Data exploration in evolutionary reconstruction of PET images

Genetic Programming and Evolvable Machines https://doi.org/10.1007/s10710-018-9330-7 Data exploration in evolutionary reconstruction of PET images Cameron C. Gray1 · Shatha F. Al‑Maliki1,2 · Franck P. Vidal1 Received: 11 December 2017 / Revised: 5 July 2018 © The Author(s) 2018 Abstract This work is based on a cooperative co-evolution algorithm called ‘Fly Algorithm’, which is an evolutionary algorithm (EA) where individuals are called ‘flies’. It is a specific case of the ‘Parisian Approach’ where the solution of an optimisation problem is a set of individuals (e.g. the whole population) instead of a single individual (the best one) as in typical EAs. The optimisation problem considered here is tomography reconstruction in positron emission tomography (PET). It estimates the concentration of a radioactive substance (called a radiotracer) within the body. Tomography, in this context, is considered as a difficult ill-posed inverse problem. The Fly Algorithm aims at optimising the position of 3-D points that mimic the radiotracer. At the end of the optimisation process, the fly population is extracted as it corresponds to an estimate of the radioactive concentration. During the optimisation loop a lot of data is generated by the algorithm, such as image metrics, duration, and internal states. This data is recorded in a log file that can be post-processed and visualised. We propose using information visualisation and user interaction techniques to explore the algorithm’s internal data. Our aim is to better understand what happens during the evolutionary loop. Using an example, we demonstrate that it is possible to interactively discover when an early termination could be triggered. It is implemented in a new stopping criterion. It is tested on two other examples on which it leads to a 60% reduction of the number of iterations without any loss of accuracy. Keywords Fly Algorithm · Tomography reconstruction · Information visualisation · Data exploration · Artificial evolution · Parisian evolution Abbreviations EA Evolutionary algorithm PET Positron emission tomography ET Emission tomography * Franck P. Vidal Extended author information available on the last page of the article 13 Vol.:(0123456789) Genetic Programming and Evolvable Machines InfoVis Information visualisation CSV Comma-Separated Values CCEA Cooperative co-evolution algorithm CT Computed tomography SPECT Single-photon emission computed tomography SNR Signal-to-noise ratio MRI Magnetic resonance imaging keV Kiloelectron volt LOR Line of response MLEM Maximum-Likelihood Expectation-Maximization EM Expectation-Maximization OSEM Ordered Subset Expectation-Maximization voxel Volume element SVG Structured Vector Graphics MAE Mean absolute error MSE Mean squared error RMSE Root mean squared error ZNCC Zero-normalised cross-correlation PSNR Peak signal-to-noise ratio SSIM Structural similarity DSSIM Structural dissimilarity TV Total variation 1 Introduction This research is related to the use of evolutionary computing in nuclear medicine, more particularly positron emission tomography (PET) reconstruction. In this paper, we investigate the use of information visualisation (InfoVis) and data exploration to understand some of the behaviours of an evolutionary algorithm (EA). In particular, we want to assess if the algorithm could have been stopped earlier to get a reasonable solution instead of waiting until the algorithm ends and using the final solution as the problem answer. The combination of visualisation and evolutionary computing is still a relatively overlooked field. Two different approaches can be distinguished: • visualisation to understand an evolutionary algorithm [30, 38, 52, 53], and • interactive artificial evolution to improve the visualisation [9, 24, 36]. First attempts were reported at the end of the 90s. Early visualisations were using relatively basic techniques that mostly relied on plotting with limited or no interactivity. During the evolutionary PET reconstruction, multiple time series are recorded hundreds of thousands of times. Comparing these time series by hand using typical scatterplots and line charts with no interactivity is not practically feasible: 13 Genetic Programming and Evolvable Machines • The order of magnitude of each time series is different. They would need to be independently normalised before plotting. • Trial and error would be needed to choose the axis of interest because it is not necessarily straightforward to do so without a deep a priori understanding of the data. • Adjusting the data range visualised in the scatterplots would also need to be performed with trial and error. • Displaying selected images would need to be done manually. The use of Parallel Coordinate Plots is very popular to visualise high-dimensional geometry and analyse multivariate data [28], which is the type of data considered here. Interactivity using the brushing technique [37] makes it feasible to easily explore parts of this high-dimensional space and visually analyse this complex multivariate dataset. This is the approach we adopted here to develop an integrated visualisation framework dedicated to our evolutionary PET reconstruction algorithm. The emergence of information visualisation and data analytics is opening new doors for its use in the evolutionary computing domain. We adopt the first approach to analyse the evolutionary process of our image reconstruction algorithm for tomography in nuclear medicine. In typical evolutionary algorithms, the best individual of the final population is the solution of the optimisation problem. Our algorithm relies on the Parisian approach where the solution to the problem is a group of individuals, e.g. the whole population or a subset of the population. The population size progressively increases to improve the resolution of the output image. The algorithm is launched with input parameters such as the initial number of individuals, the final number of individuals, the probability of operators, etc., the final solution is extracted at the end of the optimisation process then converted into a problem-specific answer. A lot of the data is generated during the evolution process, in particular data based on error metrics and correlation measurements. Traditionally all this data is discarded at the end of the evolutionary process, as only the final population is considered. Our hypothesis is that intermediate populations and internal data should not be systematically discarded as they can be reviewed offline. They can be used to analyse the performance of the population over time. When using stagnation as the stopping criterion, the final population is not necessarily the best one due to oscillations around the minimal fitness value. In such a case, past generations will have to be accessible. Also, reaching the targeted number of individuals may not be necessary if the reconstructed image stops improving. Offline (...truncated)