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How to Assign Multiple Variables in MATLAB in One Line

How to Assign Multiple Variables at Once in MATLAB

MATLAB is a powerful programming language for engineers and scientists. It’s known for its fast, efficient, and easy-to-use syntax. One of the things that makes MATLAB so popular is its ability to assign multiple variables at once. This can save you a lot of time and effort, especially when you’re working with large datasets.

In this article, we’ll show you how to assign multiple variables at once in MATLAB. We’ll also provide some examples to help you understand the process. So if you’re ready to learn how to assign multiple variables at once in MATLAB, keep reading!

In MATLAB, you can assign multiple variables to the same value at once using the following syntax:

var1 = var2 = var3 = … = value

This is a very convenient way to assign the same value to multiple variables, and it can save you time and typing.

Syntax for assigning multiple variables at once

The syntax for assigning multiple variables at once in MATLAB is as follows:

Where `var1`, `var2`, `var3`, etc. are the variables to be assigned, and `value` is the value to be assigned to the variables.

For example, the following code assigns the value `10` to the variables `x`, `y`, and `z`:

x = y = z = 10;

Benefits of assigning multiple variables at once

There are several benefits to assigning multiple variables at once in MATLAB.

• It can save you time and typing. If you have multiple variables that you need to assign the same value to, you can do so with a single line of code instead of three.
• It can make your code more readable and easier to understand. When you assign multiple variables at once, it is clear that the variables are all being assigned the same value. This can help to prevent errors and make it easier to debug your code.
• It can be used to create arrays. When you assign multiple variables to the same value, you are essentially creating an array. This can be useful for storing data or performing calculations.

Examples of assigning multiple variables at once

The following are some examples of assigning multiple variables at once in MATLAB:

• To assign the value `10` to the variables `x`, `y`, and `z`, you would use the following code:
• To assign the values `1`, `2`, and `3` to the variables `a`, `b`, and `c`, you would use the following code:

a = b = c = [1, 2, 3];

• To assign the values of the vector `v` to the variables `x`, `y`, and `z`, you would use the following code:

x = y = z = v;

Assigning multiple variables at once in MATLAB can be a very convenient way to save time and typing, and it can also make your code more readable and easier to understand. There are many different ways to assign multiple variables at once, and the best way to do it will depend on your specific needs.

3. Limitations of assigning multiple variables at once

There are some limitations to assigning multiple variables at once in MATLAB.

• You cannot assign different values to the variables. For example, if you try to assign the value `10` to the variable `x` and the value `20` to the variable `y`, MATLAB will throw an error.
• You cannot assign variables of different types. For example, you cannot assign the value `10` to the variable `x`, which is a numeric variable, and the string `”Hello”` to the variable `y`, which is a character variable.

These limitations are in place to prevent errors and to ensure that your code is readable and understandable.

4. Examples of assigning multiple variables at once

d = e = f = {‘a’, ‘b’, ‘c’};

In the first example, we are assigning the value `10` to the three variables `x`, `y`, and `z`. In the second example, we are assigning the vector `[1, 2, 3]` to the three variables `a`, `b`, and `c`. In the third example, we are assigning the cell array `{‘a’, ‘b’, ‘c’}` to the three variables `d`, `e`, and `f`.

These are just a few examples of how you can assign multiple variables at once in MATLAB. For more information, please refer to the MATLAB documentation.

In this tutorial, you learned how to assign multiple variables at once in MATLAB. You learned about the limitations of assigning multiple variables at once, and you saw some examples of how you can use this technique in your own code.

I hope you found this tutorial helpful. Thank you for reading!

Q: How do I assign multiple variables at once in MATLAB?

A: There are a few ways to assign multiple variables at once in MATLAB.

• Using the comma operator: You can use the comma operator (,) to separate multiple variable names on the right-hand side of an assignment statement. For example, the following code will assign the values 1, 2, and 3 to the variables `a`, `b`, and `c`, respectively:

a, b, c = 1, 2, 3;

• Using the cell array: You can also use a cell array to assign multiple values to multiple variables. A cell array is a special type of array that can hold multiple data types, including strings, numbers, and other arrays. To create a cell array, you can use the `cell()` function. For example, the following code will create a cell array that contains the values 1, 2, and 3:

v = cell(3); v{1} = 1; v{2} = 2; v{3} = 3;

Once you have created a cell array, you can assign it to multiple variables using the `==` operator. For example, the following code will assign the values in the cell array `v` to the variables `a`, `b`, and `c`, respectively:

a = v{1}; b = v{2}; c = v{3};

• Using the `assign()` function: You can also use the `assign()` function to assign multiple values to multiple variables. The `assign()` function takes two arguments: the first argument is a cell array of variable names, and the second argument is a cell array of values. For example, the following code will assign the values in the cell array `v` to the variables `a`, `b`, and `c`, respectively:

assign({‘a’, ‘b’, ‘c’}, v);

Q: What are the advantages of assigning multiple variables at once?

A: There are a few advantages to assigning multiple variables at once.

• It can be more concise and efficient than assigning variables one at a time.
• It can reduce the chance of making a mistake when assigning variables.
• It can make your code more readable and easier to understand.

Q: Are there any disadvantages to assigning multiple variables at once?

A: There are no significant disadvantages to assigning multiple variables at once. However, you should be aware of the following potential issues:

• If you are not careful, you could accidentally assign the wrong value to a variable.
• If you are using a cell array to assign multiple values to multiple variables, you need to make sure that the cell array has the same number of elements as the number of variables you are assigning values to.

Q: What are some common mistakes people make when assigning multiple variables at once?

A: Some common mistakes people make when assigning multiple variables at once include:

• Accidentally assigning the wrong value to a variable.
• Forgetting to use the comma operator (,) to separate multiple variable names on the right-hand side of an assignment statement.
• Using a cell array with the wrong number of elements.

Q: How can I avoid making mistakes when assigning multiple variables at once?

A: To avoid making mistakes when assigning multiple variables at once, you can follow these tips:

• Be careful when typing in the values of the variables. Make sure that you are not accidentally typing in the wrong value.
• Use the comma operator (,) to separate multiple variable names on the right-hand side of an assignment statement. This will help you to avoid accidentally assigning the same value to multiple variables.

We hope that this blog post has been helpful. If you have any questions or comments, please feel free to leave them below.

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MATLAB – Variables

Prerequisite: Getting Started with MATLAB

A variable in simple terms is a storage place that has some memory allocated to it. Basically, a variable used to store some form of data. Different types of variables require different amounts of memory and have some specific set of operations that can be applied to them. The Matlab workspace store all the variables that you create or use during a session. 

Creating Variables

To create a variable enter the name of the variable in the command window, followed by an = operator, and then assign it some values .

Example: 

  Output:

  Note:  

• If you don’t put ‘;’ after the variable name, Matlab will display its content after hitting enter.
• If you don’t give any name to your variable by default Matlab name it with an answer.
• By default, Matlab treats all variables as matrices if you write just 1 it will store it as 1×1 matrix.

To display the content of the Matlab variable you just need to type the name of the variable, and it will show its content on Command-Line:

  Example:

Multiple Assignments of Variables

We can also define the multiple variables in a single line using “;” operator.

Who command

Who lists in alphabetical order the names of all variables in the currently active workspace. To use this command variables should be present in the active workspace and memory allocated to each variable is necessary.

Assigning Vectors to variables

A vector is a one-dimensional array of numbers. Matlab allows you two types of vector: 

• Column Vector.

Row Vectors are created by enclosing numbers in square brackets separating with either space or comma.

Column Vectors are created by enclosing numbers in square brackets, separating each number with ;(semicolon).

Assigning Matrices to variables

A Matrix is a two-dimensional array of numbers. In Matlab, matrices are created by enclosing numbers in a square bracket write each row of numbers with space or comma separate and after each row put a semicolon. The below code will create a 3×3 matrix:

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Assignment statements.

The preceding MATLAB commands that assign the value of the expression after the ’=’ sign  to the variable before the ’=’ sign are assignment statements. Note that all variables in the expression after the ’=’ sign must have previously been allocated a value, or else an error occurs. For example, enter the following commands:

c=sqrt(a^2+b^2)

You will see the error message ??? Undefined function or variable ’b’ .  Consider the following:

The last line has nothing at all to do with a mathematical equation. It is a MATLAB  assignment statement that calculates x 2 −12 at x = 7 and stores the result in the variable x , thereby over-writing the previous value.

• MATLAB Assignment Mastery: A Student's Handbook

Avoiding Common Pitfalls in MATLAB Homework: A Student's Handbook

MATLAB, renowned as a potent programming language extensively applied in academic and industrial realms, is an indispensable tool for students across diverse disciplines. Whether you are venturing into MATLAB for the first time or are a seasoned user, the journey of grappling with MATLAB assignments can prove to be a formidable task. This handbook is crafted with the primary objective of providing indispensable help with MATLAB assignment , steering students through the intricacies and potential pitfalls encountered during their coursework. In recognition of the challenges that accompany MATLAB homework, the handbook seeks to empower students with valuable insights and strategies that transcend mere syntax comprehension. By delving into the nuances of common pitfalls and furnishing actionable tips, this guide aims to fortify students' grasp of MATLAB, thereby augmenting their overall proficiency in the language. With the knowledge gleaned from this resource, students can confidently navigate the intricacies of MATLAB assignments, transforming what might seem daunting into manageable endeavors. As they internalize the strategies elucidated herein, students will find themselves better equipped to not only overcome challenges but also to approach MATLAB assignments with increased efficiency and a deeper understanding of the language's intricacies.

Understanding the Basics of MATLAB

When it comes to MATLAB, a strong foundation in its basics is crucial. Many students face difficulties because they dive into assignments without a solid understanding of the fundamentals. To avoid this pitfall, start by grasping the core concepts of MATLAB, including variables, arrays, functions, and control structures.

Importance of Variables and Arrays

Assigning variables and working with arrays are fundamental aspects of MATLAB. One common mistake is not paying enough attention to variable naming conventions and mismanaging arrays. Always use meaningful variable names and understand the importance of array indexing to avoid confusion in your code.

Mastering MATLAB Functions

Functions are building blocks in MATLAB, and knowing how to use them correctly is essential. Students often struggle when they are not familiar with the available functions or misuse them. Take the time to explore MATLAB's documentation and practice using functions to gain confidence in your coding skills.

Grasping Control Structures

Control structures, such as loops and conditional statements, play a vital role in MATLAB programming. Misusing or misunderstanding these structures can lead to errors and inefficient code. Ensure that you are well-versed in how control structures work and practice incorporating them into your assignments.

Common Pitfalls in MATLAB Homework Assignments

As students advance through their MATLAB coursework, they inevitably confront distinct challenges inherent to the language. Recognizing and proactively tackling these prevalent pitfalls can markedly enhance the overall outcomes of their assignments. MATLAB, with its diverse capabilities, often presents hurdles that demand a nuanced understanding. By acknowledging the common stumbling blocks associated with the language, students position themselves to navigate through complexities more effectively. Addressing these challenges head-on not only refines problem-solving skills but also contributes to the development of robust and efficient MATLAB code. In essence, a heightened awareness of these language-specific pitfalls empowers students to approach assignments with a strategic mindset, fostering a more seamless and rewarding learning experience in the realm of MATLAB programming.

Overlooking Vectorization Opportunities

MATLAB is designed to handle vectorized operations efficiently. However, students often miss opportunities to leverage vectorization, leading to slower and less elegant code. Embrace the power of vectorization by replacing loops with array operations wherever possible, enhancing both readability and performance.

Neglecting Error Handling

Error handling is a crucial aspect of writing robust code in MATLAB. Ignoring potential errors or not implementing adequate error-checking mechanisms can result in code that is prone to crashes or produces incorrect results. Prioritize error handling by validating inputs, checking for potential issues, and providing informative error messages.

Failing to Plan and Debug Effectively

Jumping straight into coding without proper planning can lead to confusion and inefficiencies. Take the time to plan your approach, break down the problem into smaller tasks, and debug incrementally. Utilize MATLAB's debugging tools, such as breakpoints and the command window, to identify and fix errors systematically.

Ignoring Code Documentation

Documentation is often an afterthought for students engrossed in solving complex problems. However, well-documented code is essential for both the programmer and anyone else who may review or use the code. Include comments to explain your thought process, document function inputs and outputs, and maintain a clear and organized structure to enhance the readability of your code.

Strategies for Success in MATLAB Assignments

Now that we've identified the stumbling blocks commonly encountered in MATLAB assignments, let's chart a course towards strategies that will markedly augment your expertise in handling these challenges. These strategic approaches are meticulously crafted to elevate your coding proficiency, promising a more seamless and triumphant journey as you grapple with intricate problems in MATLAB. By implementing these strategies, you can not only overcome hurdles but also enhance your overall understanding and efficiency in MATLAB, ultimately empowering yourself to tackle complex assignments with confidence and precision. Embrace these tailored approaches as indispensable tools in your toolkit, paving the way for a more rewarding and successful experience in the realm of MATLAB programming.

Embracing Continuous Learning

MATLAB is a versatile language, and new features are regularly introduced. Keep yourself updated with the latest developments, explore advanced functionalities, and seek additional resources beyond your coursework. Online tutorials, forums, and MATLAB documentation can be valuable tools for continuous learning.

Collaborating and Seeking Help

Don't hesitate to collaborate with peers or seek help from instructors or online communities. MATLAB has a vast user base, and chances are someone has encountered and solved a problem similar to yours. Engaging in discussions and sharing your challenges can provide fresh perspectives and solutions.

Efficient Time Management

Effective time management is crucial when working on MATLAB assignments. Start early, break down the tasks into manageable chunks, and allocate sufficient time for testing and debugging. Avoid the last-minute rush, as it increases the likelihood of errors and decreases the quality of your work.

Building a Code Repository

Create a repository for your MATLAB code using version control systems like Git. This not only helps you track changes in your code but also serves as a backup in case of accidental deletions or errors. Additionally, maintaining a repository facilitates collaboration with peers and showcases your coding skills to potential employers.

Advanced Techniques to Enhance MATLAB Assignment Efficiency

As students advance in their MATLAB coursework, the exploration of advanced techniques becomes a critical component in elevating the efficiency and effectiveness of their assignments. Delving into more sophisticated approaches is not only advisable but imperative for a comprehensive mastery of the language. This phase of learning encourages students to move beyond the basics and embrace advanced methodologies that can streamline their coding processes.

In this context, the utilization of advanced techniques in MATLAB goes beyond the conventional methods learned in introductory stages. Students are encouraged to leverage more intricate features and functionalities offered by MATLAB to tackle complex problems with greater finesse. By incorporating these advanced techniques, students can optimize their code for improved performance, enhance the scalability of their solutions, and develop a deeper understanding of the language's capabilities.

Furthermore, embracing advanced techniques instills a sense of innovation and adaptability, essential qualities for aspiring MATLAB practitioners. It opens up new avenues for problem-solving, encourages creative thinking, and empowers students to explore the full spectrum of MATLAB's capabilities. In essence, the exploration of advanced techniques marks a crucial juncture in a student's MATLAB journey, propelling them towards a more nuanced and proficient level of programming expertise.

Leveraging MATLAB Toolboxes

MATLAB offers a plethora of toolboxes tailored for specific applications such as signal processing, image processing, and machine learning. Failing to utilize these toolboxes can lead to reinventing the wheel and unnecessary complexity in your code. Familiarize yourself with relevant toolboxes and integrate them into your assignments when appropriate, saving time and improving code quality.

Optimizing Code for Performance

Efficient code is not just about functionality but also about performance. MATLAB provides profiling tools that allow you to identify bottlenecks in your code and optimize critical sections. Techniques such as preallocating arrays, minimizing unnecessary calculations, and using built-in functions can significantly improve the speed and resource utilization of your code.

Incorporating Object-Oriented Programming (OOP) Principles

Moving from procedural to object-oriented programming (OOP) poses a notable challenge for students, yet it promises substantial benefits in the context of MATLAB assignments. Integrating OOP principles into your MATLAB projects facilitates the creation of code that is not only modular and scalable but also remarkably efficient. The essence lies in encapsulating functionality within classes and objects.

In the realm of MATLAB, adopting OOP empowers students to organize their code more systematically. Classes act as containers for related functions, providing a structured and intuitive framework. This modular approach not only enhances code readability but also simplifies the debugging process. Moreover, the reusability aspect of OOP shines through – once a class is defined, it can be utilized across multiple projects, saving time and effort.

Scalability becomes a notable advantage as well. As MATLAB assignments grow in complexity, OOP allows for the seamless addition of new features without compromising the integrity of existing code. This adaptability is particularly valuable in academic and professional settings where projects often evolve over time.

In summary, while the transition to object-oriented programming may pose initial challenges, embracing OOP principles in MATLAB assignments ultimately results in code that is well-organized, reusable, and scalable, offering a robust foundation for tackling diverse programming challenges.

Exploring Parallel Computing

For computationally intensive tasks, MATLAB provides parallel computing capabilities to leverage multicore processors. Neglecting these features can result in suboptimal performance, especially when dealing with large datasets or complex simulations. Explore parallel computing options such as parfor loops and parallelized functions to harness the full power of your hardware.

Automating Repetitive Tasks with Scripts

MATLAB allows the creation of scripts to automate repetitive tasks, making your workflow more efficient. Students sometimes overlook the potential of scripting, opting for manual execution of commands. By creating well-organized and documented scripts, you not only save time but also ensure consistency and repeatability in your assignments.

Addressing Numerical Precision Issues

Numerical precision can be a source of unexpected errors in MATLAB assignments, especially when dealing with floating-point arithmetic. Failing to address precision issues can lead to inaccuracies in calculations. Be mindful of the limitations of numerical precision, use appropriate data types, and employ techniques such as symbolic computing when dealing with sensitive numerical operations.

Integrating Data Visualization Techniques

Effective communication of results is crucial in MATLAB assignments. Neglecting data visualization techniques can make it challenging for others to understand your findings. Explore MATLAB's extensive plotting capabilities to create informative and visually appealing graphs. Incorporate annotations, legends, and proper labeling to enhance the interpretability of your visualizations.

In the journey of mastering MATLAB for assignments, proficiency goes beyond mere syntax comprehension; it necessitates the assimilation of sound programming practices. By directing attention to fundamental concepts, tackling prevalent pitfalls head-on, and employing effective strategies, students forge a pathway to confidently navigate the intricate terrain of MATLAB homework assignments. Embracing the iterative nature of learning, where each challenge becomes a stepping stone, is pivotal. The commitment to honing one's skills extends beyond solo efforts — seeking assistance when confronted with complexities is a strength, not a weakness. This collaborative spirit taps into the rich community of MATLAB users, providing diverse perspectives and solutions.

In this dynamic landscape of coding, continuous refinement is the hallmark of proficiency. Just as a craftsman hones their skills through practice, so too does a MATLAB programmer through consistent coding and problem-solving. The commitment to improvement is not only a personal endeavor but also a professional asset. As students cultivate their coding skills, they contribute to a broader culture of innovation and problem-solving within the MATLAB community.

In essence, mastery in MATLAB is a holistic journey that intertwines theoretical understanding with practical application, persistence with adaptability. By embracing this multifaceted approach, students not only conquer the challenges posed by assignments but also lay the foundation for becoming adept MATLAB programmers capable of navigating diverse coding landscapes. The call to action is clear: immerse yourself in the learning process, collaborate with peers and mentors, and let each assignment serve as a stepping stone toward a journey of perpetual growth and expertise in the realm of MATLAB programming.

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Introduction to Assignment Methods in Tracking Systems

In a multiple target tracking (MTT) system, one or more sensors generate multiple detections from multiple targets in a scan. To track these targets, one essential step is to assign these detections correctly to the targets or tracks maintained in the tracker so that these detections can be used to update these tracks. If the number of targets or detections is large, or there are conflicts between different assignment hypotheses, assigning detections is challenging.

Depending on the dimension of the assignment, assignment problems can be categorized into:

2-D assignment problem – assigns n targets to m observations. For example, assign 5 tracks to 6 detections generated from one sensor at one time step.

S-D assignment problem – assigns n targets to a set ( m 1 , m 2 , m 3 , …) of observations. For example, assign 5 tracks to 6 detections from one sensor, and 4 detections from another sensor at the same time. This example is a typical 3-D assignment problem.

To illustrate the basic idea of an assignment problem, consider a simple 2-D assignment example. One company tries to assign 3 jobs to 3 workers. Because of the different experience levels of the workers, not all workers are able to complete each job with the same effectiveness. The cost (in hours) of each worker to finish each job is given by the cost matrix shown in the table. An assignment rule is that each worker can only take one job, and one job can only be taken by one worker. To guarantee efficiency, the object of this assignment is to minimize the total cost.

Since the numbers of workers and jobs are both small in this example, all the possible assignments can be obtained by enumeration, and the minimal cost solution is highlighted in the table with assignment pairs (1, 3), (2, 2) and (3, 1). In practice, as the size of the assignment becomes larger, the optimal solution is difficult to obtain for 2-D assignment. For an S-D assignment problem, the optimal solution may not be obtainable in practice.

2-D Assignment in Multiple Target Tracking

In the 2-D MTT assignment problem, a tracker tries to assign multiple tracks to multiple detections. Other than the dimensionality challenge mentioned above, a few other factors can significantly change the complexity of the assignment:

Target or detection distribution — If targets are sparsely distributed, associating a target to its corresponding detection is relatively easy. However, if targets or detections are densely distributed, assignments become ambiguous because assigning a target to a detection or another nearby detection rarely makes any differences on the cost.

Probability of detection ( P d ) of the sensor — P d describes the probability that a target is detected by the sensor if the target is within the field of view of the sensor. If the P d of a sensor is small, then the true target may not give rise to any detection during a sensor scan. As a result, the track represented by the true target may steal detections from other tracks.

Sensor resolution — Sensor resolution determines the sensor’s ability to distinguish the detections from two targets. If the sensor resolution is low, then two targets in proximity may only give rise to one detection. This violates the common assumption that each detection can only be assigned to one track and results in unresolvable assignment conflicts between tracks.

Clutter or false alarm rate of the sensor — False alarms introduce additional possible assignments and therefore increase the complexity of data assignment.

The complexity of the assignment task can determine which assignment methods to apply. In Sensor Fusion and Tracking Toolbox™ toolbox, three 2-D assignment approaches are employed corresponding to three different trackers:

trackerGNN — adopts a global nearest data assignment approach

trackerJPDA — adopts a joint probability data association approach

trackerTOMHT — adopts a tracker-oriented multiple hypothesis tracking approach

Note that each tracker processes the data from sensors sequentially, meaning that each tracker only deals with the assignment problem with the detections of one sensor at a time. Even with this treatment, there may still be too many assignment pairs. To reduce the number of track and detection pairs considered for assignment, the gating technique is frequently used.

Gating is a screening mechanism to determine which observations are valid candidates to update existing tracks and eliminate unlikely detection-to-track pairs using the distribution information of the predicted tracks. To establish the validation gate for a track at the current scan, the estimated track for the current step is predicted from the previous step.

For example, a tracker confirms a track at time t k and receives several detections at time t k +1 . To form a validation gate at time t k +1 , the tracker first needs to obtain the predicted measurement as:

y ^ k + 1 = h ( x ^ k + 1 | k )

where x ^ k + 1 | k is the track estimate predicted from time t k and h ( x ^ k + 1 | k ) is the measurement model that outputs the expected measurement given the track state. The observation residual vector is

y ˜ = y k + 1 − y ^ k + 1

where y k +1 is the actual measurement. To establish the boundary of the gate, the detection residual covariance S is used to form an ellipsoidal validation gate. The ellipsoidal gate that establishes a spatial ellipsoidal region in the measurement space is defined in Mahalanobis distance as:

d 2 ( y k + 1 ) = y ˜ T S − 1 y ˜ ≤ G

where G is the gating threshold which you can specify based on the assignment requirement. Increasing the threshold can incorporate more detections into the gate.

After the assignment gate is established for each track, the gating status of each detection y i ( i = 1,…, n ) can be determined by comparing its Mahalanobis distance d 2 ( y i ) with the gating threshold G . If d 2 ( y i ) < G , then detection y i is inside the gate of the track and will be considered for association. Otherwise, the possibility of the detection associated with the track is removed. In Figure 1, T 1 represents a predicted track estimate, and O 1 – O 6 are six detections. Based on the gating result, O 1 , O 2 , and O 3 are within the validation gate in the figure.

Global Nearest Neighbor (GNN) Method

The GNN method is a single hypothesis assignment method. For each new data set, the goal is to assign the global nearest observations to existing tracks and to create new track hypotheses for unassigned detections.

The GNN assignment problem can be easily solved if there are no conflicts of association between tracks. The tracker only needs to assign a track to its nearest neighbor. However, conflict situations (see Figure 2) occur when there is more than one observation within a track’s validation gate or an observation is in the gates of more than one track. To resolve these conflicts, the tracker must evaluate a cost matrix.

The elements of a cost matrix for the GNN method includes the distance from tracks to detections and other factors you might want to consider. For example, one approach is to define a generalized statistical distance between observation j to track i as:

C i j = d i j + ln ( | S i j | )

where d ij is the Mahalanobis distance and ln(| S ij |), the logarithm of the determinant of the residual covariance matrix, is used to penalize tracks with greater prediction uncertainty.

For the assignment problem given in Figure 2, the following table shows a hypothetical cost matrix. The nonallowed assignments, which failed the gating test, are denoted by X. (In practice, the costs of nonallowed assignments can be denoted by large values, such as 1000.)

For this problem, the highlighted optimal solution can be found by enumeration. Detection O 3 is unassigned, so the tracker will use it to create a new tentative track. For more complicated GNN assignment problems, more accurate formulations and more efficient algorithms to obtain the optimal or suboptimal solution are required.

A general 2-D assignment problem can be formed as following. Given the cost matrix element C ij , find an assignment Z = { z ij } that minimizes

J = ∑ i = 0 n ∑ j = 0 m C i j z i j

subject to two constraints:

∑ i = 0 m z i j = 1 , ∀ j ∑ j = 0 n z i j = 1 , ∀ i

If track i is assigned to observation j , then z ij = 1. Otherwise, z ij = 0. z i 0 = 1 represents the hypothesis that track i is not assigned to any detection. Similarly, z 0 j = 1 represents the hypothesis that observation j is not assigned to any track. The first constraint means each detection can be assigned to no more than one track. The second constraint means each track can be assigned to no more than one detection.

Sensor Fusion and Tracking Toolbox provides multiple functions to solve 2-D GNN assignment problems:

assignmunkres – Uses the Munkres algorithm, which guarantees an optimal solution but may require more calculation operations.

assignauction – Uses the auction algorithm, which requires fewer operations but can possibly converge on an optimal or suboptimal solution.

assignjv – Uses the Jonker-Volgenant algorithm, which also converges on an optimal or suboptimal solution but usually with a faster converging speed.

In trackerGNN , you can select the assignment algorithm by specifying the Assignment property.

K Best Solutions to the 2-D Assignment Problem

Because of the uncertainty nature of assignment problems, only obtaining a solution (optimal or suboptimal) may not be sufficient. To account for multiple hypotheses about the assignment between tracks and detections, multiple suboptimal solutions are required. These suboptimal solutions are called K best solutions to the assignment problem.

The K best solutions are usually obtained by varying the solution obtained by any of the previously mentioned assignment functions. Then, at the next step, the K best solution algorithm removes one track-to-detection pair in the original solution and finds the next best solution. For example, for this cost matrix:

[ 10 5 8 9 7 × 20 × × × 21 1 5 × 17 × × × × 1 6 22 ]

each row represents the cost associated with a track, and each column represents the cost associated with a detection. As highlighted, the optimal solution is (7,15,16, 9) with a cost of 47. In the next step, remove the first pair (corresponding to 7), and the next best solution is (10,15, 20, 22) with a cost of 67. After that, remove the second pair (corresponding to 15), and the next best solution is (7, 5,16, 9) with a cost of 51. After a few steps, the five best solutions are:

See the Find Five Best Solutions Using Assignkbest example, which uses the assignkbest function to find the K best solutions.

Joint Probability Data Association (JPDA) Method

While the GNN method makes a rigid assignment of a detection to a track, the JPDA method applies a soft assignment so that detections within the validation gate of a track can all make weighted contributions to the track based on their probability of association.

For example, for the gating results shown in Figure 1, a JPDA tracker calculates the possibility of association between track T 1 and observations O 1 , O 2 , and O 3 . Assume the association probability of these three observations are p 11 , p 12 , and p 13 , and their residuals relative to track T 1 are y ˜ 11 , y ˜ 12 , and y ˜ 13 , respectively. Then the weighted sum of the residuals associated with track T 1 is:

y ˜ 1 = ∑ j = 1 3 p 1 j y ˜ 1 j

In the tracker, the weighted residual is used to update track T 1 in the correction step of the tracking filter. In the filter, the probability of unassignment, p 10 , is also required to update track T 1 . For more details, see JPDA Correction Algorithm for Discrete Extended Kalman Filter .

The JPDA method requires one more step when there are conflicts between assignments in different tracks. For example, in the following figure, track T 2 conflicts with T 1 on the assignment of observation O 3 . Therefore, to calculate the association probability p 13 , the joint probability that T 2 is not assigned to O 3 (that is T 2 is assigned to O 6 or unassigned) must be accounted for.

Track-Oriented Multiple Hypothesis Tracking (TOMHT) Method

Unlike the JPDA method, which combines all detections within the validation gate using a weighted sum, the TOMHT method generates multiple hypotheses or branches of the tracks based on the detections within the gate and propagates high-likelihood branches between scan steps. After propagation, these hypotheses can be tested and pruned based on the new set of detections.

For example, for the gating scenario shown in Figure 1, a TOMHT tracker considers the following four hypotheses:

Assign no detection to T 1 resulting in hypothesis T 10

Assign O 1 to T 1 resulting in hypothesis T 11

Assign O 2 to T 1 resulting in hypothesis T 12

Assign O 3 to T 1 resulting in hypothesis T 13

Given the assignment threshold, the tracker will calculate the possibility of each hypothesis and discard hypotheses with probability lower than the threshold. Hypothetically, if only p 10 and p 11 are larger than the threshold, then only T 10 and T 11 are propagated to the next step for detection update.

S-D Assignment in Multiple Target Tracking

In an S-D assignment problem, the dimension of assignment S is larger than 2. Note that all three trackers ( trackerGNN , trackerJPDA , and trackerTOMHT ) process detections from each sensor sequentially, which results in a 2-D assignment problem. However, some applications require a tracker that processes simultaneous observations from multiple sensor scans all at once, which requires solving an S-D assignment problem. Meanwhile, the S-D assignment is widely used in tracking applications such as static data fusion, which preprocesses the detection data before fed to a tracker.

An S-D assignment problem for static data fusion has S scans of a surveillance region from multiple sensors simultaneously, and each scan consists of multiple detections. The detection sources can be real targets or false alarms. The object is to detect an unknown number of targets and estimate their states. For example, as shown in the Figure 4, three sensor scans produce six detections. The detections in the same color belong to the same scan. Since each scan generates two detections, there are probably two targets in the region of surveillance. To choose between different assignment or association possibilities, evaluate the cost matrix.

The calculation of the cost can depend on many factors, such as the distance between detections and the covariance distribution of each detection. To illustrate the basic concept, the assignment costs for a few hypotheses are hypothetically given in the table [1] .

In the table, 0 denotes a track is associated with no detection in that scan. Assume the hypotheses not shown in the table are truncated by gating or neglected because of high costs. To concisely represent each track, use c ijk to represent the cost for association of observation i in scan 1, j in scan 2, and k in scan 3. For example, for the assignment hypothesis 1, c 011 = -10.2. Several track hypotheses conflict with other in the table. For instance, the two most likely assignments, c 111 and c 121 are incompatible because they share the same observation in scans 1 and 3.

The goal of solving an S-D assignment problem is to find the most likely compatible assignment hypothesis accounting for all the detections. When S ≥ 3, however, the problem is known to scale with the number of tracks and detections at an exponential rate (NP-hard). The Lagrangian relaxation method is commonly used to obtain the optimal or sub-optimal solution for an S-D assignment problem efficiently.

Brief Introduce to the Lagrangian Relaxation Method for 3-D Assignment

Three scans of data have a number of M 1 , M 2 , and M 3 observations, respectively. Denote an observation of scan 1, 2, and 3 as i , j , and k , respectively. For example, i = 1, 2, …, M 1 . Use z ijk to represent the track formation hypothesis of O 1 i , O 2 j , and O 3 k . If the hypothesis is valid, then z ijk = 1; otherwise, z ijk = 0. As mentioned, c ijk is used to represent the cost of z ijk association. c ijk is 0 for false alarms and negative for possible associations. The S-D optimization problem can be formulated as:

J ( z ) = min i , j , k ∑ i = 0 M 1 ∑ j = 0 M 2 ∑ k = 0 M 3 c i j k z i j k

subject to three constraints:

∑ i = 0 M 1 ∑ j = 0 M 2 z i j k = 1 , ∀ k = 1 , 2 , … , M 3 ∑ i = 0 M 1 ∑ k = 0 M 3 z i j k = 1 , ∀ j = 1 , 2 , … , M 2 ∑ j = 0 M 2 ∑ k = 0 M 3 z i j k = 1 , ∀ i = 1 , 2 , … , M 1

The optimization function chooses associations to minimize the total cost. The three constraints ensure that each detection is accounted for (either included in an assignment or treated as false alarm).

The Lagrangian relaxation method approaches this 3-D assignment problem by relaxing the first constraint using Lagrange multipliers. Define a new function L ( λ ) :

L ( λ ) = ∑ k = 0 M 3 λ k [ ∑ i = 0 M 1 ∑ j = 0 M 2 z i j k − 1 ]

where λ k , k = 1, 2, …, M 3 are Lagrange multipliers. Subtract L from the original object function J ( z ) to get a new object function, and the first constraint in k is relaxed. Therefore, the 3-D assignment problem reduces to a 2-D assignment problem, which can be solved by any of the 2-D assignment method. For more details, see [1] .

The Lagrangian relaxation method allows the first constraint to be mildly violated, and therefore can only guarantee a suboptimal solution. For most applications, however, this is sufficient. To specify the solution accuracy, the method uses the solution gap, which defines the difference between the current solution and the potentially optimistic solution. The gap is nonnegative, and a smaller solution gap corresponds to a solution closer to the optimal solution.

Sensor Fusion and Tracking Toolbox provides assignsd to solve for S-D assignment using the Lagrangian relaxation method. Similar to the K best 2-D assignment solver assignkbest , the toolbox also provides a K best S-D assignment solver, assignkbestsd , which is used to provide multiple suboptimal solutions for an S-D assignment problem.

See Tracking Using Distributed Synchronous Passive Sensors for the application of S-D assignment in static detection fusion.

assignTOMHT | assignauction | assignjv | assignkbest | assignkbestsd | assignmunkres | assignsd | trackerGNN | trackerJPDA | trackerTOMHT

[1] Blackman, S., and R. Popoli. Design and Analysis of Modern Tracking Systems. Artech House Radar Library, Boston, 1999.

[2] Musicki, D., and R. Evans. "Joint Integrated Probabilistic Data Association: JIPDA." IEEE Transactions on Aerospace and Electronic Systems. Vol. 40, Number 3, 2004, pp 1093 –1099.

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