- Email: [email protected]
Quantum noise in 2D projections and 3D reconstructions
Ultramicroscopy 6 (1981) 85-90 North-Holland Publishing Company
SHORT NOTE QUANTUM NOISE IN 2D PROJECTIONS AND 3D RECONSTRUCTIONS B.E.H. SAXBERG and W.O. SAXTON High Resolution Electron Microscope, Free School Lane, Cambridge CB2 3R Q, England
Received 1 August 1980
Notwithstanding the detailed mathematical justification given in [ 1], the assertion made by Hoppe's group that "a 3D reconstruction requires the same integral dose as a conventional 2D micrograph, provided the level o f significance and the resolution are identical" was the subject of considerable discussion, and indeed some disbelief, a few years ago. Little of the discussion appears to have reached print however [2,3], and Hoppe, while continuing to assert that "two-dimensional microscopy is in fact a waste of information" [4], has offered only minor additional explanation [5]. This note seeks to clarify the issue, showing how the assertion may be true or false according to what is meant by the same level of significance. Essentially, the mathematical result is that the standard error in the density reconstructed from a set of n projections (micrographs), between which a total electron dose per unit area p is shared, and which are scarmed using a microdensitometer with an aperture d square (thus limiting the resolution to about 2d) is, to within a constant of order unity, A p "~ m o / p l / 2 d 2 ,
mo being a constant relating projected mass to image contrast, and having a value around 2 X 10 -22 kg/nm 2. A short appendix outlines a derivation of this, which is based on a fully discrete reconstruction process, including interpolation effects, exactly as implemented on a computer. It is consistent with Hegerl and Hoppe's result [1] which was derived with quantisation o f angle only;it is also confirmed independently in ref. [6]. Provided the number of
distinct projections is sufficient to avoid any distortion of the signal, the standard error in the reconstructed density is independent of their number, and also independent of the size of object being reconstructed. Three times the standard error may be taken as an estimate of the smallest density difference that can be reliably detected: table 1 gives the value of this "density resolution" for a range o f p and d values. Since the result itself is clearly sound, let us address instead its interpretation. The problem lies in making a comparison of the levels of significance in 2D and 3D images. The most obvious criterion is probably the signal-to-noise ratio, and in this respect there is no doubt that a much higher total exposure is required for a 3D reconstruction with the same signalto-noise ratio as a single 2D micrograph. For a speci-
Table 1 Density resolution in 3D reconstruction, for various electron doses p and resolution elements d (expressed as multiples of the density of water) p (e nm-2) 30 100 300 10a 3 x 10a 104 3 x 104 10s
d (nm) 0.5
1
2
3
5
440 240 140 76 44 24 14 7.6
110 60 35 19 11 6.0 3.5 1.9
27 15 8.7 4.7 2.7 1.5 0.9 0.5
12 6.7 3.8 2.1 1.2 0.7 0.4 0.21
4.4 2.4 1.4 0.8 0.4 0.24 0.14 0.08
0 3 0 4 - 3 9 9 1 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland Publishing Company
B.E.H. Saxberg, W.O. Saxton / Quantum noise
86
men with diameter D and density ~-, the mean signal in a single projection micrograph is ~D; if it is recorded with a total dose Pl, the rms noise is mo/p[/2d, giving a signal-to-noise ratio of rproj
=
OP1/2Dd/m o •
The signal-to-noise ratio for a reconstruction made using a total dose P2, is however from the above rrec = -~p~/2d2/mo , SO that, for equal signal to noise ratios, the dose needed for a 3D reconstruction P2 is greater than that needed for a single projection p i by a factor (D/d) 2. But the most obvious criterion is not necessarily the most appropriate or the most useful; another criterion can be obtained by considering the ability of each method to detect and locate small blocks of mass in the specimen. The process of scanning with an aperture d square effectively divides the specimen volume into small cubic blocks of side d. By multiplying the 'density resolution' given above by the volume d 3 of such a cube, the smallest mass of a block of matter detectable in the 3D reconstruction is found to be 3mod/p 1/2. The specimen mass in each volume element can be approximated by a superposition of an integral number of such mass blocks, and the reconstruction indicates reliably how many of these blocks occupy each volume element of the 3D specimen. Now each mass block contributes 3too~ p~/2d to a density projection made with the entire dose p, and the rms noise level in the recorded projection, noted above, is a third of this. Significant detection of one such mass block therefore requires the same total dose for the 3D reconstruction as for the 2D projection. Using the detection and counting of these mass blocks as a criterion for the significance comparison, we can therefore claim the same significance for the same total electron dose. However, since the projection micrograph indicates reliably only how many of the mass blocks lie in each projection column (i.e., project into each area element d square), and no more than this, the information content of the 3D reconstruction is obviously far greater, telling us how the blocks are distributed in three dimensions rather than in only two. Is it really then possible to obtain something for nothing by substituting a 3D reconstruction made from noisy projections for a conventional 2D micro-
graph? If the density resolution given in table 1 for the electron dose attainable and resolution required is adequate - e.g., establishing the presence or absence of a heavy atom stain, or distinguishing protein from glucose or ice - then the answer is "Yes". However, the electron doses required, according to the table, are discouragingly high, and we shall frequently face Situations in which it cannot be attained. In such a case, a 3D reconstruction will be of no value, as it will be unable to make the crucial binary decision between block counts of zero and one. A single 2D micrograph on the other hand, while unable any longer to count exactly the number of blocks falling in each column, can still provide some useful information at the resolution required in the form of rough column counts, since there will usually be many blocks failing into a column. In both cases the original density resolution can be restored by accepting reduced spatial resolution of course; but only the 2D micrograph has the option of accepting a lower density resolution at the original spatial resolution.
A simulated example To lend clarity and conviction to the above, reconstructions of a simple specimen carried out from simulated data at various electron exposures are presented in fig. 1, together with single projections made with the same exposures. The specimen is assumed to be a light stain with three times the density of water; the distribution within a single 2 nm slice viewed along the tilt axis is shown top fight - formed from 2 nm cubes. It gives rise, projected vertically, to the expected image intensity distribution shown to its left, which has a maximum contrast of 12%, 3% arising from each individual cube. Simulated projections were prepared for this structure in 32 directions evenly spread over the full angular range. With a Poisson random number generator and some smoothing, data representative of shot noise-limited micrographs scanned at 0.67 nm intervals with a 2 nm square aperture were produced from the projections; the use of so many projections and this fine sampling interval eliminates any noticeable effects due to interpolation etc. from the reconstructions made subsequently. For each of a wide range of exposure values (10 s, 3 X 104, 104, 3 X 10 a,
B.E.H. Saxberg, W.O. Saxton /Quantum noise
87
12nm 12nm
10 5
3xlO 4
lO 4
It
_
%
.... :
~
.t .....
..&° J
I
Fig. 1. Simulated projections and 2D reconstructions of specimen slice density, determined with equivalent total electron doses as marked (in e/nm2); -*10% contrast levels shown on vertical axes. See text for details.
103, 300, 100 and 30 e/nm2), a set of projection data was produced in which the exposure was shared between the 32 projections, and a single vertical projection was produced with the whole exposure devoted to that projection alone. The second and subsequent rows of fig. 1 show, for each exposure in turn, the single projection and the 2D distribution within the slice as calculated from the set of 32 noisy projections. The pattern predicted previously is clearly evident. At the lowest exposure level nothing useful is to be concluded either from the projection or from the 2D
reconstruction; at 300 e/nm 2 it is possible to detect the presence of the largest peak in the projection, but as yet nothing in the reconstruction. With increasing exposure the projection is clarified. At about the point where the smallest (single cube) peaks can be identified in the projection, and the relative heights of all the peaks discerned, the stain distribution is just emerging from the noise in the reconstruction. The exposure level needed for this - about 3000 e/nm 2 - agrees well with that indicated in table 1 for the detection of the stain assumed here in 2 nm cubes. Further increase in the exposure level brings
88
B.E.H. Saxberg, W.O. Saxton /Quantum noise
the reassurance of better defmition in projection and reconstruction (showing both smoothed with a 2 nm sampling aperture, of course), but no fundamentally new information.
Supposing white uncorrelated noise in the original projection values, the discrete transform of a projection has a constant variance - say, o 2; if the maximum spatial frequency represented in the transform is kc, the total power is thus 2kco 2. After application of the Ik I ffdter function, it becomes
Conclusions kC
It is clear that, notwithstanding the equivalent statistical significance possible in 3D. reconstructions and 2D micrographs where a high electron dose can be tolerated, the 2D micrograph will in many practical cases be able to furnish partial information at higher resolution than the 3D reconstruction; this may or may not be useful, depending on the particular structural problem addressed. The total doses required for statistical significance in 3D reconstructions are disappointingly high particularly for frozen hydrated or glucose embedded specimens where the density resolution required is as low as a third of the density of water. This must inevitably cast suspicion on reconstructions made to high resolution from individual particles, even stained (e.g. ref. [7]). The use of averaging, either over lattices in the conventional manner, or over irregularly arranged units (e.g. ref. [8]), offers however a practical way of increasing effective electron doses 100-fold, and our results emphasise strongly the importance of such techniques.
2f
o2k 2 dk = 2k3co2/3 ,
0
and accordingly, the noise variance in the filtered projection value is increased with respect to the original variance by a factor of k2c/3. If n micrographs are recorded with a total electron dose per unit area p and scanned with an aperture of area d 2, the variance in the contrast of each micrograph, according to simple Poisson statistics, is n/pd 2. A factor mo --~ 2 X 10 -22 kg/nm 2 relates the integrated density p to the contrast c [9], for typical operating conditions, according to
f p =moc, so that the noise in the raw projection data has a variance nm~/pd2;on f'dtering this becomes nm2k2c/3pd 2, or nm~/12pal 4, since the cut-off spatial frequency kc will be approximately 1/2d. Adding n projection values for each reconstruction point increases the variance by a factor n; allowing for the f'mal factor rr/n applied to the reconstruction, we have a variance
Acknowledgement
Ap2 .~ (Tr/n)2n nm2o/(12pd 4) The support of the UK Science Research Council for this work is gratefully acknowledged. The manuscript was refereed by Dr. P. Schiske.
Appendix: Outline proof of reconstruction accuracy Using the f'dtered back-projection method, a slice may be reconstructed from n 1D projections by (a) applying a f'dtering process to each projection, in which its discrete Fourier transform values are multiplied by Ik 1, the modulus of the corresponding spatial frequencies; (b) for each point in the slice, adding together the filtered projection values at the points into which the point being reconstructed projects; and (c) multiplying the reconstruction by lr/n.
in the reconstruction, or an approximate rms error
AO ~- mo/pl /2d 2 , which is the required result. (In practice, the linear interpolation process used to determine filtered projection values at arbitrary points blurs the signal and the noise slightly in a space variant manner; the signal spread is insignificant beyond nearest neighbour points, and the rms noise level is reduced by about 32%.) It is not difficult to see that substantially the same result is achieved if the alternative method of direct synthesis of the 2D transform of the slice is adopted. The effect of missing projection data (blind regions in Fourier space, due to limits on the tilting angle possi-
B.E.H. Saxberg, W.O. Saxton / Quantum noise ble) are also simply assessed: the final noise variance is simply reduced in proportion to the lower number of projections used; discernibility of the specimen structure does not improve in spite of the lower noise variance, since the signal is reduced also, and distorted moreover in a way well characterised previously (e.g., ref. [4]).
References [ 1] R. HegerI and W. Hoppe, Z. Naturforsch. 3 l a (1976) 1717. [2] W. Baumeister and M. Hahn, Hoppe-Seyler's Z. Physik. Chem. 356 (1975) 1313.
89
[3] W. Hoppe, J. Gassmann, N. Hunsmann, H.J. Schramm and M. Sturm, Hoppe-Seyler's Z. Physik. Chem. 356 (1975) 1317. [4] W. Hoppe, in: Computer Processing of Electron Microscope Images, Ed. P.W. Hawkes (Springer, Berlin, 1980) ch. 4, p. 127. [5] W. Hoppe, Chem. Scripta 14 (1978-9) 227. [6] G. Kowalski, IEEE Trans. NS-24 (1977) 850. [7] W. Hoppe, J. Gassmann, N. Hunsmann, H.J. Schramm and M. Sturm, Hoppe-Seyler's Z. Physik. Chem. 355 (1974) 1483. [8] J. Frank, W. Goldfarb, D. Eisenberg and T.S. Baker, Ultramicroscopy 3 (1978) 283. [9] D.L. MiseUand I.DJ. Burdett, J. Microsc. 109 (1977) 171.
SHORT NOTE QUANTUM NOISE IN 2D PROJECTIONS AND 3D RECONSTRUCTIONS B.E.H. SAXBERG and W.O. SAXTON High Resolution Electron Microscope, Free School Lane, Cambridge CB2 3R Q, England
Received 1 August 1980
Notwithstanding the detailed mathematical justification given in [ 1], the assertion made by Hoppe's group that "a 3D reconstruction requires the same integral dose as a conventional 2D micrograph, provided the level o f significance and the resolution are identical" was the subject of considerable discussion, and indeed some disbelief, a few years ago. Little of the discussion appears to have reached print however [2,3], and Hoppe, while continuing to assert that "two-dimensional microscopy is in fact a waste of information" [4], has offered only minor additional explanation [5]. This note seeks to clarify the issue, showing how the assertion may be true or false according to what is meant by the same level of significance. Essentially, the mathematical result is that the standard error in the density reconstructed from a set of n projections (micrographs), between which a total electron dose per unit area p is shared, and which are scarmed using a microdensitometer with an aperture d square (thus limiting the resolution to about 2d) is, to within a constant of order unity, A p "~ m o / p l / 2 d 2 ,
mo being a constant relating projected mass to image contrast, and having a value around 2 X 10 -22 kg/nm 2. A short appendix outlines a derivation of this, which is based on a fully discrete reconstruction process, including interpolation effects, exactly as implemented on a computer. It is consistent with Hegerl and Hoppe's result [1] which was derived with quantisation o f angle only;it is also confirmed independently in ref. [6]. Provided the number of
distinct projections is sufficient to avoid any distortion of the signal, the standard error in the reconstructed density is independent of their number, and also independent of the size of object being reconstructed. Three times the standard error may be taken as an estimate of the smallest density difference that can be reliably detected: table 1 gives the value of this "density resolution" for a range o f p and d values. Since the result itself is clearly sound, let us address instead its interpretation. The problem lies in making a comparison of the levels of significance in 2D and 3D images. The most obvious criterion is probably the signal-to-noise ratio, and in this respect there is no doubt that a much higher total exposure is required for a 3D reconstruction with the same signalto-noise ratio as a single 2D micrograph. For a speci-
Table 1 Density resolution in 3D reconstruction, for various electron doses p and resolution elements d (expressed as multiples of the density of water) p (e nm-2) 30 100 300 10a 3 x 10a 104 3 x 104 10s
d (nm) 0.5
1
2
3
5
440 240 140 76 44 24 14 7.6
110 60 35 19 11 6.0 3.5 1.9
27 15 8.7 4.7 2.7 1.5 0.9 0.5
12 6.7 3.8 2.1 1.2 0.7 0.4 0.21
4.4 2.4 1.4 0.8 0.4 0.24 0.14 0.08
0 3 0 4 - 3 9 9 1 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland Publishing Company
B.E.H. Saxberg, W.O. Saxton / Quantum noise
86
men with diameter D and density ~-, the mean signal in a single projection micrograph is ~D; if it is recorded with a total dose Pl, the rms noise is mo/p[/2d, giving a signal-to-noise ratio of rproj
=
OP1/2Dd/m o •
The signal-to-noise ratio for a reconstruction made using a total dose P2, is however from the above rrec = -~p~/2d2/mo , SO that, for equal signal to noise ratios, the dose needed for a 3D reconstruction P2 is greater than that needed for a single projection p i by a factor (D/d) 2. But the most obvious criterion is not necessarily the most appropriate or the most useful; another criterion can be obtained by considering the ability of each method to detect and locate small blocks of mass in the specimen. The process of scanning with an aperture d square effectively divides the specimen volume into small cubic blocks of side d. By multiplying the 'density resolution' given above by the volume d 3 of such a cube, the smallest mass of a block of matter detectable in the 3D reconstruction is found to be 3mod/p 1/2. The specimen mass in each volume element can be approximated by a superposition of an integral number of such mass blocks, and the reconstruction indicates reliably how many of these blocks occupy each volume element of the 3D specimen. Now each mass block contributes 3too~ p~/2d to a density projection made with the entire dose p, and the rms noise level in the recorded projection, noted above, is a third of this. Significant detection of one such mass block therefore requires the same total dose for the 3D reconstruction as for the 2D projection. Using the detection and counting of these mass blocks as a criterion for the significance comparison, we can therefore claim the same significance for the same total electron dose. However, since the projection micrograph indicates reliably only how many of the mass blocks lie in each projection column (i.e., project into each area element d square), and no more than this, the information content of the 3D reconstruction is obviously far greater, telling us how the blocks are distributed in three dimensions rather than in only two. Is it really then possible to obtain something for nothing by substituting a 3D reconstruction made from noisy projections for a conventional 2D micro-
graph? If the density resolution given in table 1 for the electron dose attainable and resolution required is adequate - e.g., establishing the presence or absence of a heavy atom stain, or distinguishing protein from glucose or ice - then the answer is "Yes". However, the electron doses required, according to the table, are discouragingly high, and we shall frequently face Situations in which it cannot be attained. In such a case, a 3D reconstruction will be of no value, as it will be unable to make the crucial binary decision between block counts of zero and one. A single 2D micrograph on the other hand, while unable any longer to count exactly the number of blocks falling in each column, can still provide some useful information at the resolution required in the form of rough column counts, since there will usually be many blocks failing into a column. In both cases the original density resolution can be restored by accepting reduced spatial resolution of course; but only the 2D micrograph has the option of accepting a lower density resolution at the original spatial resolution.
A simulated example To lend clarity and conviction to the above, reconstructions of a simple specimen carried out from simulated data at various electron exposures are presented in fig. 1, together with single projections made with the same exposures. The specimen is assumed to be a light stain with three times the density of water; the distribution within a single 2 nm slice viewed along the tilt axis is shown top fight - formed from 2 nm cubes. It gives rise, projected vertically, to the expected image intensity distribution shown to its left, which has a maximum contrast of 12%, 3% arising from each individual cube. Simulated projections were prepared for this structure in 32 directions evenly spread over the full angular range. With a Poisson random number generator and some smoothing, data representative of shot noise-limited micrographs scanned at 0.67 nm intervals with a 2 nm square aperture were produced from the projections; the use of so many projections and this fine sampling interval eliminates any noticeable effects due to interpolation etc. from the reconstructions made subsequently. For each of a wide range of exposure values (10 s, 3 X 104, 104, 3 X 10 a,
B.E.H. Saxberg, W.O. Saxton /Quantum noise
87
12nm 12nm
10 5
3xlO 4
lO 4
It
_
%
.... :
~
.t .....
..&° J
I
Fig. 1. Simulated projections and 2D reconstructions of specimen slice density, determined with equivalent total electron doses as marked (in e/nm2); -*10% contrast levels shown on vertical axes. See text for details.
103, 300, 100 and 30 e/nm2), a set of projection data was produced in which the exposure was shared between the 32 projections, and a single vertical projection was produced with the whole exposure devoted to that projection alone. The second and subsequent rows of fig. 1 show, for each exposure in turn, the single projection and the 2D distribution within the slice as calculated from the set of 32 noisy projections. The pattern predicted previously is clearly evident. At the lowest exposure level nothing useful is to be concluded either from the projection or from the 2D
reconstruction; at 300 e/nm 2 it is possible to detect the presence of the largest peak in the projection, but as yet nothing in the reconstruction. With increasing exposure the projection is clarified. At about the point where the smallest (single cube) peaks can be identified in the projection, and the relative heights of all the peaks discerned, the stain distribution is just emerging from the noise in the reconstruction. The exposure level needed for this - about 3000 e/nm 2 - agrees well with that indicated in table 1 for the detection of the stain assumed here in 2 nm cubes. Further increase in the exposure level brings
88
B.E.H. Saxberg, W.O. Saxton /Quantum noise
the reassurance of better defmition in projection and reconstruction (showing both smoothed with a 2 nm sampling aperture, of course), but no fundamentally new information.
Supposing white uncorrelated noise in the original projection values, the discrete transform of a projection has a constant variance - say, o 2; if the maximum spatial frequency represented in the transform is kc, the total power is thus 2kco 2. After application of the Ik I ffdter function, it becomes
Conclusions kC
It is clear that, notwithstanding the equivalent statistical significance possible in 3D. reconstructions and 2D micrographs where a high electron dose can be tolerated, the 2D micrograph will in many practical cases be able to furnish partial information at higher resolution than the 3D reconstruction; this may or may not be useful, depending on the particular structural problem addressed. The total doses required for statistical significance in 3D reconstructions are disappointingly high particularly for frozen hydrated or glucose embedded specimens where the density resolution required is as low as a third of the density of water. This must inevitably cast suspicion on reconstructions made to high resolution from individual particles, even stained (e.g. ref. [7]). The use of averaging, either over lattices in the conventional manner, or over irregularly arranged units (e.g. ref. [8]), offers however a practical way of increasing effective electron doses 100-fold, and our results emphasise strongly the importance of such techniques.
2f
o2k 2 dk = 2k3co2/3 ,
0
and accordingly, the noise variance in the filtered projection value is increased with respect to the original variance by a factor of k2c/3. If n micrographs are recorded with a total electron dose per unit area p and scanned with an aperture of area d 2, the variance in the contrast of each micrograph, according to simple Poisson statistics, is n/pd 2. A factor mo --~ 2 X 10 -22 kg/nm 2 relates the integrated density p to the contrast c [9], for typical operating conditions, according to
f p =moc, so that the noise in the raw projection data has a variance nm~/pd2;on f'dtering this becomes nm2k2c/3pd 2, or nm~/12pal 4, since the cut-off spatial frequency kc will be approximately 1/2d. Adding n projection values for each reconstruction point increases the variance by a factor n; allowing for the f'mal factor rr/n applied to the reconstruction, we have a variance
Acknowledgement
Ap2 .~ (Tr/n)2n nm2o/(12pd 4) The support of the UK Science Research Council for this work is gratefully acknowledged. The manuscript was refereed by Dr. P. Schiske.
Appendix: Outline proof of reconstruction accuracy Using the f'dtered back-projection method, a slice may be reconstructed from n 1D projections by (a) applying a f'dtering process to each projection, in which its discrete Fourier transform values are multiplied by Ik 1, the modulus of the corresponding spatial frequencies; (b) for each point in the slice, adding together the filtered projection values at the points into which the point being reconstructed projects; and (c) multiplying the reconstruction by lr/n.
in the reconstruction, or an approximate rms error
AO ~- mo/pl /2d 2 , which is the required result. (In practice, the linear interpolation process used to determine filtered projection values at arbitrary points blurs the signal and the noise slightly in a space variant manner; the signal spread is insignificant beyond nearest neighbour points, and the rms noise level is reduced by about 32%.) It is not difficult to see that substantially the same result is achieved if the alternative method of direct synthesis of the 2D transform of the slice is adopted. The effect of missing projection data (blind regions in Fourier space, due to limits on the tilting angle possi-
B.E.H. Saxberg, W.O. Saxton / Quantum noise ble) are also simply assessed: the final noise variance is simply reduced in proportion to the lower number of projections used; discernibility of the specimen structure does not improve in spite of the lower noise variance, since the signal is reduced also, and distorted moreover in a way well characterised previously (e.g., ref. [4]).
References [ 1] R. HegerI and W. Hoppe, Z. Naturforsch. 3 l a (1976) 1717. [2] W. Baumeister and M. Hahn, Hoppe-Seyler's Z. Physik. Chem. 356 (1975) 1313.
89
[3] W. Hoppe, J. Gassmann, N. Hunsmann, H.J. Schramm and M. Sturm, Hoppe-Seyler's Z. Physik. Chem. 356 (1975) 1317. [4] W. Hoppe, in: Computer Processing of Electron Microscope Images, Ed. P.W. Hawkes (Springer, Berlin, 1980) ch. 4, p. 127. [5] W. Hoppe, Chem. Scripta 14 (1978-9) 227. [6] G. Kowalski, IEEE Trans. NS-24 (1977) 850. [7] W. Hoppe, J. Gassmann, N. Hunsmann, H.J. Schramm and M. Sturm, Hoppe-Seyler's Z. Physik. Chem. 355 (1974) 1483. [8] J. Frank, W. Goldfarb, D. Eisenberg and T.S. Baker, Ultramicroscopy 3 (1978) 283. [9] D.L. MiseUand I.DJ. Burdett, J. Microsc. 109 (1977) 171.