Tag Archives: image decomposition

Million to one

‘All models are wrong, but some are useful’ is a quote, usually attributed to George Box, that is often cited in the context of computer models and simulations.  Working out which models are useful can be difficult and it is essential to get it right when a model is to be used to design an aircraft, support the safety case for a nuclear power station or inform regulatory risk assessment on a new chemical.  One way to identify a useful model to assess its predictions against measurements made in the real-world [see ‘Model validation’ on September 18th, 2012].  Many people have worked on validation metrics that allow predicted and measured signals to be compared; and, some result in a statement of the probability that the predicted and measured signal belong to the same population.  This works well if the predictions and measurements are, for example, the temperature measured at a single weather station over a period of time; however, these validation metrics cannot handle fields of data, for instance the map of temperature, measured with an infrared camera, in a power station during start-up.  We have been working on resolving this issue and we have recently published a paper on ‘A probabilistic metric for the validation of computational models’.  We reduce the dimensionality of a field of data, represented by values in a matrix, to a vector using orthogonal decomposition [see ‘Recognizing strain’ on October 28th, 2015].  The data field could be a map of temperature, the strain field in an aircraft wing or the topology of a landscape – it does not matter.  The decomposition is performed separately and identically on the predicted and measured data fields to create to two vectors – one each for the predictions and measurements.  We look at the differences in these two vectors and compare them against the uncertainty in the measurements to arrive at a probability that the predictions belong to the same population as the measurements.  There are subtleties in the process that I have omitted but essentially, we can take two data fields composed of millions of values and arrive at a single number to describe the usefulness of the model’s predictions.

Our paper was published by the Royal Society with a press release but in the same week as the proposed Brexit agreement and so I would like to think that it was ignored due to the overwhelming interest in the political storm around Brexit rather than its esoteric nature.

Source:

Dvurecenska K, Graham S, Patelli E & Patterson EA, A probabilistic metric for the validation of computational models, Royal Society Open Science, 5:1180687, 2018.

Credibility is in the eye of the beholder

Picture1Last month I described how computational models were used as more than fables in many areas of applied science, including engineering and precision medicine [‘Models as fables’ on March 16th, 2016].  When people need to make decisions with socioeconomic and, or personal costs, based on the predictions from these models, then the models need to be credible.  Credibility is like beauty, it is in the eye of the beholder.   It is a challenging problem to convince decision-makers, who are often not expert in the technology or modelling techniques, that the predictions are reliable and accurate.  After all, a model that is reliable and accurate but in which decision-makers have no confidence is almost useless.  In my research we are interested in the credibility of computational mechanics models that are used to optimise the design of load-bearing structures, whether it is the frame of a building, the wing of an aircraft or a hip prosthesis.  We have techniques that allow us to characterise maps of strain using feature vectors [see my post entitled ‘Recognising strain‘ on October 28th, 2015] and then to compare the ‘distances’ between the vectors representing the predictions and measurements.  If the predicted map of strain  is an perfect representation of the map measured in a physical prototype, then this ‘distance’ will be zero.  Of course, this never happens because there is noise in the measured data and our models are never perfect because they contain simplifying assumptions that make the modelling viable.  The difficult question is how much difference is acceptable between the predictions and measurements .  The public expect certainty with respect to the performance of an engineering structure whereas engineers know that there is always some uncertainty – we can reduce it but that costs money.  Money for more sophisticated models, for more computational resources to execute the models, and for more and better quality measurements.

Recognizing strain

rlpoYou can step off an express train but you can’t speed up a donkey. This is paraphrased from ‘The Fly Trap’ by Fredrik Sjöberg in the context of our adoption of faster and faster technology and the associated life style. Last week we stepped briefly off the ‘express train’ and lowered our strain levels by going to a concert given by the Royal Liverpool Philharmonic Orchestra, including pieces by Dvorak, Chopin and Tchaikovsky. I am not musical at all and so I am unable to tell you much about the performances or compositions, except to say that I enjoyed the performances as did the rest of the audience to judge from the enthusiastic applause. A good deal of my enjoyment arose from the energy of the orchestra and my ability to recognise the musical themes or acoustic features in the pieces. The previous sentence was not intended as a critic’s perspective on the concert but a tenuous link…

Recognising features is one aspect of my recent research, though in strain data rather than music. Modern digital technology allows us to acquire information-rich data maps with tens of thousands of individual data values arranged in arrays or matrices, in which it can be difficult to spot patterns or features. We treat our strain data as images and use image decomposition to compress a data matrix into a feature vector. The diagram shows the process of image decomposition, in which a colour image is converted to a map of intensity in the image. The intensity values can be stored in a matrix and we can fit sets of polynomials to them by ‘tuning’ the coefficients in the polynomials. The coefficients are gathered together in a feature vector. The original data can be reconstructed from the feature vector if you know the set of polynomials used in the decomposition process, so decomposition is also a form of data compression. It is easier to recognise features in the small number of coefficients than in the original data map, which is why we use the process and why it was developed to allow computers to perform pattern recognition tasks such as facial recognition.

decompositionSources:

Wang W, Mottershead JE, Patki A, Patterson EA, Construction of shape features for the representation of full-field displacement/strain data, Applied Mechanics and Materials, 24-25:365-370, 2010.

Patki, A.S., Patterson, E.A, Decomposing strain maps using Fourier-Zernike shape descriptors, Exptl. Mech., 52(8):1137-1149, 2012.

Nabatchian A., Abdel-Raheem E., and Ahmadi M., 2008, Human face recognition using different moment invariants: a comparative review. Congress on Image and Signal Processing, 661-666.