You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


1. Field of the Invention
The present invention relates to a coded aperture imaging system, and more particularly relates to a coded aperture imaging system for generating and displaying an image signal which is representative of an image of a source of non-focusable radiation, such as a gamma ray or an x-ray emitting source. More particularly, the invention relates to a uniformly redundant array coded aperture used in a coded aperture imaging system, and a method of generating the coded aperture.
2. Description of the Prior Art
Nonfocusable radiation sources, such as gamma ray and x-ray emitting sources, are most commonly found in nuclear power plants, medical radiological laboratories, and other nuclear material handling facilities. Thus, routine monitoring of potential highly radioactive areas must be performed during nuclear power plant operation. Environmental restoration and waste management of radiation contaminated sites also require detecting and locating contaminants to assure restoration of the environment to a safe level. Additionally, in order to perform repairs and maintenance within nuclear power plants, the area to be worked in must be scanned by health physics personnel to determine an amount of exposure the radiation workers will encounter while performing their job. Worker radiation exposure is closely monitored via personally worn dosimetry and radiation workers who have reached government specified radiation exposure limits are prevented from further exposure and can no longer perform their jobs for a specified period of time. Thus, it is extremely beneficial, and it is the goal of all nuclear power plant operators, to minimize personal radiation exposure so that radiation exposure limits are not reached.
The health physics personnel responsible for scanning a potentially highly radioactive area often wear protective clothing and a respirator to protect them against airborne radiation. The scanning of suspected high radiation areas with a well known device called a Geiger-Mueller counter is necessary to locate sources of high radiation. Geiger-Mueller counters are relatively small and carried by the worker performing the scanning process. In some instances, the Geiger-Mueller counter is attached to a pole so that the worker can maintain a safe distance from radioactive equipment when scanning. The worker manually scans the entire area to determine "hot spots" or areas of high radiation.
Upon completion of the scanning process, the worker creates a map of the radiation hazards which have been identified by the scanning process using the Geiger-Mueller counter. Appropriate shields are placed around those high radiation sources to protect the workers. Based upon the results of the scanning process, radiation workers performing repairs and maintenance are instructed how long they can remain in the area and what equipment should be avoided due to high levels of radiation being emitted. Furthermore, the scanning process is time consuming because the sweep rate, i.e., the rate at which the Geiger-Mueller counter handpiece is moved over a radioactive source, is relatively slow. In some instances, the worker must get close to the gamma ray emitting source to obtain proper readings since typical Geiger-Mueller counters are not very sensitive.
The two most important factors in limiting radiation exposure are time of exposure and distance from the source of radiation. Presently available scanning techniques using a Geiger-Mueller counter are inefficient in minimizing time of radiation exposure and maximizing distance from the source of radiation. Additionally, the scanning process using a Geiger-Mueller counter provides crude "hot spot" location.
Prior art system designs utilizing the technique of coded aperture imaging have attempted to improve upon the disadvantages of this early scanning technique. Generally, by providing a system which generates a visual representation of the non-focusable radiation source, coded aperture imaging systems allow detection and location of such sources while minimizing, or even eliminating, harmful exposure by personnel. In particular, coded aperture imaging, besides providing the above identified advantage, also yields accurate images of sources of non-focusable radiation. Coded aperture imaging is itself a well known technique involving the use of an aperture composed of transparent and opaque cells. A shadow is cast through these cells on a position sensitive detector in response to the aperture being exposed to gamma or x-ray radiation sources. An image of the source is reconstructed from the information obtained during this exposure process.
However, such prior art coded aperture imaging systems have historically exhibited severe limitations. For example, very often, position sensitive detectors of coded aperture imaging systems are plagued by substantial instrument noise. In addition, the systems often operate in a high background radiation environment, and high energy photons emanating from outside the instrument field of view are detected by the position sensitive detector without first passing through, or being modulated by, the coded aperture. Thus, detected photons not modulated by the coded aperture and inherent non-modulated instrument noise degrade the signal-to-noise ratio of the instrument, resulting in poor image quality and low sensitivity.
Some prior art coded aperture imaging systems utilize an active shield technique to minimize the detrimental effect of non-modulated photons. Under this approach, if the incoming high energy photon flux is low enough to allow the instrument to count individual photon arrival, an active shield and an active coded aperture can be used to provide a means for coincidence/anti-coincidence identification. Thus, non-modulated photons can be individually identified and then excluded, or vetoed, during image reconstruction. However, if the photon flux increases such that many photons arrive at the detector simultaneously, accurate arrival time of each photon cannot be established thereby making separation of non-modulated photons from the desired modulated photons very difficult.
Other prior art coded aperture imaging systems have overcome this limitation by utilizing a normal/complementary coded aperture approach. Under this approach, the radiation source is first modulated by a coded aperture, called a normal aperture, and then by the complement of the coded aperture, called a complementary aperture. The complementary aperture is created by exchanging the location of the cells on the normal aperture. In other words, all cells that were transparent on the normal aperture are opaque on the complementary aperture, and all cells that were opaque on the normal aperture are transparent on the complementary aperture. If each aperture is exposed to the source for an equal time duration and a cancelling process is performed between the images modulated by each aperture, the detrimental effects of non-modulated photons and instrument background noise can be significantly reduced. This result is obtained even in the presence of a high energy photon flux. Coded aperture imaging systems that use this above approach have employed coded apertures formed from either rectangular or hexagonal uniformly redundant arrays (URA).
Rectangular URA coded apertures, such as those disclosed in U.S. Pat. No. 4,209,780 to Fenimore, et al. and U.S. Pat. No. 5,036,546 to Gottesman, et al., have been used for coded aperture imaging. However, use of these types of apertures requires the physical implementation of two separate apertures, one functioning as the normal aperture and the other as the complementary aperture. Systems utilizing rectangular URAs must include means for switching between apertures during the imaging process. This switching process can lead to system inaccuracies, namely, alignment inaccuracy which will negatively affect the image reproduction performance of the system. Also, if the aperture switching is accomplished manually, one of the major benefits that a coded aperture imaging system has over a Geiger-Mueller counter, namely, protection of personnel from radiation exposure, is completely lost. Further, manual aperture switching is time-consuming and detracts from the additional benefit of being able to quickly identify and contain dangerous sources of radiation.
A hexagonal URA was subsequently developed to overcome the need for two separate apertures, as disclosed by M. H. Finger and T. A. Prince, in an article entitled "Hexagonal Uniformly Redundant Arrays for Coded Aperture Imaging", Proceedings of the 19th International Cosmic Ray Conference, OG 9.2-1, pages 295-298 (1985). A single hexagonal URA (HURA) coded aperture, which is nearly anti-symmetric about its hexagonal axis, can function both as a normal aperture and a complementary aperture upon 60.degree. rotation. A HURA coded aperture is congruous for a hexagonal or circular position sensitive detector. However, a severe disadvantage exists in the use of such an aperture with a Cartesian-configured position sensitive detector. Since matching of the hexagonal geometry of the aperture with the Cartesian geometry of the detector requires significant geometric manipulation, considerable data formatting must be performed by the imaging system. Because most state-of-the-art detectors are of a Cartesian configuration, the use of a HURA coded aperture is limited.
Accordingly, there is a need for a portable, reliable, coded aperture imaging system which can automatically map out "hot spots" in an area to be monitored. Furthermore, it would be advantageous to have a device which can be permanently mounted in an area to provide a map of the radiation environment in an area with no radiation exposure to human beings. Still further, there is a need for an imaging system which incorporates a uniformly redundant array coded aperture, having an automatically complementary property, which can easily be interfaced with a readily available Cartesian-configured (rectangular or square) position sensitive detector.