Patent Publication Number: US-8126248-B2

Title: Procedures for the presentation of medical images

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority of German application No. 10 2007 046 941.3 filed Sep. 28, 2007, which is incorporated by reference herein in its entirety. 
     FIELD OF THE INVENTION 
     The invention relates to a method for presentation of medical images by a reproduction facility of a diagnostic device with suppression of the noise, as well as to an x-ray diagnostic device for executing the method. 
     BACKGROUND OF THE INVENTION 
     US 2006/0120507 A1 discloses such an x-ray diagnostic device for executing the method for angiography, which is shown for example in  FIG. 1 , which features a C-arm  2  supported to allow it to rotate on a stand  1 , at the ends of which an x-ray radiation source, for example an x-ray emitter  3 , and an x-ray image detector  4  are arranged. 
     The x-ray image detector  4  can be a rectangular or square flat semiconductor detector, which is preferably made of amorphous silicon (a-Si). 
     Located in the optical path of the x-ray emitter  3  is a patient support table  5  for recording images for example of a heart of a patient to be examined. Connected to the x-ray diagnostic device is an imaging system  6  which receives and processes the image signals of the x-ray image detector  4 . The x-ray images can then be viewed on a monitor  7 . 
     During recording of such 2D data x-ray images are frequently produced which have a very high amount of noise and in which the signal-to-noise ratio is thus also bad. This can make diagnosis more difficult. 
     If 3D data sets are to be created, the rotatably-supported C-arm  2  with x-ray source  3  and x-ray detector  4  is turned, so that, as shown schematically in  FIG. 2  looking down on the axis of rotation, the x-ray source  3  shown in this diagram by its beam focus  3  as well as the x-ray image detector  4  move around an object  9  to be examined on a planetary track  8 . The planetary track  8  can be followed completely or partly for creating a 3D data set. 
     The object  9  to be examined can for example be the body of an animal or a human being, but can also be a phantom body. 
     The x-ray source  3  emits a ray bundle  10  emanating from the ray focus of its radiographic source, which hits the x-ray image detector  4 . 
     The x-ray source  3  and the x-ray image detector  4  thus each circulate around the object  5 , so that the x-ray source  3  and the x-ray image detector  4  lie on opposite sides of the object  9  in relation to each other. 
     In normal radiography or fluoroscopy using such an x-ray diagnostic device the medical 2D data of the x-ray image detector  4  will be buffered in the imaging system  6  if necessary and subsequently reproduced on the monitor  7 . 
       FIG. 3  now shows the known execution sequence for recording 3D-data which is created by means of an x-ray device  11  described in  FIGS. 1 and 2 . The raw data  12  is fed to a reconstruction  13 , which computes from it the volume data set or 3D data set  14 . 
     In the creation of such 3D data, the reconstruction can be adversely affected as a result of noise in the raw data, which leads to a reduction in the ability to detect contrasts and structures in the 3D data set. 
     In a reconstruction of the 3D data, said data comprises a data set containing density information of the location (x, y, z) for example. This density information can now contain a noise component which reduces the 3D image quality. 
     The only option available to the doctor for increasing the signal-to-noise ratio (S/N ratio) and thus the image quality is to increase the dose, but this results in both the patient and the doctor being subjected to a higher exposure to radiation. There are also statutory regulations restricting the maximum dose. 
     Although the S/N ratio can be improved by a smoothing filter, for example Crispyl, which finds edges in the image and smoothes the image along these edges, a meaningful noise reduction is not guaranteed by this. 
     SUMMARY OF THE INVENTION 
     The object underlying the invention is to embody a method and a device of the type stated at the outset such that effective noise reduction can be brought about in a simple manner both for 2D and also 3D images. 
     The object is inventively achieved for a method through the following steps: 
     a) One-off calibration of the signal-dependent noise, 
     b) Separation of the signal and noise components in the image, 
     c) Adapting the two components in accordance with parameters that have been set and 
     d) Composing the signals. 
     The separate processing of the signal and noise components enables the noise components to be at least reduced in a simple and effective manner. 
     It has proved to be advantageous for the separation of the image components to be undertaken by a filter with non-linear wavelet decomposition. 
     The method can advantageously feature the following steps:
     S1) One-off calibration of the signal-dependent noise,   S2) Decomposition of the image into bandpass images,   S3) Use of a local signal estimator for determining the local signal component in the image,   S4) Adapting the local signal strength as a function of the local measured signal strength and of the calibrated noise component and   S5) Reconstruction of the image.   

     According to the invention the one-off calibration in accordance with step a) or S1 can depend on local frequency and intensity. 
     Especially advantageously the one-off calibration in accordance with step a) or S1 can be a signal-dependent calibration of the quantum noise and of the electrical noise. 
     It has been shown to be advantageous for the decomposition of the image into bandpass images in accordance with step S2 to be carried out by means of non-linear lowpass filters of different frequencies, the output signals of which are subtracted from one another. 
     In an advantageous manner the local signal component in accordance with S3 can be made up of a noise signal and a useful signal. 
     Especially advantageously the local signal estimator can be a local kernel which determines the standard deviation (ó) in this kernel, so that the total standard deviation (σ total ) can be measured and the σ signal  standard deviation a signal can be estimated as a result of the standard deviation of the noise component (σ noise ) known from the calibration. 
     Inventively the method steps can be applied to 2D data or 3D data, with the method being applied to raw data reconstructed after filtration into a volume data, or alternatively applied to volume data which is composed after filtering into a volume data set. 
     The object is inventively achieved for a device by the image system having a device for noise calibration, a noise filter which separates the image signal into useful signal and noise signal, a correction stage which adapts these two components in accordance with set parameters and a reconstruction stage which assembles the data into a data set. 
     The result able to be achieved is that 2D or 3D data sets are separated according to frequency, filtered separately and corrected as well as subsequently composed, so that at least one noise-reduced data set is obtained. 
     In an advantageous manner the noise filter can separate the image signal, using a non-linear wavelet decomposition, into useful signal and noise signal. 
     It has been shown to be advantageous for the noise filter to be a 3-dimensional noise filter which separates a volume data set into bandpass volume data using a non-linear lowpass filter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention is explained below in greater detail on the basis of the exemplary embodiments shown in the drawing. The figures are as follows: 
         FIG. 1  a known x-ray diagnostic device for carrying out the method, 
         FIG. 2  a view of the track of a detector and a radiographic source around an object to be examined in a view along the axis, 
         FIG. 3  a schematic diagram of a known overall sequence, 
         FIG. 4  a schematic diagram of a first inventive overall sequence, 
         FIG. 5  a schematic diagram of a further inventive overall sequence, 
         FIG. 6  an inventive method sequence, 
         FIG. 7  a step phantom for calibrating the x-ray diagnostic device in accordance with  FIG. 1 , 
         FIG. 8  a section from an x-ray image of a step phantom, 
         FIG. 9  a inventive embodiment with bandpasses and 
         FIG. 10  an inventive embodiment implemented with lowpasses. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 4  now shows schematically the inventive execution sequence of the recording of 3D data, which is created by means of the x-ray device  11  described in  FIGS. 1 and 2 . The raw data  12  is first subjected to a noise filtering  15  of the raw data  12  and then fed to the reconstruction  13 , which calculates the volume data set or 3D data set  14  from it. 
       FIG. 5  is a schematic diagram of a further inventive execution sequence of the processing of 3D data which is created by an x-ray device  11  described in  FIGS. 1 and 2  and has been processed in accordance with the execution sequence depicted in  FIG. 3 . The 3D-data record  14  is subjected to noise filtering  16  and then fed to a composition  17  which computes the filtered 3D data record  18  from this. 
       FIG. 6  provides a more detailed description of the execution sequence of the filtering method. In step S1 a one-off, especially local-frequency-dependent and intensity-dependent calibration of the signal-dependent noise, for example of the quantum noise and of the electrical noise, is performed. Subsequently in step S2 the image is separated into bandpass images, typically using nonlinear lowpass filters. In step S3 a local signal estimator is used to determine the local signal component which comprises a noise component and a useful signal component. In step S4 the local signal strength is adapted depending on the local measured signal strength and the calibrated noise component. Finally in step S5 the image is reconstructed by additive composition. 
     The signal estimator is a local kernel, for example here 3×3, which defines the standard deviation (ó) in this kernel. This allows the total standard deviation (σ total  to be measured; the standard deviation of the noise component (σ noise ) is known from the calibration and this allows the signal standard deviation σ signal  to be estimated. 
     The filter method for noise removal, the suppression of the noise, consists of the following steps: 
     One-off local frequency and intensity-dependent calibration of the signal-dependent quantum noise and of the electrical noise,
     S1) Separation of the image into bandpass images using non-linear lowpass filters,   S2) Use of a local signal estimator for determining the local signal component made up of a noise component and a useful signal,   S3) Adapting the local signal strength depending on the local measured signal strength and the calibrated noise component and   S4) Reconstruction of the image.   

     The filter method is based on separating the image into individual bandpass images. A wavelet decomposition is used, in which the number of bandpass coefficients is identical to that of the input image. 
     To compute the lowpass images an unsharp filter is used, the kernel of which is enlarged in accordance with band by the insertion of holes. 
     The difference between two consecutive lowpass images produces the bandpass image. 
     If l 0  is the input image and l i+1  the lowpass-filtered image of l i , then the bandpass image h i  defined as:
 
 h   i   =l   i   −l   i+1  
 
for i≧0.
 
     There now follows a precise description of the method steps for the 2D imaging: 
     Decomposition of the images into bandpass images using a non-linear lowpass filter 
     An image pixel is referred to below as I(x, y), with (x, y) being the coordinates of the pixel. 
     A 2D window (3×3) around the central pixel (x, y) is considered. This window consists of the pixels (x′,y′), with |x′−x|≦1,|y−y′|≦1. This window is used to compute the filtered value I′(x, y). This operation occurs for all pixels of the image matrix. 
     The individual filtered pixel values are defined as follows 
     In the first step all values of the 2D window are transferred into a one-dimensional field:
 
 x   K =( x   1   ,x   2   , . . . ,x   N ) t  
 
In this example N=9.
 
     Al pixels are now sorted in the field x K  according to their difference from I(x, y) and a new 1D field is obtained:
 
 x   k =( x   (1)   ,x   (2)   , . . . ,x   (N) ) t ,
 
with x (1) =I(x, y) being the central pixel in the 2D window is, and x (i),I= 2 . . . N with |x (1) −x (i) |≦|x (1) −x (j) |, j=i . . . N.
 
     The second step consists of the filtering. The filtered pixel value is computed as follows: 
             y   =         a   t     ⁢       x   k     /   γ       =       1   γ     ⁢       ∑     i   =   1     N     ⁢           ⁢       a   i     ⁢     x     (   i   )                     
With a=(a 1 ,a 2 , . . . , a N ) t : Weighting factors and
 
             γ   =       ∑     i   =   1     N     ⁢           ⁢       a   i     .             
Here a i =1 is used for i=1 . . . M and a i =0 for i&gt;N. This implies γ=M.
 
     The value M and thus also γ are parameters and describe the smoothing of the decomposition. 
     After the image has been filtered by the above steps and the band l m=0  created in this way, the band l m+1  is recursively created. This is done by enlarging the kernel, i.e. the coordinates x·2 m , y·2 m  with xε{−1;0;1} and yε{−1;0;1} are now used for the window with the filtered data from l m . 
     These operations are executed recursively until m=4. 
     The bandpass-filtered image of the step m is computed as follows:
 
 h   m   =l   m   −l   m+1 .
 
     Determination of the Calibrated Noise Values 
     Let the dynamic range of the detector have n elements [0, . . . , n−1], onto which the intensity values are mapped in a linear fashion. Through the step phantom  21  depicted in  FIG. 7 , to which unattenuated x-ray radiation  20  is applied, the calibrated noise components are now determined. 
     Practice has shown that a subdivision of the dynamic range into eight steps  22  to  29  is sufficient; however a finer subdivision can be used. 
     The material of the step phantom can be Pb, Cu or aluminum. 
     The following steps of absorption of the input signal must be present (+5%): 
     Step  22  features the full absorption of 100%, step  23  typically of 87.5%, step  24  of 75%, step  25  of 62.5%, step  26  of 50%, step  27  of 37.5%, step  28  of 25%, step  29  of 12.5% and an unattenuated surface  30  no absorption (0%). 
     The surfaces of the steps  22  to  29  must be adapted, that with four bandpasses a (2*2*2 4-1 =) 64 times 64 pixel homogeneous surface is available for calibration. The x-ray radiation  31  attenuated by the step phantom  21  is detected by the x-ray image detector  4 . 
       FIG. 8  shows a section from a recorded x-ray image of a step phantom  21  in accordance with  FIG. 3 . 
     The data record is separated into its bandpass images h m  and lowpass images l m  and the intensity-dependent noise values (I) are determined within the various absorption surfaces for the respective band: 
     
       
         
           
             
               
                 
                   σ 
                   noise 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       I 
                       mn 
                     
                     , 
                     x 
                     , 
                     y 
                   
                   ) 
                 
               
               = 
               
                 
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                         - 
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                   ⁢ 
                   
                       
                   
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                           h 
                           m 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
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                             , 
                             
                               y 
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                               j 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
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                         ⁡ 
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                       
                     
                      
                   
                 
               
             
             ; 
           
         
       
     
     The nth (here n=1 . . . 8) intensity I mn  of the band m is measured from the lowpass image l m+1  at the coordinates (x, y):
 
 I   mn   =l   m+1 ( x,y );
 
     From the discrete measured intensities with the associated noise values the missing intensities of the dynamic range with their noise values are determined by interpolation. To do this splines are applied for example through the eight measured points. 
     Specific functions are identified as splines. An nth-degree spline is a function which is composed in sections from polynomials with maximum degree n. In this case at the points at which two polynomial sections join, the so-called node, specific conditions are imposed, for example that the spline is always able to be differentiated (n−1) times. 
     Splines are used for interpolation. By being defined in sections splines are more flexible than polynomials and still relatively simple and smooth. 
     If the spline involves a sectional linear function, the spline is referred to as linear (a polygon shape is then involved); in a similar way there are square, cubic etc. splines. 
     Adapting the Bandpass Coefficients Using a Local Signal Estimator and the Calibrated Noise Values 
     The coefficient in the decomposition is composed depending on the non-linear decomposition additively from a signal component and a noise component. The above-mentioned signal estimator allows the factor to be estimated with which the set of coefficients must be reduced in order to get rid of the noise component. This factor is f=(σ total −σ noise )/σ total . The bandpass coefficient is now multiplied by this factor. 
     The bandpass coefficients are also adapted to the calibrated noise component and the signal (σ signal ) measured by the local signal estimator which is composed of a noise component and a signal component. 
     The following assumption is used as a starting point: 
     The overall signal in a bandpass is composed of a useful signal and a noise component:
 
σ total   m ( x,y )=σ signal   m ( x,y )+σ noise   m ( I );
 
     The overall signal can be measured: 
     
       
         
           
             
               
                 
                   σ 
                   total 
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                 ⁡ 
                 
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     The intensity is measured in this case from the lowpass image l m+1  at the coordinates (x, y):
 
 I=l   m+1 ( x,y );
 
     The factor f m , by which the overall signal must be multiplied in order to obtain the useful signal, is: 
     
       
         
           
             
               
                 
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     The new coefficient of the bandpass m is computed as follows:
 
 h′   m ( x,y,z )= f   m ( x,y,z )· h   m ( x,y,z );
 
     The image is then reconstructed again from the m=1 . . . n bandpasses: 
     
       
         
           
             
               
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     The principle of the processing circuit is shown in greater detail in  FIG. 9 . The calibration values are read into a memory  40  after completion of calibration (S1). The spectral image decomposition (S2) is undertaken by bandpasses  41  to  44 , to which the input signal is fed. The outputs of the bandpasses  41  to  44  are connected to multiplication stages  45  to  48  which adapt the local signal strength (S4) by multiplication of the bandpass signal by the intensity Imn. The outputs of the multiplication stages are connected to an addition stage  49  for reconstruction (S5). The output signal of the addition stage  49  can be displayed via the monitor  7 . 
     The structure of the processing circuit in accordance with  FIG. 10  essentially corresponds to that depicted in  FIG. 9 . The four bandpasses  41  to  44  have merely been replaced by five lowpasses  50  to  54 , the output signals of which are converted by adjacent lowpasses  50  to  54  by subtraction stages  55  to  58  into a bandpass-filtered signal, which is then further processed as a result of the multiplication stages  45  to  48  and addition stages  49  and reproduced by means of the monitor  7 . 
     The method can also be applied to a 3D volume. Here too it is based on a one-off calibration of the noise of the reconstructed volume and of a subsequent 3-dimensional noise filter. 
     The noise filter includes the following steps:
         Separating the 3D volume into bandpass volumes using a non-linear lowpass filter.   Using a local signal estimator for determining the local signal component, which is composed of a noise component and a useful signal.   Adapting the local signal strength depending on the local measured signal strength and the calibrated noise component.   Composition of the volume.       

     Decomposition of the 3D Volume into Bandpass Volumes 
     For decomposition of the 3D volume a wavelet decomposition is used, in which the number of the bandpass coefficients is identical to that of the input volume. 
     To compute the lowpass volume an unsharp filter is used, the kernel of which is enlarged in accordance with band by the insertion of holes. 
     The difference between two consecutive lowpass volumes produces the bandpass volume. 
     If l o  is the input volume and l i+1  is the lowpass-filtered volume of I, then the bandpass volume h i  is defined as:
 
 h   i   =l   i   −l   i+1  
 
for i≧0.
 
     In the decomposition step the sum of the absolute deviation from the central pixel of 4 pixels in 11 different directions in the band k is computed: 
     
       
         
           
             
                 
             
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                       ) 
                     
                   
                 
                  
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   s 
                   5 
                   k 
                 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     ∈ 
                     
                       { 
                       
                         
                           - 
                           2 
                         
                         , 
                         
                           - 
                           1 
                         
                         , 
                         1 
                         , 
                         2 
                       
                       } 
                     
                   
                   
                       
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                    
                   
                     
                       
                         l 
                         
                           k 
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           
                             z 
                             + 
                             
                               i 
                               · 
                               
                                 2 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         l 
                         
                           k 
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                         ) 
                       
                     
                   
                    
                 
               
             
           
         
       
       
         
           
             
               
                 s 
                 6 
                 k 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   
                     { 
                     
                       
                         - 
                         2 
                       
                       , 
                       
                         - 
                         1 
                       
                       , 
                       1 
                       , 
                       2 
                     
                     } 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         y 
                         , 
                         
                           z 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                 
                  
               
             
           
         
       
       
         
           
             
               
                 s 
                 7 
                 k 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   
                     { 
                     
                       
                         - 
                         2 
                       
                       , 
                       
                         - 
                         1 
                       
                       , 
                       1 
                       , 
                       2 
                     
                     } 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           - 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         y 
                         , 
                         
                           z 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                 
                  
               
             
           
         
       
       
         
           
             
               
                 s 
                 8 
                 k 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   
                     { 
                     
                       
                         - 
                         2 
                       
                       , 
                       
                         - 
                         1 
                       
                       , 
                       1 
                       , 
                       2 
                     
                     } 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           y 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           z 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                 
                  
               
             
           
         
       
       
         
           
             
               
                 s 
                 9 
                 k 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   
                     { 
                     
                       
                         - 
                         2 
                       
                       , 
                       
                         - 
                         1 
                       
                       , 
                       1 
                       , 
                       2 
                     
                     } 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           y 
                           - 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           z 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                 
                  
               
             
           
         
       
       
         
           
             
               
                 s 
                 10 
                 k 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   
                     { 
                     
                       
                         - 
                         2 
                       
                       , 
                       
                         - 
                         1 
                       
                       , 
                       1 
                       , 
                       2 
                     
                     } 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           - 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           y 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           z 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                 
                  
               
             
           
         
       
       
         
           
             
               
                 s 
                 11 
                 k 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   
                     { 
                     
                       
                         - 
                         2 
                       
                       , 
                       
                         - 
                         1 
                       
                       , 
                       1 
                       , 
                       2 
                     
                     } 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           - 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           y 
                           - 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                         
                           z 
                           + 
                           
                             i 
                             · 
                             
                               2 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       l 
                       
                         k 
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                 
                  
               
             
           
         
       
     
     The minimum deviation is determined from this:
 
 S   min :=min( S   1   k ( x,y,z ), S   2   k ( x,y,z ), S   3   k ( x,y,z ), S   4   k ( x,y,z ), S   5   k ( x,y,z ), S   6   k ( x,y,z ), S   7   k ( x,y,z ), S   8   k ( x,y,z ), S   9   k ( x,y,z ), S   10   k ( x,y,z ), S   11   k ( x,y,z ))
 
     The pixel value of the mth band I m (x, y, z) is determined from the previous pixel value I m−1 (x, y, z) as follows: 
     
       
         
           
             
               
                 l 
                 m 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     
                       - 
                       1 
                     
                   
                   1 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       
                         - 
                         1 
                       
                     
                     1 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         
                           - 
                           1 
                         
                       
                       1 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         w 
                         
                           i 
                           , 
                           j 
                         
                         m 
                       
                       ⁢ 
                       
                         
                           l 
                           
                             m 
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               + 
                               
                                 i 
                                 · 
                                 
                                   2 
                                   
                                     m 
                                     - 
                                     1 
                                   
                                 
                               
                             
                             , 
                             
                               y 
                               + 
                               
                                 j 
                                 · 
                                 
                                   2 
                                   
                                     m 
                                     - 
                                     1 
                                   
                                 
                               
                             
                             , 
                             
                               z 
                               + 
                               
                                 k 
                                 · 
                                 
                                   2 
                                   
                                     m 
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ∑ 
                   
                     i 
                     = 
                     
                       - 
                       1 
                     
                   
                   1 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       
                         - 
                         1 
                       
                     
                     1 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         
                           - 
                           1 
                         
                       
                       1 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       w 
                       ijk 
                       m 
                     
                   
                 
               
             
           
         
       
     
     The weight w ijk   m  of the mth band is calculated as follows: 
     
       
         
           
             
               
                 w 
                 ijk 
                 m 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               { 
               
                 1 
                 
                   max 
                   ⁡ 
                   
                     ( 
                     
                       
                         S 
                         min 
                       
                       , 
                       
                         
                            
                           
                             
                               
                                 
                                   
                                     l 
                                     
                                       m 
                                       - 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         + 
                                         
                                           i 
                                           · 
                                           
                                             2 
                                             
                                               m 
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                       , 
                                       
                                         y 
                                         + 
                                         
                                           j 
                                           · 
                                           
                                             2 
                                             
                                               m 
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                       , 
                                       
                                         z 
                                         + 
                                         
                                           k 
                                           · 
                                           
                                             2 
                                             
                                               m 
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                            
                         
                         
                           
                             ; 
                             i 
                           
                           , 
                           
                             j 
                             = 
                             0 
                           
                         
                         
                           
                             ; 
                             i 
                           
                           , 
                           
                             j 
                             ≠ 
                             0 
                           
                         
                       
                     
                     ) 
                   
                 
               
               } 
             
           
         
       
     
     The bandpass-filtered volume of stage m is computed as follows:
 
 h   m   =l   m   −l   m+1  
 
     The calibrated noise values are now determined as already described. 
     The data set is separated into its bandpass volumes h m  and l m  and within the different absorption cubes the intensity-dependent noise values are determined: 
     
       
         
           
             
               
                 
                   σ 
                   noise 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     
                       - 
                       2 
                     
                   
                   2 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       
                         - 
                         2 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         
                           - 
                           2 
                         
                       
                       2 
                     
                     ⁢ 
                     
                        
                       
                         
                           
                             h 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 + 
                                 i 
                               
                               , 
                               
                                 y 
                                 + 
                                 j 
                               
                               , 
                               
                                 z 
                                 + 
                                 k 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             h 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                         
                       
                        
                     
                   
                 
               
             
             ; 
           
         
       
     
     The missing intensities of the dynamic range with their noise values are determined by interpolation from the discrete measured intensities with the associated noise values. 
     Adapting the bandpass coefficients using a local signal estimator and the calibrated noise values 
     The following assumption is used as a starting point: 
     The overall signal is composed of a useful signal and a noise component:
 
σ total   m ( x,y,z )=σ signal   m ( x,y,z )+σ noise   m ( I );
 
     The overall signal can be measured: 
     
       
         
           
             
               
                 
                   σ 
                   total 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     
                       - 
                       2 
                     
                   
                   2 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       
                         - 
                         2 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         
                           - 
                           2 
                         
                       
                       2 
                     
                     ⁢ 
                     
                        
                       
                         
                           
                             h 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 + 
                                 i 
                               
                               , 
                               
                                 y 
                                 + 
                                 j 
                               
                               , 
                               
                                 z 
                                 + 
                                 k 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             h 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                         
                       
                        
                     
                   
                 
               
             
             ; 
           
         
       
     
     The factor f m , by which the overall signal must be multiplied to obtain the useful signal, is: 
     
       
         
           
             
               
                 f 
                 m 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     σ 
                     total 
                     m 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
                 - 
                 
                   
                     σ 
                     noise 
                     m 
                   
                   ⁡ 
                   
                     ( 
                     I 
                     ) 
                   
                 
               
               
                 
                   σ 
                   total 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
     
     The new coefficient of the bandpass m is computed as follows:
 
 h′   m ( x,y,z )= f   m ( x,y,z )· h   m ( x,y,z )
 
     The image is then reconstructed again from the n bandpasses: 
     
       
         
           
             
               
                 f 
                 out 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 
                   l 
                   
                     n 
                     + 
                     1 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
               + 
               
                 
                   ∑ 
                   
                     m 
                     = 
                     0 
                   
                   n 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     h 
                     m 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The application of a 3-dimensional noise-reduction filter provides the user with an improved 3D quality. 
     The proposed method for noise reduction 2D data with the aid of a calibrated noise component allows the noise in medical images to be almost completely eliminated. 
     This produces the following advantages for the examining physician and the patient
         Improved 2D image quality:
           Better and faster diagnosis are possible.   
           Savings in dosage with the image quality remaining the same:
           The exposure to radiation of the doctor and the patient can be reduced while retaining the same image quality.   
               

     Upstream connection of a signal-invariant noise filter with the method described above produces the following advantages for 3D applications:
         Increase in the 3D image quality through an improved reconstruction by means of noise filters:
           Less noise and better contrast recognition in the reconstructed layers.   
           Savings in dosage with the image quality remaining the same:
           A reduced-dose imaging protocol with the same image quality can be used for the examiner and the patient.   
           New examinations:
           With very fast image protocols (e.g. 200 images/s):   Physical limits of x-ray tubes do not allow the required dose to be applied. The increased reconstruction quality makes this option possible for the customer (doctor and patient).   For very long imaging protocols:   By the introduction of new trajectories (e.g. by robot arm) for example physical limits of current x-ray tubes can be reached. The increased reconstruction quality makes this option possible.   
               

     The tomographic imaging device involved in the invention can for example be x-ray C-arm systems, biplanar x-ray devices or computer tomographs. 
     Instead of the stand  1  shown, floor or ceiling tripods can also be used, to which the C-arms  2  are attached. The C-arms  2  can also be replaced by a so-called electronic C-arm  2 , in which there is an electronic coupling of x-ray source  3  and x-ray image detector  4 . 
     The C-arms  2  can also be guided on robot arms which are attached to the ceiling or the floor. The method can also be conducted with x-ray devices, in which the individual, image-generating components  3  and  4  are each held by a robot arm which is arranged on the ceiling and/or floor in each case.