Interpolation algorithms for digital mammography systems with multiple detectors

Interpolation Algorithms for Digital Mammography Systems with Multiple Detectors Hong Liu, PhD, Ge Wang, PhD, Jinghong Chen, MS Laurie L. Fajardo, MD

nomial, and one-dimensionaI and two-dimensional cubic

Results. These interpolation algorithms offered similar

The development of electronic imagers for digital mammography is an active area (1-5). Limited by the current hardware technology, a detector with high spatial resolution covers only a small field (usually less than 10 x 10 cm). To acquire a full-field digital mammogram (larger than 18 x 24 cm), one must use a scanning apparatus or a multiple detector design (6-10). Representative systems use multiple small electronic detectors that are abutted together (10). The resultant images, however, contain seams due to the gaps between the abutted detectors. The missing information in the seams can be estimated by using interpolation. Various interpolation algorithms, such as local weighting, polynomial, and cubic spline interpolations, are widely used in other applications, but to our knowledge they have not been evaluated for use in digital mammography systems with multiple detectors. Herein we present results of an investigation of the typical interpolation algorithms for filling in the seams in large-field digital mammograms. We will first describe our methods of data acquisition and the eight interpolation methods. Then we will compare these methods in terms of interpolation accuracy and geometric distortion by using real data. Finally, we will discuss the effect of the different interpolation methods on image quality and diagnostic performance, as well as other relevant issues.

Acad Radio11999; 6:170-175

From the Department of Radiology, University of Virginia, Charlottesville, VA 22908 (H.L., J.C., L.L.F.), and the Department of Radiology, University of Iowa, Iowa City (G.W.). Received April 10, 1998; revision requested August 11 ; revision received September 17; accepted September 22. Supported in part by the Whitaker Foundation (Biomedical Engineering Program) and by the National Cancer Institute (CA69043). Address reprint requests to H.L. ©AUR, 1999

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Table 1 Data Usedfor Two-dimensiOnal-PolynomialInterpolation 254

255

./-2 }-1

A(254, j - 2) A(254, j - 1)

~1 j+2

A(254, j+ 1) A(254, j+ 2)

256

257

258

259

A(255, j - 2) A(255, j - 1)

A(258, j - 2) A(248, j - 1)

A(259, j - 2) A(259, j - 1)

A(255, j+ 1) A(255, j+ 2)

A(258, j+ 1) A(258, j+ 2)

A(259, j+ 1) A(259, j+ 2)

Note.--A represents pixel value; therefore, A(254, j - 2) is the pixel value of the pixel located at the 254th column and the (j - 2)th row. j is the row index of the pixel.

MATERIALS AND METHODS Patient mammograms and phantom images were acquired with two prototype digital mammography systems in our experiments. The two prototypes are chargecoupled device systems, one fiberoptically coupled (7) and one lens coupled (4). The patients were imaged with x-ray techniques of 25-33 kVp and 60-160 mAs, depending on the thickness or density of the breast. Before any simulated seam was introduced or any interpolation algorithm applied, the digital images were manipulated by a preprocessing procedure called "flat-fielding" to correct the vignetting effect introduced by the lens, optical fiber, and other components in the optoelectronic imaging chain (2). Therefore, the profile of a homogeneous object on either side of the seam should be flat. For convenience, we used a 512 x 512-pixel array (pixel size, 63 × 63 ~tm), which is a portion of the original image. Then, each 512 × 512 image was evenly divided into two subimages by a 2-pixel-wide vertical seam generated digitally. This seam width is representative in practice, although in some systems the seam width can be irregular, particularly at the intersection of two seams. The following eight interpolation algorithms were used in this study to fill in the seam.

Algorithm 1: Nearest Neighbor In this algorithm, a missing pixel is set to the value of its nearest measured pixel. This method is computationally the simplest, and its rationale is that the nearest point should be the most relevant. Algorithm 2: One-dimensional Weighting This algorithm uses four horizontal neighbors of the 2 seam pixels of interest, and linearly combines them with each of the relative weighting factors inversely propor-

tional to the distance to the unknown pixel. This weight definition is consistent with that in the linear and bilinear interpolation schemes.

Algorithm 3: Two-dimensional Weighting This algorithm uses four data points from each side of the seam. The four points include 2 nearest measured pixels along the row of the missing pixel, and 2 diagonal pixels in the column adjacent to the seam. This method provides more local data points for interpolation than the above single line method. The weighting factors are similarly defined.

Algorithm 4: Mean Value This method approximates a missing pixel value with the mean of its adjacent measured pixel values. Either one-dimensional or two-dimensional mean value interpolation can be performed.

Algorithm 5: One-dimensional Polynomial This algorithm uses 4 adjacent measured pixels along the row of a missing pixel, 2 pixels on each side of the seam. The interpolating polynomial is given by Lagrange's classic formula (11), which passes through all the known data points. Algorithm 6: Two-dimensional Polynomial To perform two-dimensional polynomial interpolation for a value of a missing pixel, one uses some measured pixels above and below the missing pixel row. The basic idea of two-dimensional interpolation is to decompose the problem into two steps of one-dimensional interpolation. As shown in Table 1, where the size of the interpolation grid is 4 x 4, to estimate image values marked by question marks we first perform vertical interpolations (ie, along the columns of Table 1) to obtain those values

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Table 2 Relative Errors of the Eight=Interpolation Algorithms with Three Clinical Images Shown in Figures 1-3 Relative Errors (%) for Algorithms 1-8 Image

1

2

3

4

5

6

7

8

1 (Fig 1) 2 (Fig 2) 3 (Fig 3)

3.43 6.79 3.20

2.97 5.68 2.77

3.00 5.17 2.53

3.00 5,62 2.75

4.06 6.37 3.67

4.20 6.84 3.82

4.21 6.59 3.81

4.15 7.15 3.78

a. b. c. Figure 1. American College of Radiology (Reston, Va) phantom image used for accuracy analysis. (a) Original image, (b) image with a seam, and (c) interpolated image obtained with algorithm 1 (nearest neighbor duplication).

marked by asterisks and then perform a horizontal interpolation (ie, along the row) by using the vertically interpolated values marked by asterisks to estimate values marked by question marks. Note that in two-dimensional polynomial interpolation we interpolate for values marked by asterisks, instead of using the measured values as in one-dimensional polynomial interpolation. We include this two-dimensional polynomial interpolation scheme for smoother fitting of image features in the seam. Algorithm 7: One-dimensional Cubic Spline

In one-dimensional cubic spline interpolation, we use not only neighboring known points but also the associated first derivative (11). The free boundary conditions were assumed in this study. The same four neighboring points were used as in algorithm 5. Algorithm 8: Two-dimensional Cubic Spline

Similarly, two-dimensional cubic spline interpolation is decomposed into two steps of one-dimensional cubic spline interpolation. The same local points were used as in algorithm 6.

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IESULT5 Absolute and Relative Errors

Each of the eight interpolation algorithms was applied to a series of clinical digital mammograms to estimate missing data at the simulated seams. The effectiveness of the interpolation algorithms was evaluated quantitatively in terms of absolute and relative errors. The results for three digital mammograms are summarized in Table 2. The errors were all below 10%, indicating that the interpolated pixel values deviated less than 10% from the true pixel values. It is interesting that for a given image, the eight algorithms offer similar accuracy. The polynomial and spline interpolation algorithms did not provide significantly better accuracy, as one might expect. This may reflect the fact that the intensity profiles of breast images are usually irregular and difficult for analytic modeling. To compare the accuracies of the interpolation algorithms in the context of different clinical features, we selected regions of interest that contain clinically important features from the three digital mammograms (Figs 1-3). The size of the ROIs is 64 x 64 pixels. The average rela-

Table 3 Relative Errors of the Eight ln~rpolation Algorithms in Regions of Interest of Clinical Features

Relative Errors (%) for Algorithms 1-8 Image

1

2

3

4

5

6

7

8

4 5 6

5.0 6.52 1.95

5.39 5.56 1.57

7.08 6.17 3.80

5.60 5.88 1.52

4.47 5.24 2.57

4.57 5.43 2.68

4.61 5.68 2.68

5.86 6.72 2.64

Note.--Image 4 includes a cluster of microcalcifications (100-500 gm), image 5 contains several larger microcalcifications, and image 6 has masses.

a. b. c. Figure 2. Digital mammogram with a cluster of microcalcifications used for accuracy analysis. (a) Original image, (b) image with a seam, and (c) interpolated image obtained with algorithm 6 (two-dimensional polynomial interpolation).

a. b. c. Figure 3. Digital mammogram with masses used for accuracy analysis. (a) Original image, (b) image with seam, and (c) interpolated image obtained with algorithm 4 (mean value interpolation).

five errors in these ROIs are shown in Table 3. The results demonstrate that the interpolation algorithms were more effective in estimating larger, lower-contrast features (masses) than smaller, higher-contrast features (microcalcifications).

The range of the absolute interpolation errors along the seams was quantified in Table 4. Roughly speaking, the average absolute errors vary from four to eight for our 8-bit images.

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Table 4 Absolute Errors of the EighHnterpolation Algorithms with Three Clinical Images Absolute Errors for Algorithms 1-8

a.

Image

1

2

3

4

5

6

7

8

1 (Fig 1) 2 (Fig 2) 3 (Fig 3)

4.64 6.76 4.28

4.02 5.66 3.71

4.06 5.15 3.38

4.06 5.60 3.68

5.49 6.35 4.91

5.68 6.81 5.11

5.69 6.57 5.10

5.61 7.12 5.06

b.

c.

Figure 4. Magnified views of a mammogram with microcalcifications. (a) Original image, (b) image with a seam, (e) interpolated image obtained with algorithm 1 (nearest neighbor duplication), and (d) interpolated image obtained with algorithm 5 (one-dimensional polynomial interpolation).

G e o m e t r i c D i s t o r t i o n of C l i n i c a l F e a t u r e s

Diagnostic decisions are made by evaluating the shape and configuration of mammographic features such as microcalcifications or masses. Therefore, it is important to evaluate whether interpolation algorithms introduce geometric distortion. We found that geometric distortion in the appearance of microcalcifications is more apparent with nearest neighbor and mean value interpolation methods (algorithms 1 and 4) as evidenced by irregularities introduced in borders of the microcalcifications. On the other hand, the microcalcifications interpolated with weighting, polynomial, or cubic spline interpolation methods (algorithms 2, 3, and 5-8) have smoother borders that more closely resemble the original mammogram. Figure 4 illustrates a cluster of microcalcifications before (Fig 4a) and after (Fig 4b) introduction of a simulated seam, and two representative interpolated images (Fig 4c, 4d) obtained by using nearest neighbor and one-dimensional polynomial al-

174

d.

gorithms, respectively. Also, if a small microcalcification falls exactly within the seam, none of the above interpolation algorithms can recover the signal. The failure to image a few calcifications of a cluster, however, generally does not substnatially alter diagnostic performance. As far as interpolation of missing pixels of masses is concerned, the eight methods performed similarly.

In our experiments, when the pixel values of a region of interest are close to the upper or lower limits of the dynamic range--for example, 0 or 255 for the 8-bit case (0 or 4,095 for the 12-bit case)--Lagrange or cubic spline algorithms may introduce pixel values less than 0 or higher than 255 due to the requirement for a continuous derivative(s). Therefore, the interpolated image will have defects, necessitating truncation or a local weighting method to correct the defected points. The absolute and relative errors provide data on the degree to which interpolated pixel values reproduce true pixel values for digital mammograms. In comparing the geometric appearance of interpolated clinical features, we found that the borders of breast calcifications sometimes became irregular after interpolation with nearest neighbor and mean value algorithms. In contrast, the weighting, polynomial, and cubic spline interpolation algorithms produced less geometric distortion of the calcifications. Furthermore, there was little observed difference between the images obtained with weighting, polynomial, and cubic spline algorithms. Therefore, the one-dimensional weighting interpolation method seems an excellent choice for both effectiveness and efficiency. Additionally, our interpolation methods can also be used to correct defective pixels, particularly column defects in the imager, which can be particularly troublesome. In conclusion, weighting, polynomial, and cubic spline interpolation algorithms are effective for estimating lost information in the seams of large-field mammography images acquired with multiple digital detectors. For typical mammograms, these interpolation algorithms offer a less than 10% interpolation accuracy. The weighting,

polynomial, and cubic spline interpolation algorithms may create less geometric distortion than the nearest neighbor and mean value algorithms. All algorithms are more powerful in handling larger, lower-contrast features, such as masses, than smaller, higher-contrast features, such as microcalcifications. Small microcalcifications completely within the seams cannot be recovered regardless of the interpolation algorithm used. Hence, seams should be avoided or minimized in the design of digital breast imaging systems. IEFERENCE~

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