# US 10,120,051 B2 · Method for estimating a convolution product

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- Number: US 10,120,051 B2
- Document ID: US10120051B2
- Status: Granted
- Kind: B2
- Application number: 15/339,733
- Filed: 2016-10-31
- Published: 2017-05-25
- Granted: 2018-11-06
- Earliest claimed: 2015-10-29
- Assignees: Universite Grenoble Alpes
- Applicants: Universite Grenoble Alpes
- Inventors: Laurent Lamalle
- Web page: https://patentgrep.com/patent/US10120051B2

## CPC

- G01R 33/56: Image enhancement or correction, e.g. subtraction or averaging techniques {, e.g. improvement of signal-to-noise ratio and resolution}
- G01R 33/482: {using a Cartesian trajectory}
- G01R 33/5608: {Data processing and visualization specially adapted for MR, e.g. for feature analysis and pattern recognition on the basis of measured MR data, segmentation of measured MR data, edge contour detection on the basis of measured MR data, for enhancing measured MR data in terms of signal-to-noise ratio by means of noise filtering or apodization, for enhancing measured MR data in terms of resolution by means for deblurring, windowing, zero filling, or generation of gray-scaled images, colour-coded images or images displaying vectors instead of pixels (image data processing or generation, in general G06T)}
- G06F 17/15: Correlation function computation {including computation of convolution operations (arithmetic circuits for sum of products per se, e.g. multiply-accumulators G06F7/5443; digital filters, e.g. FIR, IIR, adaptive filters H03H17/00)}
- G06T 5/20: using local operators
- G06T 7/0012: {Biomedical image inspection}
- G06T 12/20: Inverse problem, i.e. transformations from projection space into object space
- G01R 33/4824: {using a non-Cartesian trajectory}
- G06T 2207/10088: Magnetic resonance imaging [MRI]
- G06T 2207/30016: Brain
- G01R: MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES (indicating correct tuning of resonant circuits H03J3/12)
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- G06T: IMAGE DATA PROCESSING OR GENERATION, IN GENERAL

## Abstract

The invention is a method for estimating a product of convolution of a measurement function, representing measurements, defined according to first coordinates, with a convolution kernel. The convolution product is estimated at second coordinates, different from said first coordinates. The measurement function represents measurements performed on the basis of a detector, in particular a detector of physical quantities. An exemplary application of the method relates to Magnetic Resonance Imaging, the method allowing a redistributing of measured data, not uniformly distributed, according to a uniform mesh, prior to the application of a fast Fourier transform algorithm.

## Claims

### Claim 1

A Method for generating an image by processing a measurement function (ƒ), the measurement function being representative of measurements acquired by a detector, the measurement function being defined in a starting space, according to first coordinates (ri), the method including the estimation, at second coordinates (uj) of said starting space, of a value of a product of convolution (f*g) of the measurement function (ƒ) with a convolution kernel (g), defined on a support, called the convolution support (Ω), the method comprising:
acquiring measurements by a detector;
establishing the measurement function (ƒ), according to first coordinates (ri) of the starting space on the basis of the measurements (15) arising from the detector; and
defining second coordinates in the starting space (uj),
wherein the method further comprises, for each second coordinate (uj):
a) on the basis of the convolution support (Ω), defining a centered convolution support (Ωj), associated with said second coordinate (uj), in such a way that said second coordinate (uj) constitutes the center of said centered convolution support (Ωj);
b) defining a mesh of said centered convolution support (Ωj), to associate a surface element (Si,j) with each first coordinate (ri) included in said centered convolution support (Ωj);
c) determining a weighting factor (∂i,j) associated with each first coordinate (ri) included in said centered convolution support (Ωj);
d) computing a product of convolution of said measurement function (ƒ) with said kernel (g) according to said centered convolution support (Ωj), by weighting a value ƒ(ri) of the measurement function, at each first coordinate (ri) included in said centered convolution support (Ωj), with the weighting factor (∂i,j) associated with said first coordinate (ri), determined during step c);
e) estimating a value of said convolution product (f*g(uj)) at said second coordinate (uj) as a function of the computation carried out during step d);
wherein steps a) to e) are implemented, for each second coordinate (uj), by at least one microprocessor,
wherein during step b), the mesh is defined in such a way that each surface element (Si,j) is contained in said centered convolution support (Ωj) and in such a way that the weighting factor (∂i,j) associated with one and the same first coordinate (ri) varies as a function of the second coordinate (uj), and
wherein the method further comprises performing an inverse Fourier Transform or a Fourier Transform on the estimated values of said convolution product at the second coordinates and generating an image representing measurements based on an output of the inverse Fourier Transform or based on an output of the Fourier Transform.

### Claim 2 (depends on 1)

The method according to claim 1, in which during step d), the convolution product comprises a sum of the products:
of a value ƒ(ri) of the measurement function at a first coordinate (ri),
of a value (g(uj−ri)) of the convolution kernel at said first coordinate referenced to the second coordinate, and
of the weighting factor (∂i,j) determined during step c), the latter being associated with said first coordinate (ri) and with said second coordinate (uj), and
wherein said products are computed for each first coordinate (ri) included in the centered convolution support (Ωj) associated with a second coordinate (uj).

### Claim 3 (depends on 1)

The Method according to claim 1, further comprising, prior to steps a) to e), establishing an initial mesh of the starting space, to associate, with each first coordinate (ri), an initial surface element (Si), and in which during at least one step b), each surface element (Si,j) associated with a first coordinate (ri) is determined by truncating said initial mesh by said centered convolution support (Ωj), so that each initial surface element (Si) extending beyond said centered convolution support is modified to be delimited by said centered convolution support.

### Claim 4 (depends on 1)

The Method according to claim 1, further comprising during at least one step b), meshing said centered convolution support is such that at least one surface element (Si,j) is defined independently of any initial mesh.

### Claim 5 (depends on 4)

The Method according to claim 4, wherein said step b) further comprises:
determining, in said centered convolution support (Ωj), first particular coordinates (ri0) at which the convolution kernel (g) takes a particular value; and
defining particular surface elements (Si,j0) associated with the first particular coordinates.

### Claim 6 (depends on 1)

The Method according to claim 1, wherein during step c), the weighting factor (∂i,j) associated with each first coordinate (ri) is determined by computing an area of the surface element (Si,j) associated with said first coordinate (ri), defined during step b).

### Claim 7 (depends on 1)

The Method according to claim 1, wherein during step c), the weighting factor (∂i,j) associated with each coordinate (ri) is determined by computing an integral of the convolution kernel (g) over the surface element area (Si,j) associated with said first coordinate (ri).

### Claim 8 (depends on 1)

The Method according to claim 1, wherein during step c), each centered convolution support (Ωj) is subdivided into a plurality of elementary surfaces (zp,j), the computation of the weighting factor (∂i,j) associated with each first coordinate (ri) comprising a sum of the elementary surfaces (zp,j) included in the surface element (Si,j) associated with said first coordinate (ri), said sum being optionally weighted by a value of said convolution kernel (g), in each of said elementary surfaces.

### Claim 9 (depends on 8)

The Method according to claim 8, wherein the number (Pj) of elementary surfaces (zp,j) in a centered convolution support (Ωj) is determined as a function:
of the number of first coordinates (ri) included in said centered convolution support (Ωj), or
of the distance between two first adjacent coordinates (ri, r′i) belonging to said centered convolution support, or
of the evolution of the convolution kernel (g) in said centered convolution support.

### Claim 10 (depends on 1)

The Method according to claim 1, wherein each second coordinate (uj) defines a uniform mesh of the starting space.

### Claim 11 (depends on 1)

The Method according to claim 1, wherein the starting space is a frequency space or a spatial space or a temporal space.

### Claim 12 (depends on 1)

The Method according to claim 1, wherein said microprocessor comprises a plurality of elementary microprocessors, and wherein steps a) to e) are implemented simultaneously by various elementary microprocessors, each elementary microprocessor performing steps a) to e) for one or more second coordinates (uj).

### Claim 13 (depends on 1)

The Method according to claim 1, wherein the convolution kernel (g) is:
a symmetric function with respect to the center of its support (Ω), and/or
a function whose value is at a maximum at the center of its support (Ω).

### Claim 14 (depends on 1)

The Method according to claim 1, wherein the measurements, on a basis of which the measurement function (ƒ) is established, are acquired by a detector of electromagnetic waves of a magnetic resonance imaging (MRI) system.

### Claim 15 (depends on 1)

The method according to claim 1, wherein the convolution products result in uniformly distributed data in a temporal frequency or spatial frequency domain, the inverse Fourier Transform used in an estimation of the measurements in a temporal or spatial domain, or
the convolution products result in uniformly distributed data in the temporal or spatial domain, the Fourier Transform used in an estimation of the measurements in the temporal frequency or spatial frequency domain.

### Claim 16 (depends on 1)

A Method for estimating a Fourier transform of a measurement function (ƒ) representative of measurements acquired by a detector, and defined in a starting space, according to first discrete coordinates (ri), said method comprising:
estimating a product of convolution (f*g) of said measurement function (ƒ), with a convolution kernel (g), according to a method according to claim 1, at second discrete coordinates (uj), the latter defining a regular mesh of the starting space;
applying, in the inverse Fourier Transform or the Fourier Transform, a fast Fourier transform to said estimated convolution product (f*g) at said second discrete coordinates (uj).

### Claim 17 (depends on 1)

A non-transitory computer readable recording medium comprising instructions for the execution of steps a) to e) of the method according to claim 1 by a microprocessor.