Method to remove station-induced error pattern from measured object characteristics and compensate the measured object characteristics with the error

A method of separating station-induced errors caused by an object holder from measurements of an object having a first surface using a first sensor and a second sensor to measure the first surface when the object is placed in a first plane substantially parallel to the first surface, and wherein the first sensor and the second sensor are located on the opposite sides of the first plane along a first line substantially perpendicular to the first plane, said method comprising the steps of:

• 1) measuring with the first sensor a multiplicity of first points on the first surface of the measured object at a first orientation to obtain a first matrix (M) of measurement points in polar coordinates having nodes on a plurality of concentric circles, each number representing the distance between a first point and the first sensor;
• said first matrix (M) containing a surface matrix (S) and a station-induced error matrix (E);
• 2) rotating the measured object independently of the object holder about a first axis substantially parallel to the first plane by a first angle (α) to placed the measured object at a second orientation, thereby generating a rotation matrix (R);
• 3) measuring with the first sensor a multiplicity of second points on the first surface of the measured object at the second orientation to obtain a second matrix (M′) of measurement points, each representing the distance between a second point and the first sensor, each second point having a corresponding first points;
• 4) obtaining a first difference matrix (M′−M) between the second matrix (M′) and the first matrix (M), thereby obtaining the product of a second difference matrix (R−I) and the surface matrix (S);
• 5) flipping the measured object about a second axis substantially perpendicular to the first axis by 180 degrees to place the object at a third orientation;
• 6) measuring with the second sensor the first surface of the measured object at the third orientation to obtain a multiplicity of third points to obtain a third matrix (M″) of measurement points, each third point having a corresponding second point;
• 7) obtaining a first average value (s1) of a select measurement point (m1″) in the third matrix and the corresponding measurement point (m1) in the first matrix;
• 8) removing the second difference matrix (R−I) from the surface matrix (S) by LU decomposing the second difference matrix, wherein L is a lower triangular matrix and U is an upper triangular matrix, and multiplying the first inverse matrix (L−1) to the first difference matrix (M′−M), wherein the first inverse matrix (L−1) is the inverse of the lower triangular matrix (L);
• 9) obtaining an auxiliary matrix (λ) from the product of the first inverse matrix and the first difference matrix;
• 10) obtaining a remaining surface component (Sr) for a circle from the auxiliary matrix (λ), the second inverse matrix (U−1) and the first average value (s1), wherein the second inverse matrix is the inverse of the upper triangular matrix (U);
• 11) repeating steps 7 to 11 for all circles of the measured points.

A method of removing station-induced errors caused by an object holder from measurement data of an object held by the object holder obtained by a device, said method comprising the steps of:

• 1) measuring the object at a first orientation with respect to the object holder to obtain a first matrix of measurement points containing the station-induced errors and the map of the proper shape of the object's surface;
• 2) rotating the object independently of the holding device to place the object at a second orientation with respect to the object holder;
• 3) measuring the object at the second orientation to obtain a second matrix of measurement points containing station-induced errors and the surface map of the object as processed by a rotation transformation represented by a rotation matrix;
• 4) removing the station-induced errors from the measurement data by subtracting the first matrix from the second matrix;
• 5) flipping the object and measuring the object at a third orientation with respect to the object holder to obtain a third matrix of measurement points;
• 6) obtaining the surface characteristics at selected measurement points from the first matrix and the third matrix;
• and
• 7) removing the rotation transformation by applying an inverse of the difference between a rotation matrix and an identity matrix to the difference of the second matrix and the first matrix.