A coupled biventricular finite element and lumped-parameter circulatory system model of heart failure Numerical modeling of the cardiovascular system is becoming an important tool for assessing the influence of heart disease and treatment therapies. In the current study we present an approach for modeling the interaction between the heart and circulatory system. This was accomplished by creating animal specific biventricular finite element models, which characterize the mechanical response of the heart, and coupling them to a lumped parameter model that represents the systemic and pulmonic circulatory system. In order to minimize computation time, the coupling was enforced in a weak (one-way) manner, where the ventricular pressure-volume relationships were generated by the finite element models and then passed into the circulatory system model to ensure volume conservation and physiological pressure changes. The models were first validated by tuning the parameters, such that the output of the models matched experimentally measured pressures and volumes. Then the models were used to examine cardiac function and the myofiber stress in a healthy canine heart and a canine heart with dilated cardiomyopathy. The results show good agreement with experimental measurements. The stress in the case of cardiomyopathy was found to increase significantly, while the pump function was decreased, compared to the healthy case. The total run time of the simulations is less than many fully-coupled models presented in the literature. This will allow for a much quicker evaluation of possible treatment strategies for combating the effects of heart failure, especially in optimization schemes that require numerous finite element simulations. Recent statistics from the American Heart Association (2011) indicate that Congestive Heart Failure (CHF) affects 5.7 million Americans. CHF is characterized by the hearts inability to pump blood efficiently into the body, due to either impaired myocardial contraction and global dilation of the ventricles (systolic heart failure), or decreased relaxation of the ventricles (diastolic heart failure). Several treatment strategies have been investigated to combat the adverse remodeling that leads to the progression of CHF (Gorman et al. 2004 , Thompson et al. 2010a , Thompson et al. 2010b ). Some of these treatments involve the improvement of blood flow to damaged regions of the heart, while others include reshaping the heart by means of surgical procedures and implantation of medical devices. Examples of implantable medical devices include the Acorn CSD and Copasys (Acker et al. 2006 , Grossi et al. 2010 ). These devices were designed to improve pump function by reducing ventricular volume and thus reduce wall stress. The effect of these devices on regional ventricular mechanics and global ventricular function, however, is currently not fully understood.. The mechanics of the heart have been widely studied using the finite element (FE) method. Left ventricular (LV) FE models, which incorporate myocardial contractility, have been described and used to determine the influence of myocardial infarction on structure and function (Dang et al. 2005a , Guccione et al. 2001 , Walker et al. 2005a ). Additionally, the efficacy of various surgical procedures has been simulated (Dang et al. 2005b , Walker et al. 2005b , as well as the effect of cardiac support devices (Jhun et al. 2010 , Wenk et al. 2009 ). Models that include volumetric growth (Kroon et al. 2009a ) and myofiber remodeling (Kroon et al. 2009b ) have also been implemented with the FE method, and coupled to lumped-parameter models of the circulatory system. Most recently, a non-invasive method for determining myocardial material properties was developed using magnetic resonance imaging (MRI), FE modeling, and optimization (Sun et al. 2009 , Wenk et al. 2011 ), which can be used to characterize weakly contracting regions of the heart. These single ventricle models have provided much insight on cardiac mechanics, but do not account for the interaction between the left and right ventricles.. To enhance LV models, biventricular FE models have been developed to simulate the effects of ventricular dilation and electrical dyssynchrony on ventricular function and regional mechanics (Kerckhoffs et al. 2010) , as well as the effects of external defibrillation on heart function (Rodriguez et al. 2006 ). In addition, biventricular FE models have been used as a framework in which to incorporate multiscale models for eccentric and concentric cardiac growth (Goktepe et al. 2010 ). The effects of surgical procedures for pulmonary valve and patch replacement in patients with tetrology of Fallot were also investigated with a biventricular FE model (Tang et al. 2008 ). Kerckhoffs et al. (2007b) developed a coupled biventricular FE and lumped systems model to evaluate the effects of ischemia on pump function, using an implicit scheme to simulate the results. These models, however, are computationally expensive and have not been used to simulate the effect of cardiac devices including surgical ventricular remodeling and passive constraint.. In the current study, we present a method to weakly couple a biventricular finite element model and a lumped parameter model of the circulatory system. To achieve this end, we used the biventricular FE model to represent the mechanical response of the heart, and then input the response into the circulatory system model to ensure physiological pressure changes and continuity of the total system volume during the cardiac cycle. By using a weak (one-way) coupling scheme, the simulations were conducted in much less time than previous approaches described in the literature. This is an important aspect of this work, since it allows for numerous simulations to be conducted in a short period of time. This is essential for the computational efficiency of optimization schemes, which require a large number of simulations to evaluate an objective function (Sun et al. 2009 ). In order to validate our method, two FE models were constructed, based on MRI and experimental hemodynamic measurements in a normal healthy dog and the same dog with pacing tachycardia-induced dilated cardiomyopathy. The primary goal of the current study is to evaluate the effects of heart failure on stress and pump function in the heart. In the near future, this approach will be used to optimize the design of devices used to treat systolic heart failure. Specifically, the method will be used to simulate the effect of passive constraint devices, such as the Acorn CSD and Coapsys device. A computationally efficient method for weakly coupling a biventricular FE model and lumped parameter model was developed and tested on animal-specific models of a normal canine and a canine with dilated cardiomyopathy. There was good agreement between the model simulations and experimentally measured volumes for both the normal and diseased case. The biventricular FE model and circulatory model were found to be in good agreement for a state of elevated heart rate, which was not directly measured during experiments. This indicates that the coupling scheme is able to represent the ventricular response over a range of physiological states.. One of the advantages of this approach is the relatively short total run times of the simulations. Once biventricular mechanics have been characterized with the FE models, via the pressure-volume relationships, this information can be used directly in the circulatory system model, which takes only minutes to run. This is mainly due to the fact that the FE simulations are set up to model only the end-diastolic and end-systolic states. Using this information, the circulatory model is able to simulate the entire cardiac cycle in a short period of time. Other studies have reported simulation times that take on the order of 100 hours (Kerckhoffs et al. 2007b ), due to the fully coupled nature of the model. These long run times would make it extremely cumbersome to evaluate multiple simulations during an optimization. We have utilized a computationally efficient finite element formulation in previous optimization studies in which we determined the contractile properties of myocardium in animal-specific models of anteroapical and posterobasal infarction (Sun et al. 2009 , Wenk et al. 2011 ).. In the present study, there is no need to assume a non-physiological preload (McLeod 1966), which allows us to capture the nonlinear effects of the ventricular mechanics. The lumped parameter model ensures that the pressure and volume changes in the circulatory system are properly coupled and conserve the total system volume. Once a particular state of interest is calculated with the circulatory system model, these pressures can be passed back to the FE model in order to calculate the stress and strain in the myocardial wall.. It can be seen from the results that the diastolic compliance decreases due to CHF (i.e., the wall becomes stiffer). As the heart dilates it becomes more difficult to fill, as evidenced by the increase in the passive stiffness parameter, C. Increased wall stiffness due to CHF has also been found in experimental studies by Wu et al. (2002) and Nagueh et al. (2004) . In this case, canines underwent rapid pacing to induce dilated cardiomyopothy, similar to the present study, which causes myocardial stiffening, which were modulated by altered Titin expression. At sarcomere lengths comparable to those computed in the present study, the passive stiffness was increased by 5-6 times.. During the systolic phase of the cardiac cycle the strength of contraction is weaker in the LV due to CHF. The values of E ES and V 0 are in close agreement with those found in experimental measurements with canines (Maughan et al. 1979 , Maughan et al. 1985 . Weaker contraction is indicated by a decrease in the active contraction parameter, T max , as well as a decrease in E ES . This is a primary characteristic of CHF (Gorman et al. 2004 , Potter et al. 2007 ). As the heart dilates further, and myofiber stress increases, this will lead to further dysfunction of the contracting myocytes until the heart is no longer able to pump enough blood to the body to maintain even a resting metabolic state. As the heart becomes weaker, higher end-diastolic pressures are required to produce a stroke volume that is equivalent to that in a healthy heart, as seen in Figure 7 .. The magnitude of average myofiber stress calculated in the current study is in close agreement with values calculated for canine ventricles in other FE studies. In Guccione et al. (1995b) , an FE model of a healthy canine was validated with experimental strain at midventricle. The average ED myofiber stress was found to be approximately 1.5 kPa, compared to 1.10 ± 0.59 kPa in the current study. In Kerckhoffs et al. (2010) , stress was reported for a healthy baseline and dilated canine heart. The average ES myofiber stress in the healthy heart was approximately 12 ± 5.3 kPa, compared to 14.5 ± 6.4 kPa in the current study, and was approximately 20.4 ± 9.9 kPa in the dilated heart, compared to 19.9 ± 7.69 kPa in the current study. In both studies, the stress in the myofiber direction was found to be elevated in the entire heart due to cardiomyopathy.. There are a few limitations associated with the current study. Only a single animal data set was used to develop the coupled FE/circulatory system model. Focused animal experiments are needed to confirm the results at several physiological states, as well as multiple times during the cardiac cycle. Only cine MRI data was available for this study. In the future, tagged MRI will be obtained to further validate the FE models by matching the predicted deformation to experimentally measured strains. The one-way coupling may not be as accurate as a fully-coupled model. This is primarily due to the fact that the coupling does not happen at every time step. Both fully-coupled and weakly coupled methods will be bounded by the same EDPVR and ESPVR, which means the ED and ES states should be in close agreement. However, the exact path of the pressure-volume loop may differ between the two methods for a given heart beat. Depending on the application, the tradeoff of fast runtimes could outweigh the need for higher accuracy, especially when conducting computationally intensive optimization simulations. The active constitutive model does not contain strain rate dependence. The current model represents the steady state response at ES. There are numerous finite element models of regional ventricular mechanics that have focused only on the LV end-diastole and end-systole, which have been sufficient to quantify the efficacy of novel surgical procedures or medical devices for treating dilated or ischemic cardiomyopathy or heart failure (Dang et al. 2005a , Walker et al. 2005a , Sun et al. 2009 , Wenk et al. 2011 ). The purpose of the present work is to describe a methodology for adding the RV, and ensuring volume conservation between ventricles. However, we are in the process of adding strain rate dependence in a new LV material model that is being developed in LS-DYNA.. In summary, the coupling approach presented in this work showed good agreement with experimentally measured pressures and volumes. The one-way scheme was able to characterize both a normal and dilated cardiomyopathy heart, in a relatively shorter runtime. In the future, this approach will be used to evaluate the effects of a passive constraint device, which wraps around the heart, to prevent dilation and help remodel the heart towards a normal state. It is anticipated that the fast runtime will allow for computationally efficient optimization of the key attributes of the constraint device. In this study, we developed a weakly coupled FE/lumped-parameter system to model the cardiovascular system. The model consists of two separate units: a biventricular FE model for the heart and a lumped parameter model for the cardiovascular tree. The coupling between the two units is based on previous experimental observations indicating that the ventricular output is well characterized by the end-systolic pressure-volume relationship (ESPVR) and end-diastolic pressure-volume relationship (EDPVR) (Suga and Sagawa 1974) . We characterized the mechanical response of the heart to various pressure loads using the biventricular FE model and then, used this data to drive the circulatory system model (to ensure that pressure changes in each ventricle were coupled and physiologically realistic). A flow chart of the coupling scheme is shown in Figure 1 . The key features of each model component are described in the following sections.

O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Motivating and assisting physical exercise in independently living older adults: A pilot study Background: With age reaction time, coordination and cognition tend to deteriorate, which may lead to gait impairments, falls and injuries. To reduce this problem in elderly and to improve health, well-being and independence, regular balance and strength exercises are recommended. However, elderly face strong barriers to exercise. Objective: We developed Active Lifestyle, an IT-based system for active and healthy ageing aiming at improving elderly's balance and strength. Active Lifestyle is a proactive training application, running on a tablet, which assists, monitors and motivates elderly to follow personalized training plans autonomously at home, while integrating them socially. The objective is to run a pilot study to investigate: i) the feasibility of assisting the autonomous, physical training of independently living elderly with the Active Lifestyle system, ii) the adherence of the participants to the training plans, and iii) the effectiveness of the motivation instruments built into the system. Methods: After three introductory meetings, 13 elderly adults followed personalized twoweeks strength and balance training plans using the Active Lifestyle app autonomously at home. Questionnaires were used to assess the technological familiarity of the participants, the feasibility aspects of the physical intervention, and the effectiveness of the motivation instruments. Adherence to the exercise plan was evaluated using the performance data collected by the app during the study. Results: A total of 13 participants were enrolled, of whom 11 (85%) completed the study (mean age 77 ± 7 years); predominantly females (55%), vocational educated (64%), and their past profession requiring moderate physical activity (64%). The Active Lifestyle app facilitated autonomous physical training at home (median=7 on a 7-point Likert scale), and participants expressed a high intention to use the app also after the end of the study (median=7). Adherence with the training plans was 73% (89% on the balance exercises and 60% on the strength exercises). The outcome from our questionnaires showed that without the app the participants did not feel motivated to perform exercises; with the support of the app they felt more motivated (median=6). Participants were especially motivated by being part of a virtual exercise group and by the capability to automatically monitor their performance (median=6 for both). Conclusions: This study shows that the Active Lifestyle app prototype has valuable potential to support physical exercise practice at home and it is worthwhile to further develop it into a more mature system. Furthermore, the results add to the knowledge base into mobile-based applications for elderly, in that it shows that elderly users can learn to work with mobile-based systems. The Active Lifestyle app proved viable to support and motivate independently living elderly to autonomously perform balance and strength exercises. Health status is an important indicator of quality of life among older persons [1, 2] . Functional performance, chronic conditions, and diseases, which directly influence fitness, are related to the perceived health among middle-aged and older adults [2] [3] [4] . Chronic diseases are, furthermore, leading causes of death and disability in both developed and developing countries [5, 6] . Inactivity is at the origin of several chronic diseases [7] . Regular physical activity or exercise substantially prevents the development and progression of most chronic degenerative diseases [8] [9] [10] , is of benefit to frail and older persons, and is the only therapy found to consistently improve sarcopenia, physical function, cognitive performance and mood in older adults [11] . For older people, a sedentary lifestyle also increases the risk of falls, whereas physically active older people have a reduced risk of falls with injuries [12] [13] [14] . In summary, it is evident that to increase older adults' quality of life and fitness, we need to encourage the elderly to become more physically active [15] [16] .. There are a variety of barriers that make it hard for elderly to maintain or increase their physical activity level. Neighborhoods and communities may be poorly designed or perceived as being unsafe, thus preventing elderly from leaving home [17] [18] . Older adults may have trouble getting to specialized facilities (e.g., community center for the aged) and physical training programs offered in such institutions [17] [18] . General health care professionals (e.g., nurses, family physicians) may lack the time or expertise to address problems of physical inactivity among their older patients, and often lack information about quality programs, training materials, and how to make referrals to community resources [19] . Furthermore, elderly often express the desire for training support at home [20] .. Specialized healthcare professionals (i.e., sport medicine, gerontologists, physiotherapists and human movement scientists) strongly recommend physical training programs that focus on muscle strength and balance [21] to promote health, well-being, and functional independence of the elderly. Structured exercise training has a positive impact on older adults and may be used for the management of frailty [22] [23] [24] . Home environmental interventions based on different forms of assistive technology devices have, in this context, the potential to overcome some of the barriers to start training and maintain physical independence for independent living elderly [25] . However, the effectiveness of such an approach has not yet been demonstrated to a large extent. Modern exercise equipment may not always be suitable for elderly individuals, who might be concerned about the intensity of training sessions and may rather express to have a preference for more traditional therapy approaches [26] . New treatments usually have to go through a series of phases to test whether they are feasible, safe and effective [27] . It seems, therefore, necessary to perform a pilot study to assess the feasibility of applying assistive technology devices in an elderly population with the aim to encourage performance of targeted physical exercise. Findings of such a study can inform a larger scale main Phase III study [27] .. The objective of this study is to run a Phase II pilot study according the model for complex interventions advocated by the British Medical Research Council [28] with an iPad-based app (short for application) called Active Lifestyle, a software for the autonomous physical training of strength and balance for independently living elderly. We aim to investigate i) the feasibility of assisting the autonomous, physical training of independently living elderly with the Active Lifestyle system, ii) the adherence of the participants to the Active Lifestyle training plans, and iii) the effectiveness of the system's motivation instruments. We conclude that pilot studies with explicit feasibility objectives are important foundational steps in preparing for large trials. On-going formal review of the multifaceted issues inherent in the design and conduct of pilot studies can provide invaluable feasibility and scientific data for IT developers and rehabilitation specialists alike, and may be highly relevant for furthering the development of theory-driven, mobile-based rehabilitation.. This study shows that Active Lifestyle has potential to support physical exercise at home and it is worthwhile to further develop it into a more mature system. The results add to the knowledge base on mobile-based applications for elderly, in that it shows that elderly users can learn to work with mobile-based systems. The Active Lifestyle app proves feasible to support and motivate independently living elderly adults to autonomously perform balance and strength exercises at home. The Active Lifestyle intervention in a main study is deemed feasible with some need for protocol modifications regarding recruitment strategies, motivational instruments and information on physical performance measures. In our further work we will more specifically examine the effectiveness of Active Lifestyle on measures of physical functioning, with longer training periods, and larger sets of participants. Additionally, we will investigate how the users of the Active Lifestyle app react to social and individual motivation factors, and compare their adherence and motivation against a control group. The aim of this study was to investigate i) the feasibility of the Active Lifestyle system for physical interventions, ii) the adherence of the participants to the pre-defined training plans, and iii) the effectiveness of the motivation instruments built into the system.. We demonstrated the feasibility of acquiring acceptable attrition and adherence rates for independently living elderly to the experimental training. Our target of about 70% recruitment rate (as observed by [35] for similar studies) for the total sample frame, however, by far not met. Those individuals that responded and visited an information session, however, showed a large inclusion and adherence rate. These findings indicate the importance of recruitment strategies and information sessions to create awareness of the need for strength and balance training and the necessary trust for people to also enroll in a study. The differing adherence for the strength and balance training components should be addressed in a revised version of our app. It might be that the balance part required less effort and was easier to perform in contrast to the strength exercises. Many participants made remarks about the load of the strength exercises, e.g., some participants had difficulties to walk or had had a surgery on the knee or hip and, therefore, could not perform the recommended number of sets.. The ankle weights were a barrier to the adherence to the strength exercises because, apart from the effort required to lift the weights, most participants reported difficulties to wear and wrap them around their ankles. One man bought a new pair of weights that were easier to wear and close. Another woman mentioned "tying the weights is almost the hardest!". Compliance with the intervention was excellent. Eleven of initially thirteen included elderly individuals completed the training. This is far more than the rate that could have been expected [35] . It should be noted, however, that the mean compliance rate for interventions in independent living elderly that was determined for several studies summarized in a systematic review [32] mainly focused on studies with far longer time periods. Our data should, therefore, be replicated in another study where the intervention is applied during several training sessions over several weeks. It can be expected that because of such a longer period less favorable compliance and retention rates will be achieved. Our findings warrant, however, such follow-up studies based on these first results.. Regarding motivation, we noticed that all participants felt motivated by making something grow with their effort and by receiving a simple reward (e.g., a new picture of the flower metaphor). We infer that positive reinforcement using the growing flower metaphor worked successfully, even if approximately half of the participants (55%) would have preferred other metaphors. The selection of a metaphor (e.g., flower vs. car) is indeed very subjective and depends on the individual preferences of each participant.. About the low rate of writing activities, we hypothesize two possible reasons. Firstly, participants did not know each other, so they did not feel comfortable to write. Secondly, a minor part of them did not have good typing skills. For instance, one woman once wrote a nonunderstandable not correctly written message on the board. When she was asked about the sending messages option at the Bulletin Board or at the inBox, she said "I would write more, but for that I need to learn how to do it better." This 81 years old participant, however, never used the Internet or a computer before.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


DENOISING AND ENHANCEMENT OF MAMMOGRAPHIC IMAGES UNDER THE ASSUMPTION OF HETEROSCEDASTIC ADDITIVE NOISE BY AN OPTIMAL SUBBAND THRESHOLDING Mammographic images suffer from low contrast and signal dependent noise, and a very small size of tumoral signs is not easily detected, especially for an early diagnosis of breast cancer. In this context, many methods proposed in literature fail for lack of generality. In particular, too weak assumptions on the noise model, e.g., stationary normal additive noise, and an inaccurate choice of the wavelet family that is applied, can lead to an information loss, noise emphasizing, unacceptable enhancement results, or in turn an unwanted distortion of the original image aspect. In this paper, we consider an optimal wavelet thresholding, in the context of Discrete Dyadic Wavelet Transforms, by directly relating all the parameters involved in both denoising and contrast enhancement to signal dependent noise variance (estimated by a robust algorithm) and to the size of cancer signs. Moreover, by performing a reconstruction from a zero-approximation in conjunction with a Gaussian smoothing filter, we are able to extract the background and the foreground of the image separately, as to compute suitable contrast improvement indexes. The whole procedure will be tested on high resolution X-ray mammographic images and compared with other techniques. Anyway, the visual assessment of the results by an expert radiologist will be also considered as a subjective evaluation. In this paper, we address the problem of denoising and contrast enhancement of mammographic images by wavelet thresholding using DDWT. It is well known that noise corrupting X-ray images has not a trivial modelization and principally it is signal-dependent. So, assuming a very general noise model, we have firstly applied a robust method in order to estimate noise variance for a heteroscedastic additive noise. Then, we have theoretically derived relations to directly set thresholds either for denoising and for enhancement, specifically for DDWT. The thresholds we propose are both level and noise dependent, according to the assumption that noise amount decreases as level increases and that noise power is strongly influenced by the wavelets filters used. It is also important to underline that the use of DDWT represents a key issue in this context in order to optimize noise reduction, avoiding loss of details and geometric distortion. The method we propose can be applied to many medical images containing small details such as microcalcifications, since simultaneously performs a selective enhancement of small objects and reduces noise according to the gray-value of each pixel, without altering the aspect of the background. This topic is crucial in every Computer Assisted Diagnosis System (CADx) for helping radiologists in an early diagnosis of breast cancer avoiding a misleading or a miss of a tumoral disease.

O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Mean field and propagation of chaos in multi-class heterogeneous loss models We consider a system consisting of N parallel servers, where jobs with different resource requirements arrive and are assigned to the servers for processing. Each server has a finite resource capacity and therefore can serve only a finite number of jobs at a time. We assume that different servers have different resource capacities. A job is accepted for processing only if the resource requested by the job is available at the server to which it is assigned. Otherwise, the job is discarded or blocked. We consider randomized schemes to assign jobs to servers with the aim of reducing the average blocking probability of jobs in the system. In particular, we consider a scheme that assigns an incoming job to the server having maximum available vacancy or unused resource among d randomly sampled servers. We consider the system in the limit where both the number of servers and the arrival rates of jobs are scaled by the same large factor. This gives rise to a mean field analysis. We show that in the limiting system the servers behave independently -a property termed as propagation of chaos. Stationary probabilities of server occupancies are obtained from the stationary solution of the mean field. We further characterize the rate of decay of the stationary tail probabilities. Numerical results suggest that the proposed scheme significantly reduces the average blocking probability of jobs as compared to static schemes that probabilistically route jobs to servers independently of their states. Consider jobs with different resource requirements, arriving at a multi-server system consisting of a large number of parallel servers. Each server has a finite resource capacity and therefore A. Mukhopadhyay, and R. R. Mazumdar are with the department of Electrical and Computer Engineering, University of Waterloo, Canada. Email: {arpan.mukhopadhyay, mazum}@uwaterloo.ca A. Karthik is with Qualcomm, Bangalore, India. Email: k4ananth@uwaterloo.ca F. Guillemin is with Orange Labs, France. Email: fabrice.guillemin@orange.com A preliminary version of this work was presented at the 27th International Teletraffic Congress (ITC 27), Belgium, 2015. can process only a finite number of jobs at a time. Different servers are assumed to have different capacities. Upon arrival, each job is routed/assigned to a server where it is either accepted or blocked depending on the availability of resource requested by the job at the server.. If accepted, the processing of the job begins immediately at the server. The objective is to design job routing/assigning schemes that reduce the average blocking probability of jobs in the system. Such a model arises frequently in the context of cloud computing systems that provide infrastructure as a service [1] , [2] . A cloud service provider sells computing resources to its users in terms of virtual machines (VM's), that are computing instances consisting of various resources such as CPU, memory, storage etc. To meet different user demands the cloud operator allows users to choose from various types of VM's which differ by the amounts of resources they hold. We model situations [3] , [4] , where there is one bottleneck resource (e.g. memory).. Hence, in our case the different VM's correspond to different amounts of the same resource.. Each user requests a VM of specific type (e.g. large or small) for a required amount of time. The VM request is then assigned to a physical machine (PM) or server where the request is either accepted or blocked depending on the availability of resource at the server. If accepted, the user holds the VM for the duration of its service after which it is released. Therefore, to maintain a certain quality of service, a cloud service provider should aim at reducing the average blocking probability of users which measures the fraction of time a user is denied its requested resources.. The problem can be cast as a stochastic knapsack or a bin packing problem, where the blocking behavior is determined by the policy that assigns jobs or VM requests to servers. In particular, we consider a model where each server has the capacity to serve a finite number VM requests at a time and different servers can have different capacities. We refer to this as the heterogeneous loss system model. There are other possible abstractions to model clouds that involve buffering of jobs in an infinite queue as was analyzed in [5] , [6] . However the loss model is particularly relevant for the Infrastructure-as-a-service (IaaS) paradigm offered by Amazon's EC2 [1] and Microsoft's Azure [2] where each server can process only a finite number of jobs at a time.. We consider a randomized scheme to assign VM requests to servers based on random sampling of d servers from the entire system. We show that assigning each request to the server having the maximum vacancy amongst the d ≥ 2 sampled servers yields dramatic reduction in the blocking probability as compared to that in static assignment schemes (d = 1) where assignments are made independent of server states. Although, comparing the states of all servers would ideally result in the best performance, such comparison involve high communication overhead due to large size of server farms. Moreover, we show that sampling too many servers is unnecessary since sampling only a few servers results in nearly optimal in terms of minimizing the average blocking probability of requests.. Related Literature: The routing scheme that we consider is a loss model analog of the powerof-d scheme considered in [7] - [9] for first-come-first-serve (FCFS) queues and in [10] for heterogeneous processor sharing (PS) servers. In the power-of-d scheme, each incoming job is assigned to the server having the least number of unfinished jobs among d servers, sampled randomly at the arrival instant of the job. Turner [11] , [12] studied this scheme for a system of Erlang servers having infinite capacities in the large system limit. It was shown that in the large system limit, the system behavior can be characterized by a mean field limit, which satisfies a system of differential equations. The resulting tail distribution of server occupancies was shown to have a fast rate of decay even for small values of d. However, the existence and uniqueness of the equilibrium point of the mean field were not established. Xie et al. [4] recently analyzed the the power-of-d scheme in an Erlang loss system with identical (homogeneous) servers in the mean field limit. Existence, uniqueness, and global asymptotic stability of the equilibrium point of the mean field were established when there is a single class of customers in the system. For the case with multiple classes of customers, the authors derived a recursive relationship among the tail probabilities of the number of occupied resource units at each server assuming asymptotic independence of the servers in the limiting system. In the current paper, we generalize these results to the scenario where the servers have heterogeneous capacities.

B I O B I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Infinite ternary square-free words concatenated from permutations of a single word We answer a question of Harju: An infinite square-free ternary word with an n-stem factorization exists for any n ≥ 13.We show that there are uniform ternary morphisms of length k for every k ≥ 23. This resolves almost completely a problem of the author and Rampersad. Fixed points of morphisms have been used to create words avoiding powers and patterns since the turn of the last century [2] . Analysis of morphisms is simpler when the morphisms are highly symmetrical. Morphisms of Thue, Pleasants and Keränen [2, 9, 6] , for example, may be expressed on alphabets {0, 1, . . . , n} such that the image of each letter i is obtained from the image of 0 using the cyclic shift 0 → 1 → 2 · · · → n → 0, repeated i times. Evidently, these morphisms are k-uniform for some k, and their fixed points are obtained by concatenating blocks of length k, each of which is equivalent to the first block under some permutation of alphabet letters.. Following Harju [4] , we say that a finite word p over alphabet Σ is a stem of length n of w if w = pw 1 w 2 w 3 · · · where for each i there exists a morphism µ i on Σ extending a permutation * The author is supported by an NSERC Discovery Grant.. of the letters of Σ, with w i = µ i (p). In this case the word w has an n-stem factorization.. Harju poses this problem: Are there infinite square-free ternary words with an n-stem factorization for each n ≥ 13?. In the paper of Harju it is already shown that such words exist for n = 14, 15, 16. Mike Müller has pointed out to the author [8] the existence of such words for n = 20, 21 and 22:. For n = 20, a square-free word with a stem of length 20 is generated by the morphism. It is squarefree by Crochemore's characterization and each image is a concatenation of images of the stem 01210201021012102120 under various permutations.. For n = 21, one could take. For n = 22, here is a solution from Müller:. We give solutions in the cases n ≥ 13, n = 14, 15, 16, 20, 21, 22. For each of these cases, we give an n-uniform square-free ternary morphism, where the images of letters are permutations of each other. This solves all but finitely many cases of the question of the author and Rampersad [3] : Do there exist k-uniform squarefree ternary morphisms for all k ≥ 11?

O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O


A new framework for urban sustainability assessments: Linking complexity, information and policy Urban systems emerge as distinct entities from the complex interactions among social, economic and cultural attributes, and information, energy and material stocks and flows that operate on different temporal and spatial scales. Such complexity poses a challenge to identify the causes of urban environmental problems and how to address them without causing greater deterioration. Planning has traditionally focused on regulating the location and intensity of urban activities to avoid environmental degradation, often based on assumptions that are rarely revisited and producing ambiguous effects. The key intellectual challenge for urban policy-makers is a fuller understanding of the complexity of urban systems and their environment. We address this challenge by developing an assessment framework with two main components: (1) a simple agent-based model of a hypothetical urbanizing area that integrates data on spatial economic and policy decisions, energy and fuel use, air pollution emissions and assimilation, to test how residential and policy decisions affect urban form, consumption and pollution; (2) an information index to define the degree of order and sustainability of the hypothetical urban system in the different scenarios, to determine whether specific policy and individual decisions contribute to the sustainability of the entire urban system or to its collapse. Cities are spatial patterns that persist in time, in which no single constituent remains in place (Holland, 1995) . Urban systems emerge as distinct entities from the complex interactions among social, financial, and cultural attributes, and information, energy, and material stocks and flows that operate on different temporal and spatial scales. Urban environmental problems (e.g., air pollution, open space fragmentation and excessive fuel consumption) create the pressing need for urban sustainability. Such complexity poses a challenge to identify the causes of urban environmental problems and how to address them without causing greater deterioration. Environmental planning has traditionally addressed these problems with policies regulating the location and intensity of urban activities, often based on assumptions about urban and environmental dynamics that are rarely revisited (Alberti, 1999; Chin, 2002; Ewing, 1994 Ewing, , 1997 Neuman, 2005) . Given the complexity of urban systems and the environment that supports them, the key intellectual challenge of urban sustainability is a fuller understanding of the dynamic spatial interactions among the components of the coupled urban-environmental system. Such understanding can inform urban decision-makers of the environmental consequences of responding to urban needs.. We seek to contribute to this understanding by developing an assessment framework with two main components. The first includes the development of a simple agent-based model of a hypothetical urbanizing area that integrates data on spatial economic and policy decisions, energy and fuel use, air pollution emissions and assimilation. We use our model to test how different residential preferences and landscape characteristics shape the development of urban areas, in turn affecting energy use and pollution patterns, and how different policies can affect this relationship. The second component of our assessment framework involves 0198-9715/$ -see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compenvurbsys.2008.08.003 defining the stability, degree of order and sustainability of the hypothetical urban system using information theory and indexes that can explain the system-wide states and trends. This integrated framework enables us to make meaningful comparisons among urban scenarios, adding to the existing array of sustainability assessment tools, such as the ecological footprint, life-cycle analysis, and integrated land-use transport GIS models (Deakin, Mitchell, Nijkamp, & Vreeker, 2007 provide a summary of some of these methods). The specific contribution of the proposed framework is that it explicitly connects policy, decentralized decision-making and environmental quality, and analyzes both temporal and spatial trends of these interactions. In this manner, scholars and policy-makers can explore how policy influences behaviour, the impact of such behaviour on environmental variables, and whether this impact contributes to the sustainability of the entire urban system or to its collapse.. Our paper also seeks to question and clarify the assumptions about the relationship between specific land-use patterns and energy use and concentration of pollutants. Empirical findings are inconclusive (Alberti, 1999; Garde, 2004; Neuman, 2005) , suggesting not that either position is necessarily wrong, but that this relationship may be nonlinear or may vary with the scale of analysis. For example, increases in density support energy efficiency by shortening trips and reducing household energy requirements. Beyond a certain area size, however, high urban densities generate negative externalities due to congestion and concentration of pollutants beyond the environment's assimilative capacity, while low densities allow for pollution dispersion. This observation depends on the scale of measurement, however. Low densities may be accommodated in different layouts, e.g., uniformly distributed or clustered in multiple centres, which may significantly affect energy use and pollution. While the density is equally low at a regional scale in both scenarios, the local high densities in the second layout support the decentralization of activities and shorter trips. In contrast, the first layout imposes longer distances and lower household energy efficiency, resulting in overall higher energy consumption and congestion, more so if trip destinations are located outside the region and if densities are such that public transportation modes cannot be supported. The implications for policy are different in each scenario, as the justification for intervention varies to reflect the different market failures (congestion versus distortions of transportation costs and land prices).. Finally, we also seek to expand the suite of metrics that can help gauge the sustainability of an urban system in its present state, or possible future states as behaviour, technology and policy change. While agent-based models can provide a variety of spatial and temporal output measures computed at different scales, we seek to illustrate the multi-dimensional insights that can be derived when they are combined with a new dimension of impact that accounts for the long-term sustainability and stability of the entire system based on the outputs of the agent-based model. Such combination of multiple dimensions can give a clearer picture of the consequences of specific policies and individual behaviours, as the various scales of impact are evaluated together.. In sum, the purpose of our paper is to illustrate how the integration of agent-based modelling and the Fisher Information Index can help identify policy and behavioural lever points to improve environmental quality, test the assumptions about how urban form relates to environmental quality, and provide multi-dimensional assessments of complex urban-environmental systems. The next section describes the components of the agent-based model we developed to address these questions, followed by the simulation results from different policy and behavioural scenarios and their analysis with Fisher Information Index. We conclude with general observations about urban sustainability based on the various measures and about the applicability of this assessment framework to specific urban areas. Before it can effectively be used for policy, the proposed framework requires further testing and fine-tuning, creating scenarios with the factors currently present in USAFE but that have not been tested for this paper. Tests will explore the influence of alternative zoning and school improvement layouts, crime reduction, infrastructure decisions (e.g., road construction, water and sewer coverage) and forestation. Technological scenarios will test the effect of different sources of power generation, fuel efficiency and emission factors. We will analyze the effect of urban patterns on the other pollutants included in the model but not tested here, and use the Fisher Information Index applied in both time and space to assess the sustainability of the different urban scenarios.. We also seek to apply this framework to specific case studies, for which we will collect spatial data to adjust USAFE's input maps and parameter values. In this effort, we will also adjust farm decisions to reflect more realistic fuel use and resulting emissions.. An initial extension to the model is planned to show forest clearing and re-growth. Development is both attracted to forested areas and is the cause of forest clearing. Such forces make it difficult to anticipate the effectiveness of forest protection policies, for example, on carbon assimilation. A second extension of the model will link crime to population characteristics. Both crime and residential density increase with development pressures, and also discourage further development. Currently, forest cover, zoning restrictions and crime rates are fixed as initial conditions. By allowing them to vary with development, USAFE will enable us to explore the path-dependence typical of urbanization processes that are only weakly represented in the current version of USAFE. A final extension of the model will compute the utility of residents to determine the distributional effects of policies, accounting for private and public gains (e.g., location amenities, carbon sequestration) and private and public losses (e.g., transportation costs, pollution). We expect that the Fisher Index will be especially powerful in assessing the outputs in these scenarios.. As much as improvement is necessary, increasing the detail and complexity in the model will be at the expense of understanding. This is an inherent challenge of using complexity-based modelling for policy, extensively discussed in the literature ( Bankes, 1993 ( Bankes, , 2002 Bankes, Lempert, & Popper, 2002; Clarke, 2005; Manson, 2007; Miller & Page, 2007; O'Sullivan & Torrens, 2001) . Too much detail leads to excessive complication and comprehensiveness, resulting in ''black boxes" whose assumptions and implications are hard to understand and question (Lee, 1973) . Too little detail, on the other hand, can also fail to capture the important aspects of the system we seek to understand and propose policies for. Determining the right balance between realism and interpretability in any modelling endeavour is an art, a creative process that requires continuous exploration, experimentation and adjustment within the specific context in which the model is being developed. Through this process, we seek to provide powerful insights about the interactions that shape urban areas, and how they translate not only into environmental quality but also into the stability and sustainability of the urban system over time.

O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Integrating transcriptional and protein interaction networks to prioritize condition-specific master regulators of enrichment analysis to the growth rate cutoff for yeast strainsWe selected the 53 strains of yeast that had a lower growth rate in rapamycin than under normal growth conditions. To run GMIT, we proceeded as described in the Methods section, redoing every step for the whole dataset of 53 strains, including data binning, filtering the prior network, and computing conditional mutual information. We also ran PANDA on this expanded dataset, redoing the randomization and the network inference following the identical protocol described in the Methods. As we had done with the smaller 30-strain dataset, we then computed the degree of each transcription factor in the GMIT or PANDA networks, ranked them and combined it with the rank of the degree in either the full protein interaction network or the yeast two-hybrid network. The results, including ROC curves, AUC values, p-values from the Kolmogorov-Smirnov test, and odds ratios for the top 20% TFs ranked by each metric, are shown in Supplementary Figure 1 . In every case but one, the enrichment for driver TFs among the top-ranked regulators is highest for the combined metric. The exception is the combination of the PANDA network with the Y2H network, where the odds ratio for the combined metric is the same as the odds ratio for the PANDA network alone; however, even in this case, the pvalue and AUC of the overall ranking (P = 0.02, AUC = 0.74) performs better than each network individually (P = 0.1, AUC = 0.6 [PANDA]; P = 0.03, AUC = 0.72 [Y2H]). Therefore, the improvement in enrichment appears to be a general feature of the combined network score, and it is independent of changing the number of samples included in the expression dataset, the type of network inference method, the type of protein interaction network, or even changing all of these factors at the same time.We calculated the differential expression rank R D of all the transcription factors by ranking the B-statistic computed using limma. We then combined it with the PPI rank to compute the score S D = (R D + R P )/2. For the rapamycin data, we plotted the ROC curves and odds ratios corresponding to differential expression, degree in transcriptional network (either GMIT or PANDA), degree in the PPI network (either the full network or Y2H only), the combined network metric S, and the combined differential expression and PPI metric S D . For the viral oncogene data, we followed the same procedure, except with only one type of PPI network (Y2H).The results for rapamycin are shown in Supplementary Figure 4 . In every case, combining the differential expression ranking with the PPI degree resulted in an improvement in the enrichment in driver TFs. The improvement was comparable to the combined network score, as reflected in the ROC curves as well as the odds ratios. One exception was that the odds ratio for the top 20% ranked TFs was higher for the combined network metric in the case of the Y2H protein interaction network. However, in general, the combined differential expression metric performed well in the context of rapamycin response in yeast.The results for the viral oncogene data are shown in Supplementary Figure 7 . The combination of differential expression and protein interaction network did not perform as well as the combined network degree, in the case of both GMIT and PANDA networks, and both in terms of the specificity of the top 10% of the TFs as well as over the entire ranking (as visualized in the ROC curves). One possible reason is that the set of differentially expressed drivers has different characteristics than the set of high degree drivers in the PPI network, so

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A Spatially Augmented Reality Sketching Interface for Architectural Daylighting Design Abstract-We present an application of interactive global illumination and spatially augmented reality to architectural daylight modeling that allows designers to explore alternative designs and new technologies for improving the sustainability of their buildings. Images of a model in the real world, captured by a camera above the scene, are processed to construct a virtual 3D model. To achieve interactive rendering rates, we use a hybrid rendering technique, leveraging radiosity to simulate the interreflectance between diffuse patches and shadow volumes to generate per-pixel direct illumination. The rendered images are then projected on the real model by four calibrated projectors to help users study the daylighting illumination. The virtual heliodon is a physical design environment in which multiple designers, a designer and a client, or a teacher and students can gather to experience animated visualizations of the natural illumination within a proposed design by controlling the time of day, season, and climate. Furthermore, participants may interactively redesign the geometry and materials of the space by manipulating physical design elements and see the updated lighting simulation. T HE particular area of architectural design we explore is daylighting: the use of windows and reflective surfaces to allow natural light from the sun and sky to provide effective and interesting internal illumination (Fig. 1) . Appropriate daylighting strategies can reduce energy consumption for electric lighting and create more aesthetically interesting and comfortable architectural spaces. The overall aim of a successful daylighting design is to increase the amount of useful daylight in an architecturally satisfying way while avoiding the problematic aspects of natural illumination including the risk of glare or overheating.. A heliodon is a traditional daylighting analysis device in which a small-scale physical model (often 1=4 00 ¼ 1 0 ) is affixed to a platform and rotated relative to a fixed light source that represents the sun (Fig. 2a) . Alternatively, in more complex devices, the light source may be mechanically moved around a stationary platform. By studying the distribution of light within the model, the designer gains instantaneous and intuitive qualitative feedback on direct sun penetration and the corresponding indirect illumination.. Our new virtual heliodon, shown in Figs. 2b and 2c, shares many of the features of the traditional heliodon and includes a number of important additional advantages. With a heliodon the user must awkwardly peer through windows (possibly blocking light) or place tiny video cameras within the model; in contrast, the virtual heliodon does not require a physical ceiling on the model, allowing easy inspection of the interior spaces. Furthermore, initial construction and revision of models for the virtual heliodon is faster than for heliodon models because the corners of the model need not precisely align. In the virtual heliodon we "fill in" cracks between walls that in a traditional heliodon would allow light to incorrectly leak into the model. Editing surface materials for a heliodon model is labor intensive and impossible if appropriate scale versions of the material are not available (e.g., Venetian blinds), but with our system surface and window materials can be changed digitally. Traditional heliodons only simulate direct illumination from the sun and do not automatically adjust the intensity with altitude. Our system performs a quantitatively accurate simulation of illumination from both the sun and sky and also models climate variations.. We begin with an efficient, interactive, and accurate global illumination algorithm for daylighting. We draw from the body of literature on virtual and augmented reality to build a novel daylighting tool based on this rendering method. Our interactive tangible daylighting design system can be used by novice or experienced designers who need not be experts in daylighting technology or advanced graphical simulations. We have used spatially augmented reality to build a virtual heliodon system [42] , [43] that complements modern desktop architectural daylighting design software tools. The user positions a set of small-scale physical walls within the workspace to "sketch" the 3D geometry of their design. Tangible walls are chosen to project daylighting simulation on instead of pure virtual reality solutions because they are low-cost, convenient, and easy for users to rebuild. Images captured by a camera mounted above the scene are processed to detect the wall positions. Gaps between the wall are filled to construct a closed 3D mesh (Fig. 5) . The daylighting solution in the virtual 3D building is computed with the hybrid radiosity/shadow volume global illumination rendering method (shown in Fig. 3 ), accounting for illumination from both the sun and sky. This illumination solution is then displayed on the physical walls by four calibrated projectors.. The efficiency of the hybrid rendering method makes it possible for users to interact with daylighting simulations in our system. The user can explore the high-dimensional configuration space of the design by adjusting the position of wall modules to manipulate the geometry of the design. Through a wireless remote mouse, the user can vary the external conditions such as the sun position (time of day and day of the year) and weather conditions. The simulation results are recomputed and updated at interactive rates. In addition to visualizing the environment for a single point in time, the system allows time-lapse animations as the sun moves across the sky during the course of a day, throughout the season, or under different weather conditions. The novel contributions of our project include:. . Dynamic and engaging visualization of daylighting variation for different moments (time of day and day of the year). . Spatially augmented reality for visualization and exploration of natural illumination by multiple simultaneous users.. . Interactive redesign of the scene geometry and materials with online recomputation of the resulting lighting simulation. . A rich language of wall primitives and other design elements for sketching a wide array of architectural spaces. We have presented the virtual heliodon, a novel application of spatially augmented reality for architectural daylighting design. This small-scale, low-cost physical design environment is practical for academic architectural studios or professional design firms and suitable for installation in a multipurpose space such as a conference room. Our system has a number of advantages over the traditional heliodon including ease of visualizing the interior lighting conditions and ease of editing the model geometry. This new design tool supports enhanced communication between client and architect and provides a platform for education in sustainable architectural design practice. The system can improve the architect's ability to make effective use of daylighting and reduce the need for supplemental electric lighting, reducing consumption of nonrenewable energy resources. The same framework can be applied to other architectural design problems that require complex physical simulation and visualization, including passive solar heating and cooling, acoustics, aerodynamic building envelopes, and structural analysis. . For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib. The current system runs sufficiently fast for interactive daylighting visualization during iterative architectural design. Using unoptimized code, it takes approximately 12.5 seconds to generate the projection images for a model with six walls and two windows. Of this, approximately 0.05 seconds are used to process the captured image, and then additional 3.5 seconds are used to remesh the wall coordinates into a closed triangle mesh. After remeshing, it takes approximately 9 seconds to calculate form factors for the mesh with about 2,000 triangles and compute an initial lighting solution. Because the most time consuming part of the system (remeshing and form factor calculations) need only be performed once for a particular wall geometry, additional lighting solutions for this geometry (changing the materials of walls and floors, time of day, day of the year, weather conditions, or site orientation) can be done in about 1 second. This facilitates interactive time-lapse animations of the daylighting solution. Fig. 11 shows still frames from an animation of the daily variation in lighting conditions for a fixed model geometry, and Fig. 12 shows several images from an iterative design session. Preliminary feedback from architecture students about the system has been positive. We plan to conduct formal user studies with our virtual heliodon to compare it with both the traditional heliodon and state-of-the-art daylighting analysis software packages. We believe our system will result in an improved understanding of the dynamic nature of daylighting and increased awareness of and attention to sustainable architecture design. Although our current system with four overhead projectors can be used to model many room configurations, tall, narrow room designs can result in areas occluded from all four projectors. Although this results in a presentation that does not reflect the true lighting solution, such shadow areas are easily distinguished visually and we have not found them to be overly distracting.. Accurate representation of the full dynamic range of daylight is problematic. Projectors have a limited range of intensity: our four projectors together can produce a maximum 12,000 lux on the table surface; direct sunlight is 10 Â brighter. Even the brightest projector commercially available (15,000 lumens [54] ) projected onto a 3 0 Â 4 0 screen would result in only one ninth the brightness of direct sunlight. To compensate for the limited dynamic range, we will extend tone mapping research [55] to tangible displays. As an alternative, we plan to explore false-color representations which can indicate overall lighting levels, or highlight areas that experience significant glare or inadequate illumination during some parts of the day.. While our current wall detection and tracking method has proven to be very robust, the algorithm may become less reliable as many more colors are used to code different wall heights or styles. Accordingly, we plan to explore other options for wall labels, such as: bar-code sequences on wall tops, a magnetic position tracking system [56] , visible or infrared LEDs with unique colors or on/off sequences, and active systems where our projectors display structured light patterns captured with the camera or synchronized sensors that uniquely determine their 3D position [16] . Cost, robustness, ease-of-use, accuracy, speed, and how to minimize visual distractions for the designer will all be considered.. To correct for unintentional scattering of projected illumination between physical wall surfaces [11] , we will perform inverse global illumination and color/intensity compensation [20] to determine the color that should be projected onto each wall so that the resulting intensity matches the software simulation. A solution to the general problem does not necessarily exist-we cannot emulate a scene with a white wall adjacent to a fully absorptive black floor because we cannot prevent scattered light from striking and reflecting off the real-world floor. However, several characteristics of our application will allow us to compensate for most of the scattering: the diffuse surface assumption holds for most architectural scenes; neutral mid-range tones are used in most designs; gaps between the wall panels reduce surface interreflection; and the use of tone mapping will effectively reduce the desired contrast of the projections.. The mechanisms for user interaction can be further enhanced. We envision extending the system with physical control elements or gestures similar to the north arrow to adjust other aspects of the geometry or simulation; for example, a cloud shape could be used to indicate an overcast sky condition.

O B I I O O O O B I O O O O O O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Image interpolation by two-dimensional parametric cubic convolution Abstract-Cubic convolution is a popular method for image interpolation. Traditionally, the piecewise-cubic kernel has been derived in one dimension with one parameter and applied to two-dimensional (2-D) images in a separable fashion. However, images typically are statistically nonseparable, which motivates this investigation of nonseparable cubic convolution. This paper derives two new nonseparable, 2-D cubic-convolution kernels. The first kernel, with three parameters (designated 2D-3PCC), is the most generalwith constraints for biaxial symmetry, diagonal (or 90 rotational) symmetry, continuity, and smoothness. The second kernel, with five parameters (designated 2D-5PCC), relaxes the constraint of diagonal symmetry, based on the observation that many images have rotationally asymmetric statistical properties. This paper also develops a closed-form solution for determining the optimal parameter values for parametric cubic-convolution kernels with respect to ensembles of scenes characterized by autocorrelation (or power spectrum). This solution establishes a practical foundation for adaptive interpolation based on local autocorrelation estimates. Quantitative fidelity analyses and visual experiments indicate that these new methods can outperform several popular interpolation methods. An analysis of the error budgets for reconstruction error associated with blurring and aliasing illustrates that the methods improve interpolation fidelity for images with aliased components. For images with little or no aliasing, the methods yield results similar to other popular methods. Both 2D-3PCC and 2D-5PCC are low-order polynomials with small spatial support and so are easy to implement and efficient to apply. Separable bicubic convolution is one of the most popular methods for image interpolation, but it is based on the assumption of separability, which typically is not true. This paper examines the theoretical and practical issues of nonseparable, 2-D cubic convolution. This paper develops two nonseparable, 2-D cubic-convolution kernels. The first kernel, 2D-3PCC, has three parameters and is the most general 2-D, piecewise-cubic interpolator defined on with constraints for biaxial symmetry, diagonal (or 90 rotational) symmetry, continuity, and smoothness. The second kernel, 2D-5PCC, relaxes the diagonal-symmetry constraint on 2D-3PCC and thus is more general. The roadmap of relaxing constraints could be used to develop other formulations of cubic convolution. This paper also develops a closed-form solution for the optimal parameters that minimize the expected MSE over an ensemble of scenes characterized by autocorrelation. The spatialdomain formulation for optimal parameterization is computationally efficient and establishes the basis for locally adaptive filters. The closed-form solution supports any approximation of scene autocorrelation and experimental results show that parameterization is robust with respect to approximation errors. In particular, this paper presents a MRF model with affine transformation to model scene autocorrelation and derives a closed-form solution to fit the model. Experimental results show that, for a range of scenes, the model can be fit to closely estimate the scene autocorrelation using only the autocorrelation of a down-sampled image. The closed-form solution for the model fit establishes the basis for locally adaptive characterization of the scene autocorrelation.. Quantitative fidelity analyses and visual experiments indicate that 2D-3PCC and 2D-5PCC outperform several popular interpolation methods (including a spatially adaptive interpolator) for real scenes. In theoretical and experimental analyses, both 2D-3PCC and 2D-5PCC better balance error budget tradeoffs involving scene signal and aliasing and more closely resemble the unconstrained least-squares interpolator over a range of scenes with different spatial content. In these analyses, 2D-5PCC is especially effective compared to separable methods for interpolating off-axes spatial frequencies. Both 2D-3PCC and 2D-5PCC are low-order polynomials with small spatial support and so are easy to implement and efficient to apply in the spatial domain, but have higher computational cost than separable kernels.. Our ongoing research is extending nonseparable cubic convolution to image restoration and exploiting other random field models. The 2D-3PCC and 2D-5PCC kernels are derived subject to several constraints related to interpolation: the kernel value at (0,0) is constrained to one and the values at the other knots are constrained to zero. Relaxing these constraints allows the kernels to correct for acquisition artifacts and perform image restoration [25] , [26] . Parametric cubic convolution can be optimized for any random field model. The geostatistical community uses random fields widely. It is recognized that properties of optimal interpolation kernels are determined mainly by the local behavior of the semivariogram of a presumed random field [27] . Both 2D-3PCC and 2D-5PCC are designed to exploit local autocorrelation and so are candidates for efficient interpolation with respect to random fields.

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