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I understand that it is an odd question, and this is arguably not the ideal forum, but I am at a loss.

Some of you may be familiar with DICOM as a consensus or standard format for medical image transmission and storage. It includes a lot of meta-data regarding technical aspects of in this case MRI.

One of the values is unlabeled and simply reads the name of the Hungarian mathematician: 'Riesz'.

I know that in MRI there is an inverse transform from data directly captured in frequency space to image space. This process was traditionally carried out with FFT's, but lately wavelets have become commonplace.

I had been given the unsupported information that this particular manufacturer used Daubechies wavelets, but then (and if that is true, which I doubt at this point), can you help me provide some intuition as to what step in the image generation the term Riesz would make sense? Is it sensible to assume that instead of Daubechies wavelets the image is primarily generated by Riesz-Laplace transforms or high-order Riesz transforms? In this regard, could they be the wavelet substitute for the inverse FFT? For instance in the quoted article the closing statement is:

Such transforms could turn out to be valuable for the analysis and processing of 3-D biomedical data sets (e.g., X-ray tomograms, MRI, and confocal micrographs).

Or is it more likely that the name makes reference to some sort of post-acquisition uniformity or denoising algorithm as in the first answer below? What specifically is it likely to be referencing?

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2 Answers 2

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I think the answer is in the DICOM dictionary:

(0018,9064) k-space Filtering KSpaceFiltering CS 1

This corresponds to what I am reading on the metadata transmitted (no identifiers):

44 | 0018 | 9064 | CS | RIESZ

So is it a filter in k-space (i.e. frequency space) to denoise.

More support now that the keywords are at hand. From Philips:

k-space Filtering 0018,9064 CS ANAP AUTO Applied values: COSINE,
COSINE_SQUARED, FERMI,
GAUSSIAN, HAMMING, HANNING,
LORENTZIAN, LRNTZ_GSS_TRNSFM,
NONE, RIESZ, TUKEY,
USER_DEFINED

And finally wrapping it up (from here):

For “in vivo” experiments, the MRI noise is composed of physiological noise from the sample and electronic noise of thermal origin from the conductors in the MR equipment (white noise). The statistics of the white noise are spatially invariant; however, the Fourier decomposition of the physiological noise typically peaks at low temporal frequencies (< 1/l0 sec) (Solo et al., 1997). While the level of white noise can be lowered by improving the hardware, such as by advancing the transmit-receive coil, the physiological noise, which rises with increasing signal intensity, can be reduced by increasing the spatial resolution (Kruger et al., 2001; Triantafyllou et al., 2005). Because both kinds of noise degrade the quality of the MRI datasets, affecting the interpretation of the results, low-pass filters are frequently used to eliminate its high-spatial-frequency components. Generally, denoising algorithms assume that noise is an additive Gaussian parameter, because the noise contribution that arise from the scanner’s electronics are additive for each of the real and imaginary parts of the k-space data, uncorrelated and characterized by a zero-mean Gaussian probability-density function (Hoult and Lauterbur, 1979).

The Fermi filter, normally employed in GE scanners (Friedman et al., 2006; Lowe and Sorenson, 1997), and the Hamming filter (Hamming, 1982; Holland et al., 2001; Lowe and Sorenson, 1997) are commonly used for smoothing of fMRI data. These filters usually ensure that the low-frequency data remains intact, but attenuate the high frequency components of the k-space signal. Since the power of the MRI spectrum is greatest at the center of k-space, low-pass filters predominantly eliminate noise information.

These filters seem to be options in the Philips DICOM dictionary above, although it is the Riecz filter that they happen to use in their magnets (?).

In conclusion, it is a filter operating in k-space to remove salt-and-pepper noise (SPN). From here:

Salt-and-pepper noise (SPN) is a type of impulse noise, which mainly occurs during the process of acquisition and transmission. In this type of noise, some pixels of a digital image have a maximum or minimum value.

Adaptive Riesz mean filter In this section, firstly, we present two essential definitions, i.e. of pixel similarity and pixel similarity-based Riesz mean of a k-approximate matrix. Here, the pixel similarity is a similarity function used to calculate the value of the similarity between two pixels. The function produces a value close to 1, if the pixels are close. In the summability theory, the Riesz mean is widely used to converge a non-converging sequence via a sequence of non-negative real numbers which are not all zero. In the present study, Riesz-mean has been exploited to calculate the new value of the centre pixel of a k-approximate matrix by using the values of pixel similarity between the centre pixel and the other pixels therein.

Let A be an IM (Image Matrix). Then, the value

$$\mathrm{ps}\left({a}_{ij},{a}_{st}\right):= {\left(\frac{1}{1+\left|i-s\right|+\left|j-t\right|}\right)}^2$$

is called pixel similarity between ${a}_{ij}$ and ${a}_{st}$.

Let $A$ be an NIM (noise image matrix). Then the value

$$Rm\left({A}_{ij}^k\right):= \frac{\sum \limits_{\left(s,t\right)\in {I}_{ij}^k}\mathrm{ps}\left({a}_{st},{a}_{\left(k+1\right)\left(k+1\right)}\right){a}_{st}}{\sum \limits_{\left(s,t\right)\in {I}_{ij}^k}\mathrm{ps}\left({a}_{st},{a}_{\left(k+1\right)\left(k+1\right)}\right)}$$

is called Riesz mean of ${A}_{ij}^k.$ Here ${I}_{ij}^k:= \left\{\left(s,t\right):\kern0.5em {a}_{st}\ is\ a\ regular\ entry\ of\ {A}_{ij}^k\right\}.$ > Secondly, we discuss Adaptive Riesz Mean Filter (ARmF), a new filter. Moreover, it avails of the concept of the pixel similarity-based Riesz mean different from the classical mean/Cesàro mean or medians to calculate the new value of the centre pixel of a k-approximate matrix. The reason for we prefer the Riesz mean based on the pixel similarity is when an SPN removal comes into question it performs better than other different similarity/distance functions, such as p-norm, Euclid, and Hamming.


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    $\begingroup$ Please select this as the answer if it answers your question! Happy to have helped (I hope!). $\endgroup$
    – Peter K.
    Commented Jul 16, 2022 at 11:52
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Focusing on the DICOM part, I've found this paper which supports your contention that it's more to do with directional filtering.

Directional derivative images


I suspect it refers to the (Adaptive Modified) Riesz Mean Filter.

They define the Riesz Mean:

Riesz Mean

and then apply their algorithm to reduce salt-and-pepper noise using it:

enter image description here

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  • $\begingroup$ Thank you! "Salt-and-pepper" noise I believe is distinct from "thermal noise" generated by local magnetic fields in the patient, as well as the electric equipment: the latter is the unavoidable Gaussian noise. The salt-and-pepper sounds to me like the noise related to reconstructing the image from multiple coil elements or channels. Would that make sense? If so, the algorithm would possibly be intended to provide uniformity? $\endgroup$ Commented Jul 15, 2022 at 17:40
  • $\begingroup$ My other option is that it is this... I just think it is unlikely for the name of the filter to make it to the DICOM, when the wavelet type is not there... $\endgroup$ Commented Jul 15, 2022 at 18:01
  • $\begingroup$ @AntoniParellada : I suspect it's this second approach they're talking about --- directional derivatives. $\endgroup$
    – Peter K.
    Commented Jul 15, 2022 at 21:01
  • $\begingroup$ I'm not sure I follow... Are you hinting that you changed your mind, and that this makes reference to the process of transforming from frequency to image space via wavelets (Riesz's), or that the paper you are quoting further supports your impression that this is a filter for salt-and-pepper noise? $\endgroup$ Commented Jul 15, 2022 at 23:25
  • $\begingroup$ I think you had it (+1 and thank you!), although I believe I got the concrete answer through some sleuth work, and I'm posting it for reference. $\endgroup$ Commented Jul 16, 2022 at 0:29

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