Noise Estimation in Digital X-ray Images
Noise Estimation in Digital X-ray Images. Noise is an important problem for the visual quality of images and can also adversely affect image analysis. Estimating noise is a first step to control noise.
Looking at x-ray images, the negative effects of noise are especially noticeable. The reason is the compromise between the low doses on medical grounds and the behaviour of noise in x-ray images, which increases with decreasing dose. Digital x-ray detectors generate digital images with a large range of gray values. Image processing methods can be applied to these images for estimating noise and improving the images. The increasing availability of digital x-ray detectors in the future gives rise to intensify these developments.
The first step, described here, is a method for estimating the noise portion in digital x-ray images with respect to the dose. Knowledge about the sources of noise are used to develop a parametric model function whose parameters can be determined from measured data.
To get a correct set of data and also for subsequent image improvement, it is indispensable to be able to distinguish between noise and fine structures in an image. Working with sequences of x-ray images, the ability to distinguish between movement and noise is a further prerequisite for image improvement.
This contribution is based on the author’s graduate thesis to the course “Optical Technology and Image Processing”. The thesis was written at the the University of Applied Sciences in Darmstadt and carried out at Philips Medical Systems, Hamburg.
W.C. Röntgen discovered x-rays in 1895. X-rays are electromagnetic radiation between 109 and 3·1011GHz with a photon energy approximately ten thousand times higher than that of visible light. When x-rays penetrate matter, they interact with atoms resulting in effects like weakening, absorption and scattering. The weakening of the intensity can be described using the law of Lambert-Beer.
I = I0 · e-μ·d
I = Intensity
μ = material characteristic
d = thickness of penetrated material
Philips x-ray devices usually map static x-ray images logarithmically, thus making gray values in the image proportional to object thickness.
Analog detectors usually have logarithmic properties so that the logarithm of the image information is taken directly during image capturing.
The x-ray image of a typical Philips device of the type “Digital Diagnost” has a maximum size of 3,000 x 3,000 pixels with a bit depth of 14 bit.
In imaging, noise can be seen as a stochastic overlay to the image which represents the observed object. Properties of the detection system and the nature of the x-rays themselves cause the noise in x-ray images.
The number of quanta detected during exposure time is subject to statistic variations. The number varies approximately according to a Poisson distribution. Then it states that with expected number N of participating photons, the standard deviation is proportional to √N.
The method is based on the Gauss- Laplace pyramid. For the sake of clarity, the basic procedure will be explained first and subsequently some optimisations are outlined. The image is locally smoothed by a low pass filter and also a high pass filtered image is calculated, using second order derivatives (Laplace). The results L (low pass filtered image) and H (high pass filtered image) are shown in Fig. 1. In areas of constant intensity (constant gray value) with respect to L the standard deviation σ of the gray values is calculated using the data of H in the area given in L. This value σ is allocated to the gray value and repeating this procedure for a set of gray values induces a noise function. An area of constant gray value can be found and is colored red in Fig. 1a. The data in the red area in Fig. 1b. are used for calculating σ. Possible optimisations concern the selected filters and also the areas can be improved by eliminating those parts of areas which obviously belong to object boundaries. The decisive advantage of this algorithm is that no prior information about the image is required.
Areas of Direct Radiation
In areas of direct radiation, the detector is directly hit by x-rays without first penetrating an object. In these spots the detector can be saturated. Quantum noise is no longer registered by the detector; thus the signal is only loaded with the noise of the subsequent electronics. In the spots where radiation hits directly, the estimated noise does no longer represents the noise behaviour of the entire image. Areas of direct radiation are therefore masked out.
Gray Level Undersampling
The method of noise estimation is rather computation-intensive and requires a large amount of memory in particular, when large images are considered as the images produced by the detector described above (3000 x 3000 pixels). Instead of determining a set of erratic values for each gray value l of the low pass image, the set of gray values is equidistantly subdivided into a number of p predetermined classes. For all gray values in one class a common set of erratic values is determined and the estimated noise value is allocated to the gray value in the middle of each class interval.
Further optimisation of the computing time is achieved by spatial undersampling the high pass filtered image. For estimation, the input images are reduced by pick and place. In contrast to interpolation methods, this does not falsify the image information but only thins out the histogram.
Gray values or intervals of gray values with a small set of erratic values should not be used for estimating noise because the considered set of observed pixels generally has too less inner points. But a correct estimation of the standard deviation is only possible at inner points because the deviation is measured with respect to a constant level. This level has been achieved by the low pass filtering described above and the belonging filter mask determines the property “inner point”. In consequence, the number of erratic values of each gray value interval is checked. Is this number lower than a certain threshold, the estimated noise value is omitted. Measurements confirm the supposition that the set of erratic values for a gray value is approximately normal distributed. This is only true in areas of an image that do not contain any structure. If there are structures in the image, the edges of the structure affect the high pass filtered image as well. The greater the gray value step at the edge of an object, the greater the value of the gray value in the high pass image. Thus the edges of objects bias the distribution of the set of erratic values of a gray value interval, depending on their strength. Usually, the expected Gaussian distribution of the set of erratic values is then no longer given. For a quantitative description of this modification of shape the kurtosis (the standardised fourth moment of the data) is used. Each set of erratic values is checked for conformity with the Gaussian shape of their distribution. For an ideal normal distribution the value for the kurtosis equals 0. The steeper the curve, the higher the value.
In this case, the assumption of having a normal distribution is discarded if the calculated value of the kurtosis exceeds a certain threshold. For the subsequent parameter estimation of the noise function, all noise values are used whose check value is less than the fixed threshold.
Empirical and theoretical considerations give rise to assume that the functional relation between gray value and noise (the noise function) can be described approximately by an allometric function Noise(gray value)=c · Gray value a where c and a are free parameters. This can be done in a more or less good approximation for non-logarithmic and also for logarithmic images by using this function type with different parameters. The parameters c and a of the noise function are estimated from the measured noise values using nonlinear regression techniques.
The estimated noise function reflects the knowledge about the noise behaviour. Modelling the noise function by a function of two parameters reduces the whole noise description to these both parameters. Fig. 2 shows an empirical noise function and the fitted curve. The domain of the fit is to determine very carefully. It is indicated in Fig. 2 by the two black marks. Out of this range the number of inner points for a set belonging to a constant level is too small. Furthermore, other sources influence the noise function in the extreme dark and extreme bright parts of the domain, which is reflected in the diagram by the strong local deviations between empirical noise function and the fitted curve. An advantage of this method is, that all the required information can be extracted from the image without any preliminary knowledge about the image acquisition. The use of the undersampling technique makes sure that the computing time remains small. Extensive empirical studies fortify that the estimation procedure works stable. Even images showing a lot of structures have a sufficient large domain for a stable fit. So this is the first step in an image improvement process for x-ray images with the aim to reduce the noise and to enhance the contrast of the contours.
Dipl. Ing. Bernhard Pöllmann
Studied Optical Technology and Image Processing’ at the University of Applied Sciences in Darmstadt. His graduate thesis – Die Analyse des Rauschverhaltens in digitalen Röntgenbildern – was developed at Philips Medical Systems, GXR Development, Hamburg. During this time the author was supervised by Prof. Dr. Konrad Sandau.
Dipl. Ing. Bernhard Pöllmann
NeuroCheck GmbH Neckarstr. 76/1
71686 Remseck, Germany
Prof. Dr. Konrad Sandau
University of Applied Sciences Mathematics and Science Faculty
64295 Darmstadt, Germany
Tel. +49 6151 16 8659
Fax +49 6151 16 8975