern sampling trajectory to make the data sparser in terms of the main information. There is denser swapping of k-space, as shown in Figure 2, by oversampling in the vicinity of the k-space center, with normal sampling around the k-space in which d = 1, 2, · · · , 256 is the diameter of a circle region. By choosing different d values, different data densities are obtained. Under the condition of undersampling, the aliasing artifacts of nonuniform rosette sampling sequences show an almost incoherent distribution in terms of the spatial distribution, which will not seriously destroy the image structure information. Therefore, the weight factor ω d is defined to calculate the sampled circle contribution.
where I is a whole image, and Mask d is a weighted and sampled data mask in the selected circle region of the whole image. Figure 2 illustrates the weighted sampling in k-space.
Because of this weighted sampling, the important information is sampled to improve the reconstruction quality.

III. THE PROPOSED RECONSTRUCTION METHOD FOR WEIGHTED ROSETTE SAMPLING IN MRI A. MODEL FORMULATION
The constrained convex minimization method, a well-known and widely used approach in the CS-MRI literature, has motivated our model [28] that uses the convex TV norm, which is relatively robust to staircase-like artifacts that are usually dominant, and the nuclear norm, which is used as a second regularization to exploit the correlations between MR data. The physics model is based on the iterative reconstruction algorithm [29], [32], which has been shown to perform well in medical image reconstruction in recent years. In general, for nonuniform sampling trajectory imaging, with the exception of the effects of relaxation and field inhomogeneity, the MRI signal reconstruction problem in k-space can be described in 35166 VOLUME 9, 2021 discrete form (11) b = AX (11) A = e −i2π(k x r x +k y r y ) (12) where A is the Fourier sampling matrix, b represents undersampled Fourier measurements, and [k x , k y ] is a position vector of k-space related to the weighted rosette trajectory. Regarding equation (11), if the sampling trajectory is a non-Cartesian trajectory, then for the dimension of the coding matrix, N < M , where N , M is the size of the matrix. That is, A turns out to be pathological or even singular, and it is difficult to solve this ill-conditioned problem of reconstructing X from b measurements. Hence, we select the total variation and matrix nuclear norm as regularizations and formulate the reconstruction problem as arg min where µ and τ are two positive parameters. Since the MRI reconstruction model includes two regularizations, no efficient algorithm exists to solve this problem. Therefore, we consider composite splitting techniques that can decompose composite large-scale problems into two simpler regularization subproblems. Then, the subproblems are linearly combined to obtain the solution of the original optimization function. The CSD algorithm [30], [31] that is outlined in the reconstruction model solves the following optimization problems arg min where X g is a proximal map of X , updated by where (·) T is the transpose operator, and ρ is a step-size parameter. Subsequently, we split the variable X into two variables, X 1 and X 2 , and use the composite formula in (14) to obtain the subproblem as follows As mentioned above, solving for the optimal X in the original problem is transformed into solving for X 1 and X 2 in the subproblems and then obtaining the solution X by linear combination of X 1 and X 2 .
Several numerical methods have been proposed to solve (16) [33] [34], and TV norms are essentially L1 norms of derivatives, where ∇ is a difference operator, hence, the TV subproblem that is split by the CSD algorithm can be described The selection of regularization parameters µ in the process of solving subproblems requires manual adjustment by experience, which entails time and manpower consumption and cannot achieve optimal performance. A few recent works focused on adaptive parameter estimation and mainly include the generalized cross-validation (GCV) method [35], [36], the L-curve method [37], [38], the majorization minimization (MM) method [39], Morozov's discrete principles [40], [42], [44], etc. In this paper, we consider if the noise variance δ 2 can be estimated based on Morozov's discrepancy principle and adaptively select regularization parameter µ as a good choice to solve the TV restoration problem. According to Morozov's discrepancy principle that selects µ by matching the norm of the residual to some upper bound, a good solution of the TV restoration subproblem should lie in the set When the variance δ 2 of the noise is available, c = τ mnδ 2 is a constant that depends on the noise level [34], [40], [47]; if the variance δ 2 of the noise is unknown, it can be estimated using the median rule [48]. Under the discrepancy principle, the TV-regularized image restoration problem can be represented as solving a constrained optimization problem. A feasible solution for problem (19) is mentioned above that involves an inner iterative scheme; in this paper we use an ADMM [43], [49]- [51] algorithm to solve the constrained TV regularization problem with adaptively updated parameter µ. For subproblem (19), the equivalent conversion converts the regularization parameter into the fidelity coefficient to obtain a new expression To better solve the formula, auxiliary variables P and M are introduced to represent X 1 and ∇X 1, which are transformed into an augmented Lagrangian equation and then decomposed into multiple subproblems. Then, we obtain the equivalent constrained form of problem (22) as follows The augmented Lagrangian function converts the constraint problem into an unconstrained problem that can be expressed as where λ 1 and λ 2 are the Lagrange multipliers, and β 1 and β 2 are positive penalty parameters. The ADMM algorithm can be referred to as a splitting form of the augmented Lagrangian method [52]. Solving the problem by minimizing these variables separately in one round, the multiplier is updated by the ALM, and the iterative scheme to solve (23) is as follows the augmented Lagrangian coefficient is updated For the first subproblem (24), the optimization function form is as follows According to the iterative shrinkage threshold method (IST) [53], problem (29) can be solved as For the second subproblem, we not only solve variable P but also solve the updated regularization parameters µ. The optimization function related to P is as follows In (31), µ is updated in the (k + 1)th iteration based on discrepancy principle. In the above equation, only P and X 1 are related to the regularization parameter µ, and an auxiliary variable α k+1 1 is introduced, where α k+1 1 = X k 1 + λ 1 β 1 , and equation (31) is solved as follows subproblem (32) is quadratic and hence can be solved analytically as To solve P in (33), it is first necessary to update regularization parameter µ; variable P is an auxiliary variable of X 1 ; therefore, it is only necessary to discuss whether P satisfies the discrepancy principle P − X g 2 2 ≤ c, where the noise is known c = τ MN δ 2 , and τ is a predetermined parameter. Since there is no uniform criteria for choosing τ , in this paper, we assume the noise level is high, and we should set τ = 1 for this choice [40]. δ is the noise variance. If δ is unknown, in order to decrease the error between the original and the restored images, we used the wavelet transform-based median rule [42], [53], [54] to estimate the noise variance δ 2 ; more specifically, the image after performing the wavelet transform obtains the highhigh coefficients, and the real part takes the median absolute deviation to estimate the noise variance.
The update of the regularization parameter µ needs to be determined according to the range of α k+1 Since the image has already met the discrepancy principle, there is no need to constrain the regularization, and the regular parameters can be set small enough, such as zero. If P k − X k+1 g 2 2 ≥ c, we should solve the following equation Replacing P in (33) with (35), we obtain Substituting the solved µ k+1 in (36) with (33), we can obtain the updated P. Introducing the auxiliary variables P and M , based on the discrepancy principle in each iteration, we can obtain the closed form of regularization parameter µ without any manual adjustment. After completing the solution to the second subproblem, for the third subproblem according to (26), X k+1 1 can be obtained as The minimization subproblem is quadratic as follows; therefore, we have X k+1 where ∇ and ∇ T are the difference operator and inverse operation of the difference operator, respectively, and ∇∇ T is block circulant matrix that can be diagonalized by the fast Fourier transform (FFT) to reduce the computational complexity. Therefore, equation (38) can be solved in O(mnlog(mn)) for an image with a size of m × n [55].

C. SOLVING subproblem2 BASED ON NUCLEAR NORM REGULARIZATION PARAMETER ADAPTATION
To solve subproblem2 (17), a singular value thresholding (SVT) algorithm is proposed with a fixed value of τ and positive step-size δ k [56]- [58]. We consider whether the parameter for the nuclear norm problem can be adaptively selected, such that during the image reconstruction progress, the parameter can adaptively change during iterations. Therefore, we aim to recover the original signal from its rosette samples and use adaptive thresholding, as suggested in [59].
where τ k is the adaptive regularization parameter of subprob-lem2 given by (40), and τ k is also a threshold level of the k-th iteration for solving the nuclear norm minimization problem.
where β and α are two constants. This paper chooses β as the first iteration threshold, which is 5 × √ M × N ; M and N represent the initial iterative image size, Y 0 = 0 ∈ R M ×N is the initial iteration value, and the ASVT algorithm starts with Y 0 = 0 as follows where (Y k−1 , τ k ) is singular value shrinkage operator, and δ k is a positive step size. For a matrix X ∈ R M ×N with SVD X = U V T , (Y k−1 , τ k ) is defined as follows: In regard to raw data reconstruction, we address the real part and imaginary part separately.
According to the adaptive singular value thresholding algorithm shown in Algorithm1, we solve subproblem2. The

Algorithm 1 Adaptive Singular Value Thresholding Algorithm
Input: adaptive parameter estimation problem based on TV and nuclear norm regularization is split into two separate subproblems X 1 and X 2 for analysis and discussion. Given the two adaptive parameter calculation formulas, the reconstructed MR images are obtained by a linear combination of X 1 and X 2 in the iterative update Parameters are adaptively and synchronously updated with the subproblem during iteration, and the whole reconstructed problem can be solved by Algorithm2.

IV. EXPERIMENT MATERIALS AND SETUP
To validate the algorithm, all experiments are implemented in MATLAB R2014a on an Intel(R) Core(TM) i5-4460 CPU @ 3.2 GHz with 4 GB of RAM. For experiments on sampling trajectories, since the sampling method designed in this paper cannot be realized on the actual MRI instrument, the experiments in this paper are performed by collecting and reconstructing the data by writing the sampling track mask. We compare the radial, spiral, and rosette sampling trajectories with the weighted rosette sampling trajectory. Our code is tested on ten different 2D real-valued MR images and Brain and cine MR raw datasets with complex values:  Figure 4 shows four contrastive sampling trajectories.
The experimental data include real-valued MR image data and raw complex-valued human brain and cine data, the original 2D MR images are from standard medical image test VOLUME 9, 2021 library, and the data consist of an axial brain image and cine from a 3 T commercial scanner (GE Healthcare, Waukesha, WI) with an eight-channel head coil (In Vivo, Gainesville, FL) using a two-dimensional T1-weighted spin echo protocol (TE/TR = 11/700 ms, 22-cm FOV, 10 slices, 256 × 256 matrix).
The original and recovered images are shown in Figures 5-9 including a reconstruction error map. We did not show the experimental results of all the original images, but selected the better part and more competitive results to show, such as one frame of the complex raw_data Brain 256 × 256 × 8 reconstruction result is shown in Figure 5. The signal-to-noise ratio (SNR) is used as the evaluation standard for image reconstruction, the Results and Discussion section divided into three parts, one is for trajectory comparison, another is for reconstruction algorithms comparison and the last is discussion. The results are given in Table 1 under different sampling ratios of 15%, 20%. The algorithms are efficient in recovering the real and raw complex brain images, and we compare different sampling trajectories with different algorithms, such as CG [3], TVCMRI [21], RecPF [22], FCSA [23], WaTMRI [8], LaSAL2 [9], and FIRLS_TV [25]. Algorithms comparison experimental results are shown in Table 2 and the results of complex raw data are shown separately in Table 3, and we show the algorithms all suitable for complex raw data and real data training. For the multiple channel raw data, the above proposed algorithms refer to the proposed adaptive algorithm in the existing works, and the best data and our method are highlighted in bold. All sampling trajectories are written by the author according to the description of the reference, and the reconstruction algorithm uses the source code downloaded from the relevant author's website. The above algorithm parameters are empirical settings derived from multiple experiments. Figure 4 shows 4 non-Cartesian sampling trajectories, including radial, spiral, rosette, and weighted rosette.

A. COMPARISON WITH SAMPLING TRAJECTORIES
We present the numerical experimental results in this section. The test results of different images are compared under the same sampling trajectory with 15% and 20% sampling rates. The weighted rosette sampling trajectory proposed in this paper achieves an improved SNR value under various algorithms, and the chest, brain, and heart images show an obvious performance from the Table 1. However, according to the k-space information distributions of different images, the different core radii of the weighted rosette trajectory cover the more important information, which leads a difference in the reconstruction performance. Limited by the inability of using real MRI equipment, we could not calculate the sampling time, but comparisons with time of reconstruction algorithm yielded results that showed the time of different algorithms differ great, for instance FIRLS_TV algorithm preserve the fast convergence performance with 2.5s, compared to our adaptive algorithm with 19.81s, but our algorithm provides more precise results especially in the amplitude and phase of complex MR image. The experiment compared reconstruction time of four trajectories, for the same algorithm, the reconstruction time of each trajectory is within 0.5 seconds, the time gap is very small, so we did not show the result of reconstruction time of track. We found how much is the reconstruction time reduced not only related to the trajectory type, but also affected by the algorithm.
For a more specific analysis, from Table 1, the w-rosette trajectory under a lower sampling rate that was applied to all images is shown to achieve good performance in terms of the SNR and visual results. It is better in the TVCMRI algorithm, CSA algorithm and FCSA algorithm and with good reconstruction time performance. The rosette trajectory and weighted rosette trajectory can achieve better reconstruction performance while sacrificing less reconstruction time. This paper also indicates that the above rosette-type sampling trajectory has better reconstruction performance than the radial and spiral sampling trajectories.
We aims to observe the effects of several methods more intuitively. The following is a comparison of the experimental results of different images under different sampling trajectories. Due to space limitations, the article only shows experimental results at a low sampling rate of 15%. Figures 5 to 7 show the reconstruction experimental results of the raw brain with complex value image, heart image and phantom image with four sampling trajectories under the WaTMRI, TVCMRI and FCSA reconstruction algorithms.
From Figure 5-7, in the subjective analysis of the experimental results of the complex raw brain image, the spiral sampling results show obvious blurring and achieve a higher SNR but a lower SSIM value. The rosette sampling trajectory has a better reconstruction effect on the structural similarity. For the real heart and phantom graphs, streak artifacts appear  in the radial sampling reconstruction, and the image details cannot be completely reconstructed at low sampling rates. The spiral sampling trajectory is limited by the length of the sampled data. The data sampled in the k-space is incomplete at the low sampling rate, and the left part of the heart is reconstructed with distortion, resulting in artifact blurring.
The spiral sampling for the reconstruction of the phantom image in Figure 7 has vertical artifacts, but the reconstruction of the central part is better. The rosette and weighted rosette trajectories cross themselves and the origin many times with a large degree of oversampling near the origin in k-space, which is well suited to providing destructive interference of the phase accumulation over a region of the high-energy and low-spatial frequencies and will be  robust to the effect of motion, with little blurring. The reconstruction retains better smoothness, and the removal VOLUME 9, 2021 of artifacts to enhance the contrast reconstruction effect is better in areas with large gray scale changes and flowing spins.
The SNRs and sampling ratios are given in Table 1, and we test our code on eight images and highlight the best data in bold. From the Table 1, because the data of each image is different in Fourier domain, heart and cine image with more central details, results of w-rosette trajectory demonstrate superior performance.

B. COMPARISON WITH RECONSTRUCTION ALGORITHMS
In this section, to verify the effectiveness of the reconstruction algorithms, we compared our proposed adaptive parameter algorithm with CG, TVCMRI, RecPF, CSA, FCSA,  WaTMRI, LaSAL, LaSAL2, and FIRLS_TV, FIRLS_JTV, TVNNM algorithms, which are derived from TV and the nuclear norm with fixed iterative parameters. The weighted rosette proposed in this paper is selected as the sampling trajectory. For convenience of presentation, the reconstruction algorithm used in this section will be denoted by ''Proposed'' in the following. We test our code on ten different 2D MR images and complex Raw Brain data and Cine data as Figure 3 the original image shows. In the experiment, all the comparison algorithms employed low sampling ratios of 15%, 20%, 25%, and 30%, and we set the algorithm parameters as follows: the gradient descent parameter ρ = 1, the iteration threshold tol = 10 −4 , maxiter = 500, and the initial regularization parameter µ is set to 0. We chose constant step sizes given by δ = 1.2, and the nuclear norm regularization parameter τ was selected so that it is associated with the dimension of the image given by 5× √ (n 1 × n 2 ), with α = 1, and for raw data, with α = 0.6, and thus the convergence stability gradually reduces as the number of iterations increases. The SNRs and SSIMs are used to quantitatively analyze the reconstruction effects of each algorithm.
The following Figures 8-10 show experimental results of different images at the 20% sampling ratio under different contrastive algorithms. In view of the space limitations, the experimental results are shown in the same size below, and the white frame is used to mark the results of reconstruction image details. The analysis of the SNRs and SSIMs of each algorithm is shown in Table 2, and the adaptive algorithm proposed in this paper is generally superior to other algorithms, especially when the sampling rate is larger. It demonstrates that the performance achieved by combining the two regularizations is better for smooth and texture components. In Figures 8-10, the proposed algorithm is compared with the other eleven algorithms, and the reconstruction result shows  better performance for images with smooth components in TVNNM, since TVNNM can effectively recover smooth areas, but with fixed regularization parameters it shows an oversmoothed effect according to the figures of the details. The reconstruction effect of the texture component is not as good as the method proposed in this paper. LaSAL and LaSAL2 deep learning algorithms show a superior effect in real image data reconstruction, yields considerably higher SNRs and SSIMs than the reference methods, illustrates the obvious improvement in visual quality, but for multiple channel complex raw data, lower SNRs and SSIMs are demonstrated in Table 3, and from Figure 10, we see some local blurring in the results. The experiments result of FIRLS_TV and FIRLS_JTV validate the fast convergence speed due to the preconditioner, from the Figure 8-9, we can see the FIRLS_JTV algorithm is better in detail but with lower SNRs and SSIMs, also Strip artifacts are also easy to appear in Figure 10. From the Table 2 and Table 3, the adaptive method we proposed can effectively balance the TV and nuclear norm regularizations and achieve the highest SNRs and SSIMs in most cases, especially for complex raw MR data, our method is recovering more important image details while removing sampled data noise and blurring. It is not necessarily better than the existing algorithm, but we agree with its competitiveness.
Finally we adaptively update two regularization parameters, i.e., µ and τ , during the procedure of optimized reconstruction. We take the coronal brain and full body image as examples, and plot µ and τ over iterations in Figure 11. From Figure 11, it can be see that µ and τ stably converge with iterations, and for different images with different µ and τ values, such as the coronal brain, the adaptive parameters finally approach µ = 7, τ = 0.01, as Figure 11 shows. Furthermore, we also use fixed µ or τ to reconstruct the coronal brain to evaluate the effectiveness of adaptive regularization parameters in Figure 12. The final values of iterations are computed based on the discrepancy principle, and we first set µ as 0.01, 0.1, 1, 5, and 10 with fixed  τ = 5 × √ (n 1 × n 2 ), (n 1 = n 2 = 210), and inversely we set τ as 0.05 × √ (n 1 × n 2 ), 0.5 × √ (n 1 × n 2 ), 1 × √ (n 1 × n 2 ), 10 × √ (n 1 × n 2 ), and 50 × √ (n 1 × n 2 ), with fixed µ = 7. After reconstruction, we show the PSNRs and SSIMs in Figure 12. We can see that the proposed adaptive parameter method achieved good performance in SNRs and SSIMs.

C. DISCUSSION
In Results and Discussion section, the first and second experiment validate better performance of our trajectory and reconstruction algorithm. The superior performance of the proposed weighted-rosette trajectory retraces the origin in k-space multiple times, and the off-resonance can be calculated removed by iterative algorithm, which leads to a loss of image intensity, but with very little blurring. The results show not all the image suit for this trajectory, but most of image sampled by weighted-rosette trajectory can achieve more information via one scanning time, compared to other trajectory we proposed a novel trajectory and validate its advantage. For our adaptive reconstruction, the state-of-thearts are validated on practical application CS-MRI reconstruction field, although some difference can be found, and our algorithm is not as good as LaSAL2 in 2D image reconstruction, but yields adequate reconstruction results compared to other methods, and performed better than other reconstruction methods 50% of the time when using complex raw brain data, demonstrates more competitive in clinical application, that is our major contribution to this work.
Despite the improvement performance of rosette trajectory and adaptive reconstruction algorithm, our work really with limitations. First we only carry out the trajectory simulation through computer, limiting to clinical utility, due to the no cooperation agreement with hospital. Second, our rosette trajectories are more susceptible to gradient delay [60] that can influence image quality, our next study will focus on this, explore prospective methods in the future. And another limitation is that rosette trajectories are more sensitive to off-resonance-related artifacts [17], which can result in a loss of signal, we need pay more attention on the iterative reconstruction algorithm to compensate for the loss of this signal. Lastly, we will explore incorporating motion directly into the reconstruction model [61] and the fast speed of this reconstruction algorithm really matters.

VI. CONCLUSION
In this paper, we present MRI reconstruction effects of several non-Cartesian sampling trajectories based on the distribution characteristics of k-space data, and then introduce weighted rosette trajectories. We formulate the reconstruction model that consists of two regularizations, i.e., the TV and the nuclear norm. as to solve the large-scale problems, we decomposed it into two subproblems by splitting algorithm. Subsequently, we adaptively updated two regularization parameters based on the discrepancy principle and threshold change criteria during the iteration. The experimental results confirm the effectiveness of the proposed approach for MRI reconstruction. Although it has not been realized in reality, it has been verified by simulation and theory with prospective. The next step of this research is to break these limitations in our work, and explore more sophisticated algorithms in the future.