Abstract:
In a method for measuring mechanical workpieces by tomography, a workpiece and radiation penetrating the workpiece are moved relative to one another step-by-step. A two-dimensional image of the workpiece is generated in an imaging plane from the interaction of the workpiece and the radiation in each movement position of the workpiece. In addition, a three-dimensional representation of the workpiece is computed from the two-dimensional images. From at least two two-dimensional images showing a regular actual structure existing within the workpiece, points at a high-contrast transition are registered. A three-dimensional equivalent body is determined from the position of the points, and said equivalent body is compared to a predefined nominal structure.

Description:
CROSS REFERENCES TO RELATED APPLICATIONS 
     This application is a continuation of international patent application PCT/EP2009/002331 filed on Mar. 31, 2009 designating the U.S., which international patent application has been published in German language and claims priority from German patent application DE 10 2008 018 445.4 filed on Apr. 7, 2008. The entire contents of these priority applications are incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     The invention relates to a method for measuring mechanical workpieces by tomography. 
     In the production of, in particular, high-quality mechanical workpieces, it is necessary for the workpiece to be measured after or even during the production and processing in order to check if certain measurement points on the workpiece defined in advance meet predefined dimensions within some tolerances likewise defined in advance. For this purpose, use is often made of multi-coordinate measurement machines, which scan the workpiece by means of touch probes and monitor the dimensional accuracy of the workpiece surfaces in this fashion. 
     As an alternative, a different measurement approach has been used for some time now for measuring mechanical workpieces, which approach was initially used in medicine as an imaging method for examining human bodies, namely the method of computer tomography (CT). In medical applications, the body or body region to be examined is irradiated in a plane by means of a linear array of X-ray radiation sources. On the opposite side of the body, a corresponding array of X-ray detectors is situated opposite the array of X-ray radiation sources. This pair of arrays is then rotated by an angular step about an axis running perpendicular to the plane, and a further recording is produced. After the array has been rotated, step-by-step, through a total of 360°, a cross-sectional image in the plane is computed from the individual recordings, which image reproduces the density distribution in this plane. If the body and the pair of arrays are now subsequently displaced relative to one another by a linear step along the axis, a further, immediately adjacent cross-sectional image can be generated and a three-dimensional display of the body or body region can be generated from a plurality of such adjacent cross-sectional images. This measurement method is rather complicated because a human body has a very complex density distribution with density varying in large regions, and the structures to be recorded can differ significantly and can be of unforeseeable type and shape. 
     By contrast, workpieces in quality control are often objects with only two densities, namely the density of the material of the workpiece and the density of air. Furthermore, the structures to be examined during quality control are known and merely have to be examined with respect to deviations. 
     DE 10 2005 039 422 A1 describes a computer tomography method, simplified with respect to medical applications, for examining workpieces. In this method, a mechanical workpiece is situated on a rotary table between a spot-like X-ray radiation source and an areal detector array. Here, the rotational axis of the rotary table runs substantially perpendicular to the radiation direction. The workpiece is penetrated by the X-ray radiation, and a shadow image of the workpiece is created on the detector array. The workpiece is then successively rotated by an angular step, for example 800 or 1200 times, on the rotary table and additional shadow images are created. A three-dimensional image of the workpiece is then computed from the plurality of shadow images, for example according to a method of back-projection, as described in DE 39 24 066 A1. 
     Computer tomography measuring stations of this type are commercially offered, for example by the assignee of the invention under the brand name “Metrotom” (www.zeiss.de/imt). 
     With prior art measuring stations, a so-called “reconstruction” is performed, i.e. a complete three-dimensional image of the density distribution of the measurement object (mechanical workpiece) is computed with a predefined resolution of volume elements (voxels). In order to be able to do this with sufficiently high precision for workpiece measuring, a large number of individual images are required, in practice between 360 and 1080, corresponding to angular steps of the rotary table between 1° and ⅓°. However, a certain amount of time is required for each rotational step in order to accelerate the rotary table from its rest position, to move it, and to decelerate it again until it rests. Furthermore, a certain amount of time is required to calculate the three-dimensional image, even though this can already be started during the measurement acquisition, and so, overall, prior art measuring stations require between 15 and 30 minutes for a complete measurement. In many cases, this measuring time is unacceptable. 
     SUMMARY OF THE INVENTION 
     The invention therefore has the object of providing a method for measuring a mechanical workpiece by tomography, but avoids the aforementioned disadvantages. More particularly, a method is to be provided that requires a significantly shorter overall measurement time. 
     According to an aspect of the invention, there is provided a method for measuring mechanical workpieces having a defined rotationally symmetric structure, the method comprising the steps of providing a radiation source for generating radiation, providing a detector array for receiving the radiation, said detector array defining a two-dimensional imaging plane, positioning a workpiece between the radiation source and the detector array, so that the radiation penetrates the workpiece before it impinges on the detector array, moving the radiation source and the workpiece relative to one another in a step-by-step fashion, generating a number of two-dimensional images of the workpiece by means of the detector array for a plurality of movement positions of the workpiece, and computing a three-dimensional image of the structure from the two-dimensional images, wherein the step of computing comprises detecting a high-contrast transition, which may correspond to edges of the structure, in said two-dimensional images, registering points at said high-contrast transition from at least two two-dimensional images generated at two different movement positions, and determining a three-dimensional geometric equivalent body using the registered points, and comparing said equivalent body to a predefined nominal structure, wherein the geometric equivalent body only parameterizes said rotationally symmetric structure. 
     According to another aspect, there is provided a method for measuring mechanical workpieces having a rotationally symmetric structure, wherein a workpiece and radiation from a radiation source penetrating the workpiece are moved relative to one another step-by-step, wherein a two-dimensional image of the workpiece is generated in an imaging plane from the interaction of the workpiece and the radiation in each movement position of the workpiece, and wherein a three-dimensional image of the structure is computed from the two-dimensional images, wherein, from at least two two-dimensional images of the workpiece generated at two different movement positions, points which may correspond to edges of the structure are registered at a high-contrast transition in said two-dimensional images, wherein a three-dimensional geometric equivalent body is determined from the position of the registered points, wherein said equivalent body is compared to a predefined nominal structure, and wherein the geometric equivalent body only parameterizes individual rotationally symmetric elements of the workpiece, which are imaged in the two-dimensional images with clear-cut, high-contrast object boundaries. 
     It has turned out that, in contrast to natural objects in medicine, the regularity of the shape of certain structures in mechanical workpieces (bores, grooves, etc.) can be advantageously used to drastically reduce the overall measuring time by generating an equivalent body of this structure from relatively few measurement points, thereby exploiting the known regularity of the structure. Only a few, preferably rotationally symmetric, elements are measured by tomography in a calibrated recording device with a two-dimensional projection from a plurality of directions, and these are parameterized by geometric equivalent bodies. This procedure differs from conventional methods in the prior art, because the entire density distribution must be reconstructed in a voxel matrix in those cases. 
     Use is made of the fact that a large number of the elements to be examined in the reconstruction volume are regular, and in particular rotationally symmetric, and these elements typically image with clear-cut, high-contrast object boundaries in the individual projections. These object boundaries can be extracted from the individual projections with the aid of image-processing methods, for example with a suitable edge recognition. Once a plurality of projections have been recorded at different, known orientations, the position, the axial direction and further shape parameters, such as a radius or a cone angle, of the three-dimensional element can be determined therefrom. 
     The new method may dispense with conventional complete recording of all projection images of a CT scan. In principle, two projections alone can be used to approximately determine the parameters of the three-dimensional structure to be examined. 
     As a consequence, computational-time expensive CT reconstruction of an entire voxel matrix required until now may be dispensed with. First tests have resulted in this reducing the overall measuring time from the conventional 15 to 30 minutes to less than 1 minute. 
     In preferred embodiments, the regular nominal structure is a rotationally symmetric structure, and in particular a sphere, a cylinder or a cone. 
     In a preferred refinement of the method according to the invention, the radiation source and the imaging plane for the relative movement are fixed in space and the workpiece is rotated about a rotational axis in predefined angular steps. 
     The advantage of this measure lies in the fact that use can be made of established components in respect of the measuring station. 
     In a further exemplary embodiment, the coordinates of a center and a radius of the equivalent body are determined in the case of a spherical nominal structure. 
     The advantage of this measure lies in the fact that a rough approximation of the equivalent body is possible with a minimum number of parameters and hence with a minimum amount of effort. 
     In the case of both of the aforementioned exemplary embodiments, a particularly preferred refinement is comprises that the following steps are used to determine the position of the center of the equivalent body: 
     a) determining the coordinates of at least three points on the boundary of the circular, two-dimensional image of the actual structure in a first movement position of the workpiece; 
     b) determining a first center point of the circular image from the coordinates of the three points; 
     c) rotating the workpiece on the one hand and, on the other hand, the radiation source and imaging plane relative to one another by a predefined angular step from the first into a second movement position of the workpiece; 
     d) determining the coordinates of at least three points on the boundary of the moved circular, two-dimensional image of the actual structure in the second movement position of the workpiece; 
     e) determining a second center point of the moved circular image from the coordinates of the three points; 
     f) determining two center beams between the radiation source and the first and second center points; 
     g) determining the position of the smallest distance between the center beams; 
     h) dropping the connection perpendicular between the center beams at the position of the smallest distance; and 
     i) determining the coordinates of the center point of the connection perpendicular as the position of the center of the equivalent body. 
     In addition, it is particularly preferable if the following steps are followed to determine the radius (R A ) of the equivalent body: 
     j) determining a first distance between the radiation source and the center point of the circular image; 
     k) determining a second distance between the radiation source and the center point of the actual structure; and 
     l) multiplying the radius of the circular image by the ratio between the second and the first distance. 
     The advantage of these measures lie in the fact that the mentioned few parameters of the spherical structure can be determined using few operations. 
     In a refinement of the aforementioned exemplary embodiment, tangent points are determined for a plurality of the points established in step a), at which tangent points a connection beam between the radiation source and the points touches a boundary of the actual structure generating the image, and the equivalent body is fitted into the point cloud formed by the points using the values established in steps i) and l) for the position of the center point and the radius of the equivalent body as initial values. 
     The advantage of this measure lies in the fact that the accuracy of the fit can be significantly improved. 
     This holds true even more if the point cloud is formed from tangent points established in different rotational positions of the workpiece. 
     Further advantages emerge from the description and the attached drawing. It is understood that the aforementioned features and the features yet to be explained below can be used not only in the respectively specified combination, but also in other combinations or on their own, without departing from the scope of the present invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments of the invention are illustrated in the drawing and are explained in more detail in the following description, in which: 
         FIGS. 1A-C  show a technical drawing, in three orthogonal views, of a mechanical workpiece as can be measured by the method according to the invention; 
         FIGS. 2A-B  show an extremely schematic view, in a side view and plan view, respectively, of a measuring station for carrying out the method according to the invention; 
         FIGS. 3A-B  show a perspective view of the measuring station from  FIGS. 2A-B  in a first rotational position of the workpiece and in a second rotational position of the workpiece; 
         FIG. 3C  shows an illustration to explain an equation of a circle; 
         FIG. 4  shows an illustration to explain the determination of initial parameters; and 
         FIG. 5  shows an illustration of an equivalent body. 
     
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS 
       FIGS. 1A-1C  illustrate three mutually perpendicular views of a workpiece denoted by reference numeral  10 . In the illustrated simplified example, workpiece  10  is a cuboid body. The workpiece  10  consists of a known material, assumed to be homogeneous, i.e. having no variations of density therein. 
     In the illustrated exemplary embodiment, the workpiece  10  has a spherical inner structure  12 , the center point of which is denoted by K and the radius of which is denoted by R. Compared to the rest of the workpiece  10 , the structure  12  has different properties, more particularly a different density. In the illustrated example, said structure can be a spherical, air-filled cavity in the workpiece  10 . 
     The structure  12  is an actual structure produced within the workpiece  10  in some production process. Therefore, structure  12  in practice does not correspond exactly to a nominal structure predefined by the designer in either position or shape, which nominal structure for example is saved in a CAD file. It is the object of the present method to determine to what extent the actual structure  12  deviates from the desired contour. For this purpose, the actual structure  12  is measured and approximated to the best possible extent by an optimally fitted equivalent body. The characteristic data of the equivalent body, i.e. the location of the center point K in the workpiece  10  and the radius R in the illustrated exemplary embodiment, are then compared to the constructional predefined data of the nominal structure. Whether the workpiece  10  lies within predefined tolerances in respect of the structure is derived from this comparison. 
     It goes without saying that the workpiece and the inner structure can also have a different shape. The workpiece is preferably a component of a machine or an instrument of arbitrary shape, in and/or on which there are structures of basically regular geometric shape, for example of cylindrical, conical, parabolic, spiral or any other regular shape. Here, the term “regular” should be understood to mean that the shape can be determined mathematically in a manner that is acceptable for the present application in respect of the computational-time expenditure. 
       FIGS. 2A-B  and  3 A-B illustrate how the workpiece  10  is measured by tomography. 
     To this end, the workpiece  10  is arranged in a measuring station  20 . The measuring station  20  comprises a spot-like radiation source  22 , which is preferably embodied as an X-ray source and irradiates the workpiece  10  from the side. The radiation source  22  is arranged fixed in space and emits a conical radiation  24 . The radiation  24  penetrates the workpiece  10  and is incident on a circular region  26  of a detector array  28  arranged behind the workpiece  10 . 
     Here it is understood that the spot-like X-ray radiation source  22  should only be understood to be exemplary. It goes without saying that other radiation sources, for example linear arrays of radiation sources, can also be used within the scope of the present invention. Furthermore, it is also possible to use other imaging examination methods, such as NMR tomography. 
     The detector array  28  is planar and rectangular, preferably square, and has edges  29   a - 29   d . On one face, it has a plurality of mutually adjacently arranged detectors  30 . In the illustrated exemplary embodiment, the detectors  30  have a resolution of 512×512 pixels with a pixel pitch of 0.1 mm. However, it is understood that it is also possible to utilize other resolutions. A center of the detector array  28  is denoted by D. 
     The workpiece  10  is arranged on a rotary table  40  during the measurement and preferably stands on a surface  42  of the rotary table  40 . The rotary table  40  can rotate about a rotational axis  44 , which preferably runs perpendicular to the surface  42 . Rotating the rotary table  40  about the rotational axis  44  in the direction of an arrow  46  causes a rotation of the workpiece  10  about a corresponding rotational angle cp. In  FIG. 3B , the reference signs of the elements moved by the rotation have been provided with an apostrophe. 
     Here it is understood that the spatially fixed arrangement of the radiation source  22  and the detector array  28 , and also the rotatable arrangement of the workpiece  10 , should likewise only be understood to be exemplary, and this does not restrict the present invention. On the one hand, in a kinematic reversal, the workpiece can also be fixed and the radiation source can rotate together with the detector array. Furthermore, the radiation source can be fixed and it is only the radiation that is deflected with changing direction. Finally, also another type of movement can be selected instead of a rotational movement. 
     The radiation source  30  and the center D of the detector array  28  are interconnected by a connection line  53 , which defines a y-axis of a spatially fixed coordinate system x, y, z of the measuring station  20 . The origin  47  of the spatially fixed coordinate system x, y, z is located within the rotary table  40  on the rotational axis  44 . The z-axis coincides with the rotational axis  44 . In the illustrated example, the origin  47  coincides with the center point M of the workpiece  10  for the purpose of clarity. 
     The detector array  28  is arranged relative to the spatially fixed coordinate system x, y, z such that the surface of said array runs in a plane parallel to the x-z plane and edges  29   a ,  29   c  run parallel to the x-axis and the edges  29   b ,  29   d  run parallel to the z-axis. 
     In the illustrated example, the center point M of the workpiece  10  on the rotary table  40  or the origin  47  of the spatially fixed coordinate system x, y, z is also the coordinate origin of a rotating coordinate system x*, y*, z*, with the z and z* axes coinciding. The axes x* and y* rotate relative to the axes x and y, as illustrated in  FIG. 2B  by x*′ and y*′. The center point K of the spherical structure  12  is arranged at the position x k *, y k *, z k * in the rotating coordinate system x*, y*, z*, as shown in  FIGS. 1A to 1C . 
     The irradiation of the workpiece  10  by means of the radiation source  22  results in a shadow of the spherical structure  12  in the workpiece  10  falling on the detector array  28 , which shadow is in the form of a circular two-dimensional image  50  of the spherical structure  12 . If the workpiece  10  is rotated about the rotational axis  44  (z*-axis) by an angular step, the position of the image on the detector array  28  changes from 50 to 50′, as illustrated in an exemplary fashion in  FIG. 3B  for a rotational angle φ=90°. 
     As will be explained below, the position and the dimensions of the spherical structure  12 , and hence a region of interest of the workpiece  10 , are reconstructed from a plurality of images  50 ,  50 ′, . . . recorded at different angles φ by utilizing at least three measurement points  52   a - 52   c ,  52   a ′- 52   c ′ situated on a boundary of the images  50 ,  50 ′. 
     A first step determines the center point K, i.e. the position of the spherical structure  12  in respect of the spatially fixed coordinate system x, y, z, and the radius R of the structure  12 . The measurement points  52   a - 52   c ,  52   a ′- 52   c ′ are registered for this purpose. In the process, the boundaries of the spherical projection, i.e. of the images  50 ,  50 ′, are found by means of known methods for image processing or image analysis and are transformed into a suitable number of measurement points. For simplification purposes, three points are selected from these measurement points, which three points are preferably distributed approximately equidistantly over the circumference of the image. 
     In general terms, points are registered at high-contrast transitions in the images  50 ,  50 ′, which transitions can correspond to image boundaries and edges of the structures to be examined 
     In order to establish the position of the center point K and the radius R of the spherical structure  12 , the coordinates x m , and z m  of the respective circle center M k , M k ′ and the circle radius R k , R k ′ of the images  50  and  50 ′ are determined from the coordinates of the respective three measurement points  52   a - 52   c ,  52   a ′- 52   c ′ of the images  50 ,  50 ′ utilizing the known equation of the circle illustrated in  FIG. 3C :
 
( x−x   m ) 2 +( z−z   m ) 2   =R   k   2 .
 
     Now, center beams  54 ,  54 ′ from the spot-like radiation source  22  to the center points M k  and M k ′ of the images  50 ,  50 ′ are established. The center beam  54 ′ is then rotated back by the angular step φ by the radiation source  22  and the center point M k ′ being mathematically rotated back by −φ about the rotational axis  44  in order to compensate for the rotational movement of the rotary table  40 . 
     In an ideal case, this would result in two center beams that coincide in space. However, despite very precise guides and control systems, there inevitably is a residual error in practice, and so the result of the calculation is not two coinciding straight lines, but two skewed straight lines. Now, points are determined on these two straight lines at which the distance between the straight lines is at a minimum. Then the connection perpendicular is constructed between these points. The center point of the path on the connection perpendicular between the two points is then assumed to be the best approximation for the location of the center point K in the x, y coordinate system. 
     Then, as illustrated in  FIG. 4 , the radius R of the structure  12  is determined by utilizing the intercept theorem. In the process, use is made of the fact that the ratio between the distance of the radiation source  22  from the center point M k  of the image  50  and the distance of the radiation source  22  from the center point K of the structure  12  equals the ratio between the radius R k  of the image  50  and the radius R of the structure  12 . 
     Hence, initial values for a spherical equivalent body are set, namely the position of a center point K A  and a radius R A . Hence the equivalent body can already be determined approximately in respect of location and shape from these values, which result from only two measurements at different projections of the workpiece  10  on the detector array  28 . 
       FIG. 4  furthermore shows that each tangent beam  58  leading from the radiation source  22  to a point  52  on the edge of the image  50  touches the structure  12  at a tangent point  60 . The path between the center point K and the tangent point  60 , i.e. the radius R, is orthogonal on the tangent beam  58  in this case. 
     This tangent condition can be used to establish the associated tangent point  60  for each tangent beam  58  that grazes the structure  12  along a circumference. The tangent points  60  determined in this fashion can then be used for a best-fit calculation (e.g. a least-squares fit). 
     In the process, a set of improvements of the parameters of the equivalent body is calculated using a system of equations, preferably by calculating the derivatives in the determination equation of the tangent points at the locations of the individual tangent points together with the orthogonal distances between the tangent points and the equivalent body. In the process, use can be made of the known least-squares-fit method, but also of other methods (envelope fit, inscribed fit, Chebyshev fit). The tangent points are redetermined after each improvement in the parameters of the equivalent body. This optimization is carried out iteratively until the improvements fall below a predefined residual value. Hence, the optimum equivalent body is determined, and the residual distances of the tangent points to the equivalent body can be used for e.g. analyzing the shape deviations, as is conventional in 3D measuring technology. This method is known to a person skilled in the art. 
     As mentioned above, the characteristic data of the optimized equivalent body, that is to say the location of the center point K in the workpiece  10  and the radius R in the illustrated exemplary embodiment, are now compared to the data from the nominal structure predefined by construction. A good part/bad part decision is derived from this comparison, i.e. it is determined whether the workpiece  10  lies within predefined tolerances in respect of the structure. 
     Here the accuracy can be further increased by virtue of the fact that the tangent points  60  are determined in a plurality of projections or rotational positions φ of the workpiece  10 , for example in 6 or 36 projections. The best-fit method is then applied to all tangent points  60  of the plurality of projections. 
     To this end,  FIG. 5  shows an example of an optimized equivalent body  64  for a spherical structure, which was fitted into the tangent points  60  by six projections at different angles. 
     Instead of the approximation using the best-fit method, the equivalent bodies can also be determined using an envelope approximation, an inscribed approximation or a Chebyshev approximation. 
     The best-fit method described for a spherical equivalent body can also, as already mentioned, be used for a cylindrical or conical equivalent body if the structure  12  is cylindrical or conical. The initial parameters to be determined of the cylinder are the cylinder radius and the alignment of the cylinder axis. In the case of an equivalent cone, the initial parameters are the cone angle of the cone, the alignment of the cone axis and the position of the tip of the cone.