Abstract

A method of optical filtering in order to provide for enhancement of x-ray images whose quality has been degraded because of the use of extended x-ray sources (penumbra effect) is considered. This method employs spatial filtering techniques in a coherent optical system. A description of the penumbra effect follows from the mathematical description of imaging in incoherent optical systems. With the use of this model the degradation can be described in terms of the effect of the extended source on the frequency content of the resulting image. This leads to a precise definition of the ideal filter in terms of the spatial intensity distribution of the source and a general description of the filtering operations which will be required in all cases, independent of the exact source distribution. It is shown that the form of the ideal filter is such that the same optical system used in the filtering process is ideally suited for generating filters with, approximately, the required transmission characteristics. Experimental results are presented.

© 1968 Optical Society of America

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References

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  1. L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).
  2. E. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).
  3. M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill Book Co., Inc., New York, 1960).
  4. R. H. Morgan, Am. J. Roentgenol. 88, 175 (1962).
  5. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965).
  6. K. Rossman, Am. J. Roentgenol. 90, 863 (1963).
  7. C. E. K. Mees, T. H. James, Eds., The Theory of the Photographic Process (The Macmillan Co., New York, 1966).
  8. R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

1964

R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

1963

K. Rossman, Am. J. Roentgenol. 90, 863 (1963).

1962

R. H. Morgan, Am. J. Roentgenol. 88, 175 (1962).

1960

L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).

Bates, L. M.

R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

Bennett, W. R.

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill Book Co., Inc., New York, 1960).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).

Gopalarao, U. V.

R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).

Marinaro, A.

R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

Morgan, R. H.

R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

R. H. Morgan, Am. J. Roentgenol. 88, 175 (1962).

O’Neill, E.

E. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).

Palermo, C.-J.

L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).

Rossman, K.

K. Rossman, Am. J. Roentgenol. 90, 863 (1963).

Schwartz, M.

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill Book Co., Inc., New York, 1960).

Stein, S.

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill Book Co., Inc., New York, 1960).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965).

Am. J. Roentgenol.

R. H. Morgan, Am. J. Roentgenol. 88, 175 (1962).

K. Rossman, Am. J. Roentgenol. 90, 863 (1963).

R. H. Morgan, L. M. Bates, U. V. Gopalarao, A. Marinaro, Am. J. Roentgenol. 92, 426 (1964).

Inst. Radio Eng. Trans.

L. J. Cutrona, E. N. Leith, C.-J. Palermo, L. J. Porcello, Inst. Radio Eng. Trans. IT-6, 386 (1960).

Other

E. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1963).

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill Book Co., Inc., New York, 1960).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965).

C. E. K. Mees, T. H. James, Eds., The Theory of the Photographic Process (The Macmillan Co., New York, 1966).

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Figures (12)

Fig. 1
Fig. 1

Symbolic diagram of x-ray process.

Fig. 2
Fig. 2

Effect of extended source on edge sharpness: (a) point source; (b) extended source.

Fig. 3
Fig. 3

Sequence illustrating effect of penumbra as function of (a) initial effect; (b) overlapping of illumination; spatial frequency, (c) image disappearance; (d) image reversed.

Fig. 4
Fig. 4

Demonstration of optical penumbra effect.

Fig. 5
Fig. 5

Experimental verification of effect of penumbra on contrast; (a) almost ideal image; (b) less ideal; (c) zero contrast; (d) negative contrast, i.e., phase reversal.

Fig. 6
Fig. 6

Optical system for spatial filtering and reimaging.

Fig. 7
Fig. 7

Experimental results of dc filtering of blurred medical x-ray image: (a) original x ray; (b) result of dc filtering.

Fig. 8
Fig. 8

Spectrum of image of line grating exposed by extended source: (a) location of weighed harmonics; (b) frequency weighting imposed by uniform extended source.

Fig. 9
Fig. 9

Enlarged photograph and microdensitometer scan of filter for blurred image of line grating.

Fig. 10
Fig. 10

Effect of high pass filtering on blurred image of line grating: (a) unfiltered image of blurred line grating; (b) filtered image of blurred line grating.

Fig. 11
Fig. 11

Microdensitometer scans of filtered and unfiltered line grating images.

Fig. 12
Fig. 12

Comparison of frequency content of x-ray images obtained with 0.6-mm and 2.0-mm sources.

Equations (11)

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I E ( x , y ) = c - δ ¯ / 2 δ ¯ / 2 - δ / 2 δ / 2 W ( α , β ) I ( x - α , y - β ) d α d β ,
I E ( x ) = - δ / 2 δ / 2 I ( x - α ) d α .
I E ( x ) = - δ / 2 δ / 2 [ 1 + cos 2 π f ( x - α ) ] d α = δ [ 1 + sin π δ f π δ f cos 2 π f x ] .
I ( x ) = n = - C n exp - ( j 2 π n f x ) ,
C n = 1 T - T / 2 T / 2 I ( x ) exp ( - j 2 π n f x ) d x ,
I E ( x ) = δ n = - C n sin π ( n f ) δ π ( n f ) δ exp ( j 2 π n f x ) ,
ϕ E ( u , v ) = - - I E ( x , y ) exp [ - j 2 π ( u x + v y ) ] d x d y ,
ϕ E ( u , v ) = ϕ ( u , v ) H ( u , v ) ,
ϕ ( u , v ) = - - I ( x , y ) exp [ - j 2 π ( u x + v y ) ] d x d y , H ( u , v ) = - - W ( x , y ) exp [ - j 2 π ( u x + v y ) ] d x d y .
- - W ( x , y ) d x d y < ,
H ( u , v ) = | - - W ( x , y ) exp [ - j 2 π ( u x + v y ) ] d x d y | - - W ( x , y ) d x d y = H ( 0 , 0 ) ,

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