Ashkan Yousefi Zadeh

Alireza Davoud Pasha

The Role of Big Data in the Development of Safety in Fully Automated Vehicles: A Survey

Keywords: Autonomous Vehicles, Big data, Driving Safety, Artificial Intelligence, Deep Learning

Cars are the most used vehicles by people for transportation in the world that their use endangers the lives and safety of millions of people every year. It seems that the solution presented in recent years, given the advances in computer science and artificial intelligence to overcome these dangers and have a safer driving is the use of fully automated vehicles. In this paper, first, autonomous vehicles and types of them are introduced and the safety challenges of vehicles driven by humans and factors of insecurity in these vehicles have been studied. In the following, the current trends and future challenges of autonomous vehicles have been reviewed, as well as the importance and safety position of these vehicles. Eventually, the role of big data to promote safety in fully automated vehicles and its importance in developing safety and maintaining public health in this industry for the following years has been studied.

Introduction

Nowadays, one of the most notable trends in the application of artificial intelligence in mechanical engineering is autonomous vehicles (AVs), which can be one of the most challenging of these applications due to the straight and close relation to public health. Based on the J3016 standard issued by the Society of Automotive Engineers or "SAE", levels of driving automation is defined in six levels, from SAE Level Zero (no automation) to SAE Level 5 (full vehicle autonomy) [START_REF]SAE Standards News: J3016 automated-driving graphic update[END_REF]. "Level 0" or "No automation level", indicates that the driver performs all the driving tasks. "Level 5" or "Full automation level", indicates the vehicle carries out all of the driving duties under all conditions, and the driver may have the options to control the vehicle; this level of automation is the case study of this article [START_REF]Automated Vehicles for Safety[END_REF]. When the vehicle is fully autonomous, it is capable of longitudinal and lateral control; also, it can perform "Object and Event Detection and Response" or "OEDR" and can handle emergencies autonomously. The final property which distinguished fully automated vehicles versus high automated vehicles is unlimited "Operational Design Domain" or "ODD" which means the system can operate under various conditions [START_REF] Calvert | Gaps in the control of automated vehicles on roads[END_REF]. Until 2018, only Waymo had deployed vehicles for public transportation with the highest autonomy among AVs, which was "level 4" or highly automated [START_REF] Waslander | Introduction to Self-Driving Cars[END_REF]. Based on the Waymo Safety Report in 2018, every year, 1.2 million people die in traffic crashes in the world, and these numbers are annually rising around the world [START_REF] Waymo | Waymo Safety Report: On the Road to Fully Self-Driving[END_REF]. Also, The World Health Organization (WHO) estimates that around the world, the yearly road deaths are 1.35 million, with about 50 million injuries [START_REF] Threlfall | Autonomous Vehicles Readiness Index (AVRI)[END_REF]. These reports determine that about 95% of road accidents are because of human errors [START_REF] Waymo | Waymo Safety Report: On the Road to Fully Self-Driving[END_REF]. These statistics determine that. However, companies are improving the quality of the safety features on their vehicles, such as seat belts, airbags, impact absorption etc. However human errors still exist, and they are inevitable. Thus, it is obvious that using an autonomous system which can provide the least number of errors in driving with intelligent computations, can help people to drive safely. Recently, with the development of self-driving cars, safety measures take important steps, preventing collisions by using report information received from the environment outside the vehicle [START_REF] Mallozzi | Autonomous Vehicles: State of the Art, Future Trends, and Challenges[END_REF]. Since fully automated vehicles ultimately use data to control the vehicle, accuracy, and validation of these data are crucial for safety. Big data is principally applied to represent enormous datasets, and they include masses of unstructured data that require real-time analysis. Big data makes new chances for exploring new values, helps obtain an in-depth perception of the disappeared and hidden values, and brings new difficulties and challenges. Such as how to arrange and handle such datasets efficiently [START_REF] Chen | Big data: A survey[END_REF]. To have more accurate and useful data to use for controlling a fully automated vehicle, using big data is essential. In the next sections, details of this topic have been discussed. The above figure shows that manufacturers gradually increased their focus on AVs, especially since 2015. Figure 3 displays that autonomous vehicles level 4, will allocate a huge volume of ground transportation investments until 2030. This figure accurately shows that a possible reason for lack of AVs level 5 before the year 2030 is the importance of developing safety factors for fully automated vehicles, which is a timeconsuming and delicate process. As a result, research and development of these AVs must proceed in a precise form.

Figure 3: The predicted market size of Level 4 autonomous driving worldwide between 2022 and 2030 [START_REF] Wagner | Projected L4 autonomous driving market size worldwide between 2022 and[END_REF] 1st Automotive Industries Recent Advances and Future Trends (AIRAFT2020) October 2020 Iran University of Science and Technology, Tehran, Iran 3

Importance of Safety in Fully Automated Vehicles

It can be said with certainty that the most important reason for using AVs is to drive safer. As mentioned, the leading cause of road accidents is human errors. Table 1 shows the detailed analysis of accidents in Iran from 2006 to 2015. As it is obvious, the numbers of deaths and injuries are remarkable, and yearly, many people are killed while they are using cars for transportation. Based on the US National Highway Traffic Safety Administration (NHTSA) report published in 2015, the critical driver-related causes in an accident are as follows:

Some Examples of Safety Considerations of Autonomous Vehicles by Manufacturers

Ford Motor Company

Argo AI is a collaboration company which works with Ford in the development of AVs; this company uses a three-phase training plan for safety operators [START_REF] Company | FORD'S APPROACH TO DEVELOPING SELF-DRIVING VEHICLES[END_REF]. Ford's development team record and extract massive daily data from simulations and test vehicles from onboard data systems like braking and steering. in addition, they use Virtual Driver System as well as camera and sensor data. Using an autonomous vehicle data recording device and its Event Data Recorders (EDR), data from the crash will be logged into the vehicle system automatically. Ford's validation and testing methods apply data from testing to develop and enhance the system, sub-system, and component performance. In real-world testing, they produce the distances logged and near misses faced to either improve requirements or validate system safety with new data for simulations. Finally, they will test the vehicle's capacity to independently make safety fallback maneuvers to remove safety operators when the Virtual Driver System is available [START_REF] Company | FORD'S APPROACH TO DEVELOPING SELF-DRIVING VEHICLES[END_REF].

General Motors Company

General Motors (GM) safety is through an iterative design. They design and build technologies and systems, test them in simulators as well as in the field, and then check the results and feedback to the design the process. they include learnings, mainly safety data, into future productions so they will be even safer. GM does this with many iterations, which leads to new technologies and systems for autonomous vehicles. In GM's controlled deployment, autonomous vehicles drive only within defined boundaries, and only on roads for which they have acquired high-definition map data, also, they operate just under known operational situations and constraints that apply to the entire fleet. GM's autonomous vehicle has two data recording features a second robust data logging system and a conventional Event Data Recorder (EDR) as well as their vehicle's robust data recording ability, gives information on vehicle performance while normal driving and collision avoidance situations [START_REF] Motors | [END_REF].

Role of big data in the safety of fully autonomous driving

AVs entirely work by the data that they obtain by sensors such as IR (Infra-Red) Sensors, radar, Lidar, Global Positioning System (GPS), and the information they concoct through 3D cameras, etc. the data that cars collect within these references lead them to make safe driving decisions and control them [START_REF] Potter | Big Data's Role in Self-Driving Car Development[END_REF]. Table 2 indicates that each AV collects approximately 120,000 GB data per month and 1,440,000 GB data per year. It is obvious that we are left with a massive dataset and extracting useful information from this data set is very challenging. The above table indicates that a fully automated vehicle requires many infrastructures such as artificial software and a cloud server to store massive datasets from sensors. Therefore, a driverless vehicle can build strategies for many possible situations such as different weather, lighting levels on the road, driver behavior, etc. by using collected data [START_REF] Haydin | How Big Data in Autonomous Vehicles Defines the Future[END_REF]. As a result, data volumes and storing, real-time and fast analyzing, and cybersecurity are the most significant challenges that big data make. Currently, many famous companies and universities worldwide are working to address these issues. For instance, Amazon Web Services S3, Microsoft Azure Data Lake, Google Cloud Storage, and IBM Cloud are the most popular companies providing cloud storage for big data [START_REF] Marr | What Are The Best Big Data Cloud Storage Providers? Here Are The Top 6[END_REF]. So scientists and engineers can use datasets faster and enhance the performance of their designs for AVs. Moreover, in the Cybersecurity section, companies like APTIV and NVIDIA are developing software and computing platforms for cyber safety in AVs. NVIDIA utilizes AI-powered data processors and chips to produce and protect AVs. The software and 1st Automotive Industries Recent Advances and Future Trends (AIRAFT2020) October 2020 Iran University of Science and Technology, Tehran, Iran cloud-based technologies assist AVs relay driving data and in securely learn [START_REF] Daley | 10 AUTOMOTIVE CYBERSECURITY COMPANIES MAKING VEHICLES[END_REF]. Tackling and addressing the entire big data challenges in fully autonomous vehicles is a time-consuming task that will gradually be developed and modified. Hence, to acquire and develop AVs functions, manufactures must stock, manage, and process vast and massive volumes of data. The primary research and development challenges in this field is an investigation on how AVs should use the advantages of big data to behave in possible crash conditions to protect human and public health. On the other hand, for an appropriate decision making how broad the scope of data should be. Accessing open data is essential to the use of big data in fully automated vehicles. Because AVs, in addition to learning from neural networks of other vehicles manufactured by the same corporation, must proceed learning process from all other AVs as well. By using big data, AVs allow the implementation of safety and public health policies on prioritization of protection in accident situations, and these policies have to be considered and studied thoroughly [START_REF] Shaw | Automated vehicles, big data and public health[END_REF]. The synchronization and coordination of AVs in the networks of vehicles are one of the vital usages of big data in autonomous vehicles. Also, one of the most significant importance of big data refers to controller design engineer's task part which is related to using big data for autonomous vehicle control systems [START_REF] Fényes | A predictive control for autonomous vehicles using big data analysis[END_REF].

Figure 6: Baseline architecture of autonomous driving [START_REF] Waslander | autonomoose: Toward All-Weather Autonomous Driving[END_REF] Big data in control systems can be applied for prediction and estimation goals. Also, it can be a section of the control strategy, such as an optimal trajectory adaptation approach concentrating on the safety factors of AVs using a cloud database [START_REF] Fényes | A predictive control for autonomous vehicles using big data analysis[END_REF]. Model predictive control (MPC) is one of the famous controlling methods in this area. Since the feedforward controllers implement a predictive response as they provide a reference output to obtain an appropriate tracking response, especially when the expected inputs are non-zero are the main reason that feedback and feedforward controllers are used to control a system. To keep a constant-radius turn or brake command and a constant throttle to maintain a constant velocity or deceleration measures, AVs need non-zero steering commands. Therefore, feedforward control is very advantageous in developing tracking performance in fully automated driving [START_REF] Sailan | Modeling and design of cruise control system with feedforward for all terrian vehicles[END_REF].

Figure 7: Combined feedforward and feedback control for MPC, general Scheme [START_REF] Waslander | Feedforward Speed Control[END_REF] 1st Automotive Industries Recent Advances and Future Trends (AIRAFT2020)

October 2020 Iran University of Science and Technology, Tehran, Iran As mentioned and is evident from figure 6, data-driven control for a fully automated vehicle relies on data that various sensors of the AV system obtain from the environment. To help AVs to correctly understand the environment and make better decisions compared to humans in situations such as accelerating, decelerating, steering, and path planning, many artificial intelligence strategies like machine learning (ML) and Reinforcement Learning (RL) solutions could be applied [START_REF] Ma | Artificial intelligence applications in the development of autonomous vehicles: a survey[END_REF]. On the other hand, this volume of data is massive and unstructured. As a result, using traditional database models and tools are useless and cannot be managed, processed and analyzed properly. Therefore to explore, control, and handle this enormous volume of data, data-driven models are required [START_REF] Kaffash | Big data algorithms and applications in intelligent transportation system: A review and bibliometric analysis[END_REF]. A machine learning method that uses automated feature engineering for large amounts of data to train Deep Neural Networks (DNNs) is Deep learning (DL). DNNs could attain human-level precision by real-time decisions and outcomes from real-time data input. Until today, over 80 % of the whole analysis process time is approximately allocating to preparing the data, which involves labeling, feature engineering, etc. [START_REF] Ziegler | AI and Big Data Management for Autonomous Vehicles[END_REF]. Regardless of which data-driven controller approach is used, It would be essential to work with proper tools and modern techniques to manage and analyze big data in order to have a more robust and faster learning process for the system and eventually reach a better and high-performance controller that lead the vehicle to a proper decision making to have a safer driving.

Conclusion

As discussed, almost 95% of road accidents are happening due to human errors, and everyday thousands of people around the world lose their lives or get physically or mentally injured because of these accidents. by the increasing development and promotion of artificial intelligence and their applications in various technologies such as autonomous vehicles, the number of these driver related accidents can be reduced remarkably, and the number would be down to acceptable figures. These goals can be achieved when multiple parameters involved in the control of fully autonomous vehicles, both internal and external, are identified adequately by the system. The system must also be well-trained and able to know how to deal with the various scenarios that may occur while driving or be able to correctly predict the outcome for a safer decision making. Since the basis of controlling fully autonomous vehicles is dependent on the training and testing of the controller using the data received from different sensors installed in the car using different scenarios and situations, it is discussed that these vehicles must undergo a huge and massive data which are unstructured. Managing and analyzing the data to obtain an optimum and proper controller is a challenging and complicated task which must be done by modern tools and updated data science methods. As a result, big data has a significant and crucial role in having a safer driving with fully autonomous vehicles and it is essential that control developer scientists and engineers of these vehicles suitably apply these big data into designing and developing AVs to ensure the safety of the occupants of these vehicles and the pedestrians who encounter them.

Figure 1 :

 1 Figure 1: Standard levels of driving automation by SAE [1].

Figure 2 :

 2 Figure 2: Number of active patent issues from 2010 to 2019 for the biggest autonomous driving patent owners worldwide [9]

Figure 4 :

 4 Figure4: Critical driver-related causes[START_REF] Nhtsa | Critical Reasons for Crashes Investigated in the National Motor Vehicle Crash Causation Survey[END_REF]

  Error (sleep, etc.) Other 1st Automotive Industries Recent Advances and Future Trends (AIRAFT2020) October 2020 Iran University of Science and Technology, Tehran, Iran

Figure 5 :

 5 Figure 5: Argo AI training program for safety operators[START_REF] Company | FORD'S APPROACH TO DEVELOPING SELF-DRIVING VEHICLES[END_REF]

Table 1 :

 1 Analysis of road accidents in Iran[START_REF] Iran | Number of accidents and human casualties of vehicles leading to death, injury and damage in road transport[END_REF]

	Year	Total	Number of	Number of	Number of	Number of	Number of
		accidents	fatal	people	accidents	people	accidents
			accidents	killed	resulting in	injured	resulting in
					injuries		damage
	2006-2007	795205	5841	7590	123923	160152	665441
	2007-2008	758801	6460	8416	125762	158573	626579
	2008-2009	780352	6554	8645	139370	174284	634428
	2009-2010	702512	11854	18159	166456	231972	524202
	2010-2011	776517	26239	37893	388866	472132	361412
	2011-2012	396893	13665	16884	200223	254699	183005
	2012-2013	429544	15123	18569	226565	293665	187856
	2013-2014	305787	7038	8940	135008	174064	163741
	2014-2015	264557	6609	8248	126876	163450	131072

Table 2 :

 2 An approximate volume of data into Autonomous vehicles[START_REF] Krzanich | Data is the New Oil in the Future of Automated Driving[END_REF]

	Type of data resource	Volume of data
	Cameras	~20-40 MB (Per Second)
	Radar	~10-100 KB (Per Second)
	Sonar	~10-100 KB (Per Second)
	GPS	~50 KB (Per Second)
	LIDAR	~20-70 MB (Per Second)
	Total	4,000 GB (Per Day)


Hernán Mella

Joaquín Mura

Julio Sotelo

Sergio Uribe
email: suribe@uc.cl

A comprehensive comparison

Keywords: Tagging MRI, CSPAMM, DENSE MRI, Cardiac MRI, Cardiac Strain, HARP, SP-HR, SinMod, DENSEAnalysis

come

1) INTRODUCTION

The myocardial strain is a regional biomarker of the cardiac function that has been assessed for several cardiovascular diseases [START_REF] Fonseca | Three-dimensional assessment of left ventricular systolic strain in patients with type 2 diabetes mellitus, diastolic dysfunction, and normal ejection fraction[END_REF][START_REF] Ernande | Systolic Myocardial Dysfunction in Patients with Type 2 Diabetes Mellitus: Identification at MR Imaging with Cine Displacement Encoding with Stimulated Echoes[END_REF][START_REF] Auger | Imaging left-ventricular mechanical activation in heart failure patients using cine DENSE MRI: Validation and implications for cardiac resynchronization therapy[END_REF]. One of the main advantages of strain measurements over global measurements, as ejection fraction or stroke volume, is its significant sensitivity to detect early changes in cardiac function [START_REF] Cikes | Beyond ejection fraction: An integrative approach for assessment of cardiac structure and function in heart failure[END_REF].

Several non-invasive Magnetic Resonance (MR) imaging techniques have been used to estimate myocardial strain. Among them, Tagging MR imaging has been intensely used for the evaluation of strain [START_REF] Axel | Tagged magnetic resonance imaging of the heart: A survey[END_REF][START_REF] Petitjean | Assessment of myocardial function: A review of quantification methods and results using tagged MRI[END_REF], considering the conventional tag analysis (i.e., following the intersections of the tag lines) the current gold-standard MR method for the estimation of heart deformation [START_REF] Auger | Cardiac mechanics[END_REF]. One of the most used Tagging modalities is Complementary Spatial Modulation of Magnetization (CSPAMM) [START_REF] Fischer | Improved myocardial tagging contrast[END_REF], which uses two complementary SPAMM acquisitions to generate a new image with better relaxation properties. Another technique for quantifying strain is Displacement Encoding with Stimulated Echoes (DENSE) [START_REF] Aletras | DENSE: Displacement Encoding with Stimulated Echoes in Cardiac Functional MRI[END_REF], which encodes the displacement of the tissue into the phase of the magnetization vector. In the last case, a phase-cycling approach (based on the same principle that CSPAMM) can be employed to isolate the stimulated echo [START_REF] Gilson | Complementary displacement-encoded MRI for contrast-enhanced infarct detection and quantification of myocardial function in mice[END_REF]. DENSE imaging has recently been considered the new gold standard for estimating motion and strain from MRI [START_REF] Goto | Comparison of Displacement Encoding With Stimulated Echoes to Magnetic Resonance Feature Tracking for the Assessment of Myocardial Strain in Patients With Acute Myocardial Infarction[END_REF][START_REF] Wehner | Comparison of left ventricular strains and torsion derived from feature tracking and DENSE CMR[END_REF].

In CSPAMM and DENSE, the motion of the tissue cannot be directly estimated from the image, and other postprocessing methods need to be applied. Some well-known methods are Shortest-Path HARP Refinement (SP-HR) (an improved version of the Harmonic Phase analysis (HARP) [START_REF] Osman | Cardiac motion tracking using CINE harmonic phase (HARP) magnetic resonance imaging[END_REF][START_REF] Osman | Imaging heart motion using harmonic phase MRI[END_REF]) and Sine-Wave Modeling (SinMod) [START_REF] Arts | Mapping displacement and deformation of the heart with local sine-wave modeling[END_REF], which extract harmonic peaks from the k-space using bandpass filters. In DENSE, the motion is estimated by isolating the stimulated echo from k-space [START_REF] Spottiswoode | Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting[END_REF][START_REF] Kim | Myocardial Tissue Tracking with Twodimensional Cine Displacement-encoded MR Imaging: Development and Initial Evaluation[END_REF], which contains information about the displacement of the tissue.

Several articles have evaluated the behavior of Tagging and DENSE imaging techniques under different acquisition parameters [START_REF] Nessaiver | Magnitude image CSPAMM reconstruction (MICSR)[END_REF][START_REF] Sigfridsson | In vivo SNR in DENSE MRI; temporal and regional effects of field strength, receiver coil sensitivity and flip angle strategies[END_REF][START_REF] Kim | Increasing the signal-to-noise ratio in DENSE MRI by combining displacement-encoded echoes[END_REF][START_REF] Wang | Artifacts reduction in strain maps of tagged magnetic resonance imaging using harmonic phase[END_REF][START_REF] Epstein | Displacement-encoded cardiac MRI using cosine and sine modulation to eliminate (CANSEL) artifact-generating echoes[END_REF], illustrating the importance of an appropriate imaging protocol. Recent works have compared different postprocessing methods for Tagging MR images using synthetic images [START_REF] Ibrahim | Harmonic phase versus sine-wave modulation for measuring regional heart function from tagged MRI images[END_REF][START_REF] Eldeeb | Accurate harmonic phase tracking of tagged MRI using locallyuniform myocardium displacement constraint[END_REF][START_REF] Ibrahim | Regional cardiac function analysis from tagged MRI images. Comparison of techniques: Harmonic-Phase (HARP) versus Sinusoidal-Modeling (SinMod) analysis[END_REF], giving insights about the differences between SinMod and HARP methods. The estimation of motion and strain with both Tagging and DENSE images have also been compared against feature tracking [START_REF] Li | Quantification of myocardial strain at early systole in mouse heart: Restoration of undeformed tagging grid with single-point HARP[END_REF][START_REF] Cao | A comparison of both DENSE and feature tracking techniques with tagging for the cardiovascular magnetic resonance assessment of myocardial strain[END_REF], providing a picture of the reproducibility and differences in the estimation of radial, circumferential, and longitudinal strain components from each imaging modality [START_REF] Cao | A comparison of both DENSE and feature tracking techniques with tagging for the cardiovascular magnetic resonance assessment of myocardial strain[END_REF][START_REF] Augustine | Global and regional left ventricular myocardial deformation measures by magnetic resonance feature tracking in healthy volunteers: Comparison with tagging and relevance of gender[END_REF][START_REF] Wu | Feature tracking compared with tissue tagging measurements of segmental strain by cardiovascular magnetic resonance[END_REF]. Furthermore, the estimation of motion and strain using conventional methods from SPAMM and DENSE images has also been studied under controlled conditions on in-silico, in-vitro, and in-vivo experiments [START_REF] Young | Generalized spatiotemporal myocardial strain analysis for DENSE and SPAMM imaging[END_REF], showing comparable performances in all cases except for radial strain, where analysis of DENSE images showed best results.

However, most of these works suffer from a lack of analytical solutions or controlled experiments and have excluded variables such as the cardiac motion, pixel sizes, noise level, and tag spacings, among other relevant parameters.

Although there is a consensus about the expected performance of motion and strain metrics estimated from each imaging modality, the ultimate performance depends on the acquisition parameters and the postprocessing strategy used. This work aims to analyze the precision and accuracy in estimating both motion and strain, compared against simulated values, using SP-HR, SinMod, and DENSEanalysis; three different automated postprocessing tools on CSPAMM and DENSE images when subjected to several noise and resolution levels. The study firstly uses images from 2D phantoms with only in-plane motion and under ideal acquisition conditions.

Secondly, under non-ideal conditions, considering a cartesian acquisition (i.e., adding EPI-like artifacts, k space cropping, and k space filtering). Finally, the sensitivity analysis in estimating the three-dimensional cardiac motion and artifacts is performed using 3D data sets (see Figure 1 ). To achieve these goals, we developed a multi-platform open-source Python [START_REF] Van Rossum | Python 3 Reference Manual[END_REF] library to generate numerical phantoms of CSPAMM and DENSE MR images, which can be used to simulate different physiological motion conditions.

It is essential to clarify that this work compares the estimations of motion obtained with SP-HR, SinMod, and DENSEanalysis as processing tools rather than CSPAMM and DENSE as imaging sequences.

2) MATERIALS AND METHODS

2.1) Image generation

A 3D phantom consisting of millions of isochromats randomly distributed on space is generated and limited to the cardiac geometry (see Figure 1). We defined the idealized cardiac geometry as a cylinder whose in-plane motion is determined by a set of parameters and expressions given in [START_REF] Gilliam | Automated motion estimation for 2-D cine DENSE MRI[END_REF]. Additionally, we added a third motion component in the through-plane (Z) direction to achieve a displacement from base to apex up to 20 mm [START_REF] Fischer | True myocardial motion tracking[END_REF], which is given by:

Δ = 20 × -1 [mm] (1)
where is a normalized coordinate that varies from 0 to 1 from base to apex and Δ the throughplane displacement. To emulate the clockwise and anticlockwise rotation of the LV at the basal and apical levels [START_REF] Fischer | True myocardial motion tracking[END_REF], we added a scaling factor which changes the rotation of the isochromats depending on its longitudinal position, defined by:

= 1 + -1 (2)
where denotes the scaling, the imposed end-systolic endocardial rotation at the apex, and the end-systolic endocardial rotation at the base. In all our simulations > 0 and < 0.

To generate the MR images, we assigned a complex magnetization to every isochromat, which was transferred to the images using a distance-weighted sum (respect to the voxel centers). We modified the FOV and resolution of images to emulate the k space sampling of the MR scanner.

Finally, the images with the user's specifications were obtained by filtering, zero-filling, and correcting the oversampling of the generated k space. to the voxel center (red sphere). Blue spheres denote the set of isochromats inside the voxel.

2.2) CSPAMM magnetization

CSPAMM images were built by adding two complementary SPAMM images [START_REF] Fischer | Improved myocardial tagging contrast[END_REF]. During each SPAMM pre-pulse, a position encoding gradient is placed between two radiofrequency (RF) pulses with tip angles + and ± . The last RF pulse stores the magnetization in the longitudinal direction ± (depending on its polarity) to avoid relaxation. With this into consideration, both complementary SPAMM magnetizations at the time are given by:

SPAMM ( ) = cos (β) sin(α) exp - + sin(α) cos (α) 1 -exp - ± 2 sin(α) cos (α) sin (β) exp - exp(- ) + 2 sin(α) cos (α) sin (β) exp - exp(+ ) (3)
where represents the magnetization at the thermal equilibrium, = √-1 the complex unit, α the imaging flip angle, the material position of the tissue, and the encoding frequency. The sign of the last term in Equation 3 depends on RF pulses polarity during the preparation step. Thus, if two complementary SPAMM images SPAMM and SPAMM are acquired, their difference leads to the CSPAMM magnetization expression:

CSPAMM ( ) = SPAMM -SPAMM = sin(α) cos (α) sin (β) exp - exp - ( ) + sin(α) cos (α) sin (β) exp - exp + ( ) (4)

2.3) DENSE magnetization

The DENSE acquisition sequence encodes the displacement of the tissue directly on the phase of the magnetization. The preparation pulse is the same as SPAMM, but the acquisition sequence differs due to an additional gradient with the same magnitude applied in the preparation step, which rephases the static spins. In this sequence, the magnetization is also stored in the longitudinal direction to avoid relaxation. Thus, the magnetization expression at the time for the DENSE technique becomes [START_REF] Kim | Myocardial Tissue Tracking with Twodimensional Cine Displacement-encoded MR Imaging: Development and Initial Evaluation[END_REF]:

DENSE ( ) = ± 2 sin(α) cos (α) exp - exp(-Δ ) ± 2 sin(α) cos (α) exp - exp{-(2 + Δ )} + sin(α) cos (α) 1 -exp - exp{-( + Δ )} (5)
where Δ represents the displacement of the tissue and the encoding frequency and the sign of the two first terms depend on the polarity of RF pulses during the preparation step. Thus, similarly to CSPAMM, if two complementary DENSE images and are acquired, their difference leads to:

CDENSE ( ) = - = sin(α) cos (α) exp - exp(-Δ ) + M sin(α) cos (α) exp - exp{-(2 + Δ )} (6)
The last step is also called phase-cycling [START_REF] Gilson | Complementary displacement-encoded MRI for contrast-enhanced infarct detection and quantification of myocardial function in mice[END_REF].

2.4) Numerical Experiments

2D analysis

We generated a synthetic dataset using SPAMM and DENSE magnetization expressions given in Equations ( 3) and ( 5), where a different image was generated for each RF pulse polarity to obtain, after subtraction, the CSPAMM and phase-cycled DENSE images. The dataset consisted of 100 -2D slices of a short-axis view of an idealized left-ventricle with only in-plane motions with a FOV of 100×100×8 mm 3 and isotropic (in-plane) pixel sizes of 1.0, 1.5, 2.0, 2.5, and 3.0 mm. The number of isochromats used in each data was 4 million, which means that smaller voxels contained a smaller number of isochromats.

For the SPAMM images, we used encoding frequencies of 0.79, 0.63, 0.52, 0.45, and 0.39 rad mm - 1 to achieve tag spacings of 8, 10, 12, 14, and 16 mm respectively. A fixed encoding frequency of 0.75 rad mm -1 was chosen for DENSE images to achieve suitable echoes for phase-cycling correction and avoid large phase wrapping artifacts. The imaging flip angle for both images was chosen as 15° (constant through the cardiac phases), and a tissue relaxation of 0.85s was used to emulate the relaxation properties of the myocardium at 1.5T [START_REF] Fischer | Improved myocardial tagging contrast[END_REF].

We randomly choose the physiological parameters, which controls the geometry and motion, according to Gilliam et al. [START_REF] Gilliam | Automated motion estimation for 2-D cine DENSE MRI[END_REF], i.e., we choose different physiology for each synthetic data. In our case, we used a set of parameters defining 50 cases with normal deformation patterns (normal cardiac function) and 50 with abnormal patterns (regionally reduced function) [START_REF] Gilliam | Automated motion estimation for 2-D cine DENSE MRI[END_REF]. The displacement field of the defined motion is defined by

Δ ( ) = Γ( )( - ), (7)
Where Γ( ) ∈ [0, 1] is a piece-wise continuous function, derived from the standard left-ventricular volume diagram [START_REF] Mcculloch | Cardiac Biomechanics[END_REF], which weights the maximum end-systolic displacement, and and the end-systolic and end-diastolic position of the LV, respectively. is defined by the initial geometry of the phantom, whereas depends on several parameters uniformly distributed, such as the end-diastolic endocardial radius, end-diastolic wall thickness, end-systolic endocardial and epicardial twist, and end-systolic endocardial and area scaling, among others [START_REF] Gilliam | Automated motion estimation for 2-D cine DENSE MRI[END_REF].

We performed a resolution and noise sensitivity analysis to the estimation of displacements and strain obtained from CSPAMM and DENSE MR images using SP-HR, SinMod, DENSEanalysis.

The resolution sensitivity analysis considered several pixel sizes and tag spacings for both imaging modalities, whereas, for the noise experiment, we used the tag spacings that performed better in the first experiment using the following rule: given fixed pixel size and for all the tag spacings, the tag spacing used for the noise analysis was that one that minimizes the error on the circumferential strain component. The idea behind this selection is to test just those cases which performed better in the estimation of circumferential strain due to the clinical relevance of this biomarker. : scaling factor to consider the signal decay due to the reduction of the voxel size. For an in-plane isotropic pixel size (and constant slice thickness) of 3, 2.5, 2, 1.5, and 1 mm the associated factors are 1, 0.69, 0.44, 0.25, 0.11, respectively.

Real and imaginary Gaussian noise with zero mean was added to the image k-spaces. The standard deviation (SD) of the noise was estimated to achieve certain SNRs at early systole and late diastole only on 3 × 3 phase-cycled DENSE images, as described in Table 1. However, the noise was added separately before phase-cycling. The same SD was used for images with smaller pixel sizes as the signal is reduced during the generation process (fewer isochromats are inside the voxels). The best and the worst noise scenario can be commonly found on standard cartesian cine [START_REF] Kim | Increasing the signal-to-noise ratio in DENSE MRI by combining displacement-encoded echoes[END_REF] and undersampled [START_REF] Sigfridsson | In vivo SNR in DENSE MRI; temporal and regional effects of field strength, receiver coil sensitivity and flip angle strategies[END_REF] cine DENSE acquisitions with a constant flip angle. The noise SD was estimated as a fraction of the maximum magnitude of the stimulated echo in the k space, which at t=0 shares the same magnitude as the spectral peaks of the cosine modulation (see Equations 3 and 5), and therefore can also be used in SPAMM images.

3D analysis

In this case, just one set of physiological parameters was chosen (see Table 1) to generate slicefollowing versions of the CSPAMM and DENSE images [START_REF] Fischer | True myocardial motion tracking[END_REF][START_REF] Spottiswoode | 3D myocardial tissue tracking with slice followed cine DENSE MRI[END_REF]. For both imaging techniques, the slice thickness of the selective excitation was 8 mm with offsets of 12 and 6 mm for slices at basal and mid cardiac levels, whereas the imaged thickness was 30, 25, and 20 mm for slices at basal, mid, and apical cardiac levels respectively [START_REF] Spottiswoode | 3D myocardial tissue tracking with slice followed cine DENSE MRI[END_REF][START_REF] Stuber | Single breath-hold slice-following CSPAMM myocardial tagging[END_REF][START_REF] Sampath | Automatic 3D tracking of cardiac material markers using slicefollowing and harmonic-phase MRI[END_REF][START_REF] Brotman | Effect of through-plane motion on left ventricular rotation: A study using slice-following harmonic phase imaging[END_REF]. The encoding frequency used for SPAMM was 0.39 rad mm -1 (tag period of 16 mm) and for DENSE 0.75 rad mm -1 . Both images also shared the same FOV of 350 × 350 mm , and were generated using imaging matrices of 256 × 128 for SPAMM and 128 × 64 for DENSE with an oversampling factor of 2 in the measurement direction. A constant flip angle of 15° was used to simulate the acquisition of 20 cardiac phases.

Additionally, as we simulated a cartesian acquisition and only for the 3D data, we added multishot EPI-like artifacts considering a "top-down" acquisition with a receiver bandwidth of 64 , echo-train length of 9, and off-resonance frequency of 115 , as described in [START_REF] Bender | The importance of k-space trajectory on offresonance artifact in segmented echo-planar imaging[END_REF]. This setup generates a shifting artifact in the reconstructed image due to the linear accumulation of phase across the k space. Also, complex Gaussian noise was added to achieve the SNRs given by the NL1 in Table 1.

The imaging parameters were chosen according to standard values given in the literature [START_REF] Spottiswoode | Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting[END_REF][START_REF] Kim | Myocardial Tissue Tracking with Twodimensional Cine Displacement-encoded MR Imaging: Development and Initial Evaluation[END_REF][START_REF] Stuber | Single breath-hold slice-following CSPAMM myocardial tagging[END_REF][START_REF] Tecelão | Extended harmonic phase tracking of myocardial motion: Improved coverage of myocardium and its effect on strain results[END_REF][START_REF] Liu | Shortest path refinement for motion estimation from tagged MR images[END_REF][START_REF] Sigfridsson | Single-breath-hold multiple-slice DENSE MRI[END_REF].

The set of physiological parameters used in this study (following the notation of Gilliam et al. [START_REF] Gilliam | Automated motion estimation for 2-D cine DENSE MRI[END_REF]), which defines the motion at the basal level, are presented in Table 2. : endocardial radius (mm), : epicardial radius (mm), : long axis height (mm), : the skew factor which moves motion towards epicardial ( > 1) or endocardial motion ( < 1), : end-systolic area scaling, : end-systolic endocardial scaling, : end-systolic endocardial twist (°), : end-systolic epicardial twist (°), Ψ: the direction in which the motion is reduced (°), : motion reduction factor, , , : time modulation times (s) A summarized description of the 2D and 3D experiments is shown in Figure 2.

Figure 2:

Using the 2D data, displacement and strain fields are estimated for every resolution and noise level described in the Experiments section. Then, the error metrics given in Equations 8a, 8b, and 8c are evaluated. For the 3D data, displacement and strain fields are estimated and the error metric given in Equation 9 is evaluated.

2.5) Image processing

In Equation 5, the three terms (in order of appearance) are often called Stimulated, Complex Conjugate, and Relaxation echoes (see Figure 3b), centered at (0,0), (2 , 0), and ( , 0) respectively. By using the phase-cycling approach, the relaxation echo is canceled (see Equation 6). However, depending on the pixel size (i.e., k space bandwidth), energy from the complexconjugate echo could be partially or entirely sampled, leading to severe artifacts [START_REF] Kim | Myocardial Tissue Tracking with Twodimensional Cine Displacement-encoded MR Imaging: Development and Initial Evaluation[END_REF]. To correct these artifacts, we applied a Butterworth filter [START_REF] Wang | Cardiac deformation analysis using 3D SinMod from 3D CSPAMM tagged MRI[END_REF] of 10-ℎ order and fixed cutoff frequency was applied to every DENSE image. the image shows the reconstructed magnitude, whereas for DENSE and phase-cycled DENSE, the reconstructed phase. For the image simulation, a multi-shot EPI acquisition was used in all cases, with half of the lines sampled in the phase direction.

2.6) Motion estimation

Using motion estimation techniques, we tracked the CSPAMM and DENSE images to a reference domain at end-diastole. Once the displacement field ( ) was estimated, we calculated the Lagrangian strain tensor [START_REF] Fonseca | Three-dimensional assessment of left ventricular systolic strain in patients with type 2 diabetes mellitus, diastolic dysfunction, and normal ejection fraction[END_REF][START_REF] Auger | Imaging left-ventricular mechanical activation in heart failure patients using cine DENSE MRI: Validation and implications for cardiac resynchronization therapy[END_REF]. In this study, we evaluated the circumferential ( ) and radial (

) components of the tensor, which are usually used to evaluate strain in short-axis views. As our implementations of SP-HR, SinMod, and DENSE analysis only allowed 2D motion estimation, the longitudinal strain component of the tensor was not evaluated.

Motion from CSPAMM images was obtained using a free version of the SP-HR algorithm provided by the Image Analysis and Communication Lab at Johns Hopkins University [START_REF] Liu | Shortest path refinement for motion estimation from tagged MR images[END_REF], and a self-made implementation of SinMod as described by the developers in Arts et al. [START_REF] Arts | Mapping displacement and deformation of the heart with local sine-wave modeling[END_REF]. In contrast, for DENSE images, the MATLAB toolbox DENSEanalysis [START_REF] Spottiswoode | Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting[END_REF]44] was used.

In the case of SinMod analysis, no frequency windowing was applied, and resulting displacements were corrected using the quality model proposed by the authors [START_REF] Arts | Mapping displacement and deformation of the heart with local sine-wave modeling[END_REF] with a weighting matrix with 8 in the exponent. In both cases, SP-HR and SinMod, the same bandpass filter was used, as described by Arts et al. [START_REF] Arts | Mapping displacement and deformation of the heart with local sine-wave modeling[END_REF].

When using DENSEanalysis, a temporal fitting of a 10th-degree polynomial was applied [START_REF] Wang | Accurate 2D cardiac motion tracking using scattered data fitting incorporating phase information from MRI[END_REF],

and displacement resampling was done using the implementation of the gridfit function given in DENSEanalysis [START_REF] Gilliam | Automated motion estimation for 2-D cine DENSE MRI[END_REF]44], with a triangular interpolation scheme with a smoothing factor of 0.8 for noisy data. The same temporal fitting procedure was used for SP-HR and SinMod displacements.

2.7) Statistical analysis

The error was measured using the Normalized Root Square Error (nRMSE) and Directional Error (DE), defined as [START_REF] Mura | Enhancing the Velocity Data From 4D Flow MR Images by Reducing its Divergence[END_REF]:

nRMSE (%) = 100 × 1 max 1 - (8a) nRMSE (%) = 100 × 1 max || || 1 || - || (8b) DE = 1 1 - |⋅ | | | ⋅ | | (8c)
where represents the number of masked pixels in the image, nRMSE (%) and nRMSE (%) are the nRMSE for scalar and vectorial quantities respectively, is any pixelwise scalar quantity at the pixel (e.g., circumferential and radial strains), and is the displacement field at the pixel i.

The superscript ( ) denotes the exact value. The three previous error metrics were evaluated for SP-HR, SinMod, and DENSEanalysis results at end-systole, where displacements have maximum amplitude.

The error metrics defined in Equation ( 8) characterized the mean value of the pixelwise error for just one data. However, the error metrics presented in the next section are the mean value across the whole dataset (N=100) of the metrics estimated using [START_REF] Fischer | Improved myocardial tagging contrast[END_REF].

Finally, to quantify the error in the estimation of regional strain throughout the entire cardiac cycle, we introduce another metric given by:

= 100 × 1 | ( ) -( )| max | ( )| (9)
where the overline denotes the mean value across all the segments, ( ) and ( ) the estimated and exact mean strains at the time , and the number of frames (cardiac phases). Overall, the estimation obtained from DENSE images performed better than the other techniques, whereas using CSPAMM images, better results were obtained for smaller pixel sizes and wavelengths.

3) RESULTS

3.1) Sensitivity analysis: resolution

Figure 4 shows the mean nRMSE and DE values across all the analyzed data without noise. In the absence of noise, smaller pixel sizes improve the estimation of displacements and strain using the three techniques. Regarding the displacement field evaluation, the best performance was achieved by DENSEanalysis, and almost always SP-HR performed better than SinMod (see Figures 3a and3b). For standard resolutions of DENSE and CSPAMM images (in-plane isotropic voxel sizes around 3.0 and 1.5 mm respectively [START_REF] Fischer | Improved myocardial tagging contrast[END_REF][START_REF] Spottiswoode | Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting[END_REF]), the differences in the performance of DENSEanalysis and SP-HR become a bit smaller, whereas with SinMod results always showed larger errors.

A similar tendency occurs for the estimation of strain. For almost every pixel size, results obtained with DENSEanalysis postprocessing were better than SP-HR and SinMod (see Figures 3c and3d).

Although the nRMSE of the obtained with SP-HR growths for tagging periods of 8, 10, and 12 mm, in general terms, SP-HR performed better than SinMod in most of the cases. Additionally, the error did not follow a clear trend for the spacing of 8 mm. A similar behavior was observed for the component, although the nRMSE increased with the three postprocessing techniques, with errors rising to around 31% with SP-HR, 33% with SinMod, and 20% with DENSEanalysis (see Figure 4d).

Figure 5 shows the mean values and standard deviations (between data) of the most favorable cases of the analysis presented in Figure 4, i.e., combinations of tag spacings and pixel sizes where the nRMSE of the component reached the minimum value of all the curves (see Figure 4c). With SinMod, the best results were obtained using a fixed tagging period of 10 mm for every pixel size, while with SP-HR, the best performance was achieved using a tag spacing of 10 mm for pixel sizes of 1, 1.5, and 2 mm, and 14 mm for pixels of 2.5 and 3 mm. Despite the difference in the errors between the three methods, all of them shared similar deviations. 2). SP-HR gave the most significant sensitivity to noise in every case, whereas DENSEanalysis showed the smallest.

3.2) Sensitivity analysis: noise

Figure 6 shows the error metrics calculated from the noisy data using the tagging spacings given in Figure 5. As the noise level increases, the overall performance of three motion estimation techniques become worse (as expected). Moreover, the gap between noise levels becomes higher for smaller pixel sizes as the images with larger k space bandwidth are noisier (see the SNR scaling in Table 1). At any noise level and resolution, the displacement fields were better estimated using DENSEanalysis (see Figures 5a and5b). However, the difference was more evident for the magnitude rather than the direction. For NL3 case and standard isotropic in-plane pixel sizes of 1.5 and 3 mm for CSPAMM and DENSE images, the nRMSE in magnitude and DE were approximately 6.2 ± 2.4% and 6.7 ± 5.3° for SP-HR, 9.2 ± 2.6% and 7.0 ± 4.8° for SinMod, and 3.4 ± 1.4% and 4.4 ± 3.9° for DENSEanalysis. Although the estimations made with DENSEanalysis showed the best performance, the results obtained with SP-HR and SinMod were comparable with DENSEanalysis.

In terms of strains, the behavior of the three methods was similar between the and the components. The errors showed less sensitivity to noise at bigger pixel sizes in both cases, with an increasing trend as the pixel size decreases. However, the error obtained using SinMod showed less sensitivity to noise than SP-HR and DENSEanalysis at any pixel size for both strain components.

For the same pixel sizes and noise level previously mentioned, the nRMSE obtained for the and components were 10. 7 ± 10.8% and 25.5 ± 14.8% using SP-HR, 11. 9 ± 2.5% and 29.3 ± 6.5% using SinMod, and 6.4 ± 2.0% and 18.2 ± 4.6% using DENSEanalysis. These results showed a substantial increase in the error variability with SP-HR, which means that the noise highly impacted the motion estimation under different contraction conditions. As in the previous section, the estimation of the component was worse than the component for all the tested postprocessing methods but was better captured using DENSEanalysis for all noise levels.

For small motions, i.e., early systolic cardiac phases, the error metrics for estimating displacement and strain are presented in Figure 7. At lower motion levels, both motion and strain calculation were worse than at end-systolic cardiac phases. The gap in the error of displacement estimation decreased, and in the case of strain, SP-HR and SinMod performed better than DENSEanalysis for the and partially better for the . That suggests that DENSEanalysis is more sensitive to noise for lower motion levels. components. The left plot of each metric shows the results obtained using SP-HR and to the right using SinMod. Each color denotes a different noise level. As the displacement is small at the beginning of the cardiac cycle, the impact of noise on the phase of harmonic and stimulated echo images becomes more prominent. As a consequence, worse results were achieved for all the error metrics.

3.3) 3D experiment

Figure 8: Regional strains at basal, mid, and apical levels. Every point in the curves represents the mean regional strain across the segments defined by the American Heart Association [START_REF] Cerqueira | Standardized myocardial sementation and nomenclature for tomographic imaging of the heart: A Statement for Healthcare Professionals from the Cardiac Imaging Committee of the Council on Clinical Cardiology of the American Heart Association[END_REF][START_REF] Selvadurai | Definition of Left Ventricular Segments for Cardiac Magnetic Resonance Imaging[END_REF].

Analysis of CSPAMM images using SP-HR tends to fail for cardiac phases in the diastolic part of the cardiac cycle, resulting from the signal decay.

Figure 8 shows the and strain curves obtained from basal, mid, and apical short-axis slices of the phantom shown in Figure 1. In this experiment, the performance of the three methods was interrogated through the whole cardiac cycle. SP-HR showed the most unfavorable performance, especially in the apical region, where the torsion was augmented. The three methods were imprecise in estimating the component, but DENSEanalysis was closer to the reference values.

In contrast, SinMod and DENSEanalysis gave very accurate estimations of the regional through the whole cardiac cycle and at any cardiac level, while SP-HR correctly behaved at basal and mid-levels.

In Table 3, the errors estimated using the metric proposed in Equation ( 9) are presented, which measures the difference between strain curves. Differences calculated from the strain curve estimated with SP-HR increased as the slice moved from base to apex. However, the previous statement is no longer valid for SinMod and DENSEanalysis. SinMod did not exhibit any pattern, whereas DENSEanalys showed decreasing and increasing trends from base to apex. The similarity between the reference and the estimated curves was minimal, and no significant differences were observed between SinMod and DENSEanalysis. However, we noted more significant discrepancies in terms of error for the component, where DENSEanalysis accomplished the best execution (see Table 3).

Table 3:

(%) for the results of the 3D experiment. Results

show how far the regional strains estimated with the three methods are from the reference values. From base to apex, the synthetic phantom rotation increases, and higher errors were found using SP-HR and SinMod.

Postprocessing method

4) DISCUSSION

The tag analysis from Tagging MR images has been considered the gold standard for the estimation of myocardial strain. Several approaches have been developed to estimate motion from these images, being SinMod and SP-HR two of the most used methods [START_REF] Cao | A comparison of both DENSE and feature tracking techniques with tagging for the cardiovascular magnetic resonance assessment of myocardial strain[END_REF]. On the other hand, the analysis of DENSE images using processing tools as DENSEanalysis has become a powerful tool for estimating displacements and strain. Nevertheless, tagging and DENSE techniques remain a research tool [START_REF] Cao | A comparison of both DENSE and feature tracking techniques with tagging for the cardiovascular magnetic resonance assessment of myocardial strain[END_REF], and discussion about their accuracy and precision continues.

The three methods gave accurate estimations of displacements and strains (see Figure 4) in the absence of noise. As the pixel size decreases, DENSEanalysis showed a slight decay in the error metrics of magnitude, direction, and , whereas for the component, the nRMSE increases with smaller pixel sizes. A similar behavior was observed with SP-HR and SinMod for almost any tag spacing and also for the component. Concerning the tag spacings, there was a clear tendency for both SP-HR and SinMod. As the tag spacing increases, the estimation of displacement and strain deteriorates, obtaining the most significant errors with a tag spacing of 14 mm. However, for SP-HR, the smallest tag spacing did not work adequately for any pixel sizes. We can explain this behavior by comparing the pixel size, tag spacing, and amount of motion [START_REF] Osman | Cardiac motion tracking using CINE harmonic phase (HARP) magnetic resonance imaging[END_REF][START_REF] Osman | Imaging heart motion using harmonic phase MRI[END_REF], which says that HARP-based techniques tend to fail for motions larger than the tag spacing. Additionally, for this tag spacing and the imaging parameters used, the spectral peak containing motion information was too close to the k space bandwidth, leading to information loss.

We decided not to include field inhomogeneities in the generation of the images because we did not observe considerable differences in the error metrics and their behavior. This was tested in the same experiment used to evaluate the behavior of the three techniques under nose-free conditions, but adding a smooth and spatially-varying phase to each acquisition given in Equations 3 and 5.

Moreover, the average increase in the error metrics for both displacement and strain was around 2%, while keeping their behavior observed in Figure 4. The results obtained from noise-free data with field inhomogeneities are presented in Figure S1 (see supplementary material).

In our experiments, estimations made from DENSE images showed a better behavior than SP-HR and SinMod for the quantification of displacements and strain from noisy data. For the range of pixel sizes usually acquired in DENSE images (2.5 to 3 mm), DENSEanalysis was less sensitive to noise than SP-HR and SinMod, for the range of resolutions usually acquired in CSPAMM images (1.5 mm) (see Figure 6). This behavioral dependency on the voxel size is explained with the SNR reduction as the voxels become small, i.e., the kspace bandwidth becomes bigger.

Additionally, the bandpass filters used in SP-HR and SinMod did not remove the high-frequency noise. In contrast, the Butterworth filter removed the remaining energy of the complex-conjugate echo in DENSE images and eliminated high-frequency noise.

In this study, the resolution and noise sensitivity analysis were performed mainly at the end of systole, where the displacement and strain have maximum amplitude. However, due to the signal decay and the low phase SNR in both CSPAMM and DENSE images, the estimation can be biased during early systolic and late diastolic cardiac phases, which could be determinant when smaller strains need to be measured [START_REF] Auger | Imaging left-ventricular mechanical activation in heart failure patients using cine DENSE MRI: Validation and implications for cardiac resynchronization therapy[END_REF]. Figure 7 shows the errors in the estimation of displacement and strain at early systole. Compared with Figure 5 (results at end-systole), the three methods showed a worse performance, leading to higher errors in the estimation of motion and strain with increased noise sensitivity. However, SP-HR and SinMod gave better results than DENSEanalysis for the estimation of strain, showing that at smaller motion levels, SP-HR and SinMod, are more accurate and less sensitive to noise.

Although the three methods were able to estimate accurately the regional component during the whole cardiac cycle (see Figure 8) at any cardiac level, neither SP-HR, SinMod, and DENSEanalysis were able to estimate appropriately the component (see Figures 6,7 and 8), differing severely between techniques even in the absence of noise (see Figure 4). The last finding has been previously reported as an issue shared by many motion estimation techniques [START_REF] Cao | A comparison of both DENSE and feature tracking techniques with tagging for the cardiovascular magnetic resonance assessment of myocardial strain[END_REF] and needs to be further studied.

When tested under realistic acquisition and motion conditions, the motion estimated with SP-HR failed in apical levels, where the torsion of the phantom was larger than the basal and mid-levels.

Torsion augmentation implies an increment of relative displacement between frames, causing more errors in HARP-based methods [START_REF] Liu | Shortest path refinement for motion estimation from tagged MR images[END_REF]. This behavior was observed in either SinMod and DENSEanalysis. Additionally, the three methods estimated accurately the mean regional strain component (except for SP-HR at the apex) but not the (see Figure 8). Furthermore, the estimation of the made using SP-HR and SinMod differed severely from the reference values, while DENSEanalysis worked significantly better.

As a side product of this study, we developed an open-source and flexible Python library to generate synthetic CSPAMM and DENSE images from 3D phantoms. Our library also include variables such as field inhomogeneities, dynamic flip angles, and EPI-like artifacts (among others)

to each imaging mode. Another feature is that it can be easily modified to add new imaging techniques. In our case, as we are interested in studying the estimation of motion and strain, future work is the implementation of the Strain-Encoded (SENC) sequence [START_REF] Osman | Imaging longitudinal cardiac strain on shortaxis images using strain-encoded MRI[END_REF].

The estimation of motion from both CSPAMM and DENSE images was chosen to consider its similarities related to the MR pulse sequence. Although the DENSE sequence is not yet available across all platforms, its development as a research tool has converted it into a powerful technique for estimating motion. We chose SP-HR, SinMod, and DENSEanalysis as postprocessing techniques because they have implementations commercially and freely distributed, and therefore, have been widely used in the MR community [START_REF] Ibrahim | Myocardial tagging by Cardiovascular Magnetic Resonance: evolution of techniques--pulse sequences, analysis algorithms, and applications[END_REF]. Regarding the postprocessing techniques, although SinMod was implemented following as exact as possible, the steps and algorithms proposed by the developers, the possibility of some variability between our implementation and commercially available software should be considered.

In conclusion, SinMod and DENSEanalysis showed excellent and comparable results for the estimation of displacements and strain from CSPAMM and DENSE data when we used typical image resolutions and imaging parameters. In contrast, SP-HR tends to fail for large amplitude motions, although it worked well in any other case. Additionally, we showed that the three techniques could not accurately estimate the radial strain component accurately, even when motion estimated from DENSE images using DENSEanalysis showed the best performance. Overall, the estimation obtained from DENSE images performed better than the other techniques, whereas using CSPAMM images, better results were obtained for smaller pixel sizes and wavelengths. The left plot of each metric shows the results obtained using SP-HR and to the right using SinMod. The results from DENSEanalysis are shown in all the figures. Each color denotes a different SNR (see Table 2). SP-HR gave the most significant sensitivity to noise in every case, whereas DENSEanalysis showed the smallest. components. The left plot of each metric shows the results obtained using SP-HR and to the right using SinMod. Each color denotes a different noise level. As the displacement is small at the beginning of the cardiac cycle, the impact of noise on the phase of harmonic and stimulated echo images becomes more prominent. As a consequence, worse results were achieved for all the error metrics.

Figure 8: Regional strains at basal, mid, and apical levels. Every point in the curves represents the mean regional strain across the segments defined by the American Heart Association [START_REF] Cerqueira | Standardized myocardial sementation and nomenclature for tomographic imaging of the heart: A Statement for Healthcare Professionals from the Cardiac Imaging Committee of the Council on Clinical Cardiology of the American Heart Association[END_REF][START_REF] Selvadurai | Definition of Left Ventricular Segments for Cardiac Magnetic Resonance Imaging[END_REF].

Analysis of CSPAMM images using SP-HR tends to fail for cardiac phases in the diastolic part of the cardiac cycle, resulting from the signal decay.

Figure 1 :

 1 Figure 1: Representation of the synthetic geometry and slices used to generate the images. (a)

Figure 3 :

 3 Figure 3: k-space and reconstructed images for (a) SPAMM and CSPAMM, and (b) DENSE

Figure 4 :

 4 Figure 4: Results for the resolution sensitivity analysis. Mean errors in (a) magnitude, (b)

Figure 5 :

 5 Figure 5: Mean errors and standard deviations for the best performing combination of tag

Figure 6 :

 6 Figure 6: Results for the noise sensitivity analysis for an end-systolic cardiac phase. Mean

Figure 7 :

 7 Figure 7: Results for the noise sensitivity analysis for an early-systolic cardiac phase. Mean

Figure captions:Figure 1 :

 1 Figure captions:

Figure 2 :

 2 Figure 2: Using the 2D data, displacement and strain fields are estimated for every resolution

Figure 3 :

 3 Figure 3: k-space and reconstructed images for (a) SPAMM and CSPAMM, and (b) DENSE

Figure 4 :

 4 Figure 4: Results for the resolution sensitivity analysis. Mean errors in (a) magnitude, (b)

Figure 5 :

 5 Figure 5: Mean errors and standard deviations for the best performing combination of tag

Figure 6 :

 6 Figure 6: Results for the noise sensitivity analysis for an end-systolic cardiac phase. Mean errors

Figure 7 :

 7 Figure 7: Results for the noise sensitivity analysis for an early-systolic cardiac phase. Mean

Table 1 :

 1 Noise

	Name Noise level Early-systolic SNR (× )	Late-diastolic SNR (× )
	NL0	0	noise-free	noise-free
	NL1	1	33.6	8.4
	NL2	2	27.6	7.0
	NL3	3	21.4	5.5

levels and corresponding Signal to Noise Ratios in the noise sensitivity analysis images. The noise level has the same standard deviation throughout the cardiac cycle. However, from early-systole to late-diastole, the SNR decreases due to the signal model given in Equations (

4

) and

[START_REF] Petitjean | Assessment of myocardial function: A review of quantification methods and results using tagged MRI[END_REF]

.

Table 2 :

 2 physiological parameters used for the generation of images with 3D motion patterns

	Ψ

ACKNOWLEDGMENTS

The authors thankfully acknowledge the financial support of the ANID Ph. D. Scholarship 21170592 as well as the following projects and institutions: ANID-FONDECYT 1181057, ANID-FONDECYT Postdoctorado 3170737, ANID-FONDECYT de Iniciación en Investigación 11200481, ANID-Millennium Science Initiative Program -NCN17_129.