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| IS and Magnification | Handholding Vibration |

Image Stabilization and Magnification

This is a simple slightly hand waiving model of how the effectiveness of image stabilization is affected by magnification.

What is Image Stabilization?

The system uses two ultrasonic vibration gyro sensors to determine motion and a movable lens element to correct of the induced image shift. In current systems there are two sensors that provide correction in two dimensions, up/down and left/ right.

A three dimensional object has 6 degrees of freedom of movement; three displacement up/down, left/right, in/out, and three rotation; rotation in the plane of the sensor, rotation of the lens axis up/down, rotation of the lens axis left/right.

Descriptions of the IS system indicate that the two axis are correcting two of the rotation movements.

How Does Magnification Affect these?

This is tabulated in general terms below:

Developing a Model

The basic procedure is to relate acceptable blur to an acceptable level of rotational movement for degrees of freedom #5 and #6. From this and simple a simple model of handholding support and some simplifying assumptions this can be related to a degree of linear hand movement and hence to the blur introduced by all six degrees of freedom as a function of magnification where required.

Degrees of Freedom #5 and #6

Definition 1 : The amount of blur imparted to a photo is due to image displacement motion across the sensor and at infinity focus is due only to angular deflections caused by small movements of the photographers hand. This should be approximately proportional to exposure duration.

The amount of movement assuming a simple lens without IS is related to the angle of deflection Theta by:

Theta=arctan(Dsj/F/(1+M)/Gj),  for j=5&6 Eq 1

NB This assumes the lens is focused by overall linear extension. Internal focus lenses mitigated the 1/(1+M) term slightly. In the case of the 100mm macro lens the AoV at life size is about 0.5 stop larger than for an OLE lens. This is neglected in this analysis.

Assumption 1 :  tan(x)~=x for small x

then Theta~=Dsj/F/(1+M)/Gj  for j=5&6 Eq 2

Where Dsj for j=5&6 is the image displacement in the plane of the sensor, F is the focal length and M is the linear magnification defined as the ratio of the length of the image to length of the object and Gj is a scaling factor between degrees of freedom to account for the relative vibration modes and support (hand/arm) compliance (stiffness).

Rearranging Eq 2 for Dsj and normalizing this to an arbitrary circle of confusion size c gives:

Dsj~=Theta*F*(1+M)*Gj/c  for j=5&6 Eq 3

Definition 2 : When the IS is active it will provide say N stops of advantage. This is clearly reducing the size of Dsj by a factor of 2^N. 

This gives us normalized sensor side image displacement due #5 and #6, Ds5 and Ds6 respectively for IS off and IS on by making:- 

Assumption 2 : The average magnitude of hand vibrations are the same in all orthogonal directions. This is a best case simplifying assumption ignoring shutter button stabbing and mirror slap. This is reasonable if the shutter speed is above the critical speed for mirror slap effects; this is likely to be the case with medium to long telephoto lenses.

Eq 3A

Degrees of Freedom #1 and #2

The angular displacement of #5 and #6 above must be caused by small movements of the photographers hands relative to each other thus causing rotation. 

As far as the objective side of the lens is concerned what is important here is movement of the front principle node, the effective location of the camera's point of view. If we consider each hand is located d1 and d2 from the front principle plane (the front principle node is located where the optical axis and this plane intersects assuming the lens does not have tilt or shift facilities). Then we can refer the angular displacement theta to an objective displacement Do1 and Do2 so that:

tan(Theta)=Do1/d1=Do2/d2 Eq 4

Again applying Assumption 1  tan(x)~=x for small x

Theta~=Do1/d1=Do2/d2 Eq 5

In the worst case one had can be considered to be located directly (or very close to) under the front principle node, in this situation the camera and lens pivot on this had with all the rotational movement coming from one hand. This is quite a likely scenario as one hand is normally used to cup the lens whilst the other is on the grip operating the controls.

Assumption 3 : Although hand vibrations causing rotation are notionally differential and uncorrelated the asymmetry of hand position permits the direction #1 and #2 displacement vibration to be estimated from the #5 and #6 rational vibration.

Thus the simplification of the objective linear displacement Doj for j=1&2 being derivable from the angular displacement Theta and the worst case distance dp between the front principle plane and supporting hand.

Doj~=dp*Theta for j=1&2 Eq 6

Noting that on the sensor the Dsj will be obtained by multiplying by M, and normalizing this to the CoC and introducing the scaling factor so

Dsj~=dp*Theta*M*Gj/c for j=1&2 Eq 7

Note that we see immediately that at infinity focus degrees of freedom #1 and #2 have zero effect and only become significant at closer focus.

Assumption 4: Freedom directions #1 and #2 are not stabilized by the IS system.

This gives us normalized sensor side image displacement due #1 and #2, Ds1 and Ds2 respectively for IS off and IS on (the same)

Eq 7A

Degree of Freedom #3

Utilizing again Assumption 2 we can determine the objective side movement towards and away from the object is the same as Eq 6 above.

Do3~=dp*Theta Eq 8

Noting that macro depth of field is the image side conjugate of depth of focus we can simply define the CoC normalized blur effective displacement as:

Ds3~=dp*Theta/DOF=dp*Gj*Theta/(2*f*c*(M+P)/P/M²) for j=3, Eq 9

Where P is the lens Pupilary Magnification and f is the set f-stop.

Again this direction is not compensated for by the IS system so:

Eq 9A

Degree of Freedom #4

Once again asserting Assumption 2 we can determine the movement of the shutter button hand in the plane of the sensor is same as Eq 6 above. 

As the left hand is normally supporting the barrel of the lens the camera will tend to rotate by Thi rad around the optical axis with a moment defined by the distance between the optical centre and the grip end of the camera ds. So:

tan(Thi)=dp*Theta/ds Eq 10

Again applying Assumption 1 :  tan(x)~=x for small x

Thi~=dp*Theta/ds Eq 11

Now the maximum blur this introduces will be in the corner of the sensor, but zero in the centre of the sensor. We can calculate the blur at any offset L from the centre of the sensor (probably half the diagonal is a good value) to get the typical blur. Again using Assumption 1 and normalizing this to CoC gives:

Ds4~=dp*Theta/ds*L*Gj/c for j=4, Eq 12

Thus once again this quantity is the same when IS is on or off. Modeling with a short focal length lens (24mm) suggests Ds4 although insignificant with telephoto lenses becomes dominant with shorter lens. Considering that the camera and lens are probably more stable in this direction a scaling factor G4=0.1 is chosen.

Eq 12A

Combining all the Blur Effects

The pairs of orthogonal directions #1&#2 and #3&#4 will add as root of sum of squares as for vectors.

Assumption 5 : All blur sources are uncorrelated.

Some of these contributions will add together and others subtract in an unpredictable statistical way. So these need to be combined in an on average fashion. This means all the values of Ds root of sum of squares on average (central limit theorem). This yields an RMS or 1 sigma estimate of the combination of these independent sources of blur.

Thus:

Ds_all=sqrt(Ds1²+Ds2²+Ds3²+Ds4²+Ds5²+Ds6²) Eq 13-1

The range of sharpness for no camera movement induced blur assuming an ideal lens and sensor would range from a zero diameter CoC to a COC diameter of c at the edge of DOF, thus an average sharpness of c/2. 

If we consider an ideal test target then all we need to add is the lens and sensor optical resolution. This is of the order of MTF 50% of 50 lp/mm-70lp/mm by measurement. Approximately this is 1/c/60/2, calling this system limiting sharpness factor Ssys. Adding the Ds_all figure to this gives the average sharpness over the DOF with camera motion blur and a measure of sharpness S.

S=Ds_all+Ssys Eq 13-2

Ssys=1/c/Ro/2 13-3

Where Ro is the system maximum optical resolution measured at the MTF 50% point.

Value of Theta

We need to derive a reasonable value of Theta although we could work with an arbitrary one. We can do this by working equation 3 in reverse and assuming just degrees of freedom #5 and #6 are significant at infinity focus. We can relate this to an acceptable amount of blur, presumable smaller than the CoC diameter c by a factor k and also accounting for the system limited sharpness.

However the addition of the Ssys constant means the overall number of stops advantage Nt for the overall system is less than that applied to the #5&#6 degrees of freedom N. Thus we end up with a solution for Theta and N derived from simultaneous equations for the stabilized and unstabilized cases.

Soff=sqrt(2)*Ds56+Ssys=Son*2^Nt  Eq 14-1

Son=sqrt(2)*Ds56/2^N+Sys Eq 14-2

Thus solving Eq 14-1 for Ds56 using the known target Son and Nt yields

Ds56=(Son*2^Nt-Ssys)/sqrt(2) Eq 14-3

Now substituting into Eq 14-2 to get the required number of stops for the stabilized elements only:

2^N=sqrt(2)*Ds56/(Son-Ssys) Eq 14-4

Thus the angular deflection is

Theta=Ds56/F Eq 14-5

A value of Son is rather arbitrary, but a sensible value based on tests would be associated with the MTF 50% stabilized resolution Rs of about 45-50 lp/mm, using Eq 14-6 below.

Son=2*c*Rs Eq 14-6

Applying the Model

To apply the model we need to put numbers to the parameters.

a) The values for the overall system optical sharpness and stabilized sharpness  Ro=70 (lp/mm) Rs=45 (lp/mm)

b) CoC diameter c, for full frame APS-C c=0.019mm say.

c) ds the offset from the grip to the optical axis in the plane of the sensor. This can be eyeballed with a ruler giving ds=85mm (20D)

d) L the effective sensor size from the optical axis, for APS-C at frame edge L=11mm

e) dp the dominant grip point distance from the front principle plane. This is lens dependent. We can calculate the sensor to front principle plane distance easily from supplied minimum focus range and minimum focus magnification for a lens using the below formula:

Noting the guassian lens equation1/u+1/v=1/F (focal length asymmetry and pupilary magnification need not come into this) and by definition and similar triangles M=v/u, after substitution and rearranging the lens to subject distance u=F.(1+1/M), thus:

fps=R-F.(1+1/M) Eq 14

Where R is the minimum focus sensor to subject range, F the focal length and M the magnification at minimum focus.

For up to medium telephoto lenses fps > 0, i.e. the front principle plane is in front of the sensor and the rear grip point can be taken as dominant and:

dp=fps Eq 15

However, for long telefocal lenses the rear principle plane is behind the sensor and the dominant grip point becomes the left hand under the lens and so noting that the mount the flange distance for EF is 44mm and introducing lens mount to grip point distance dm: 

dp=dm+44-fps Eq 16

f) The pupilary magnification P can be eyeballed from the lens with a ruler (See definition of macro DOF) and the focal length F is known. Although both these quantities vary with M for internal focus lenses the mean values are assumed at all magnifications. The f-stop f used is only relevant for degree of freedom #3, the worst case is the lens wide open which is used, although it would normally be stopped down at least some.

g) The target stops of IS advantage Nt=2

h) The scaling factor vector 1 to 6 G=[1 1 1 0.1 1 1]

Results

We shall now proceed with some results.

When view the plots remember Dsj, Son, Soff are all measures of blur width relative to CoC, so sharpness is inversely proportional to that.

100mm f2.8 Macro

Assumed dp=110mm, P=0.5

The first two plots show the various types of degree of freedom contributions to blur and the total blur.

We can see from this that contributions from degree of freedom #3 is insignificant and #4 is relatively small. 

Due to the reduction in angle of view as the magnification increases the amount of blur increases, this is dominated by #5 and #6 in the unstabilized case with the #1 and #2 contributions becoming significant but not dominant.

In the stabilized case the #5&#6 contributions still increase with magnification but are much reduced. However the #1&#2 contributions actually dominate beyond M=0.25X as these are not stabilized.

Plot the two total blurs for clarity:

So the IS is helping even at high magnification although it's help is reduced and can be seen by comparing the degree of blur at each magnification with IS on and IS off. 

It is worth noting that with IS on the degree of blur degrades by life size to be worse than the infinity blur at IS off. From this point of view the IS could be considered to be completely ineffective at life size.

So, a 2 stop at infinity focus IS system is providing less than a stop at life size and significantly starts degrading above 0.1X.

Plotting the two levels of blur (IS on and IS off) compared to each of their infinity values shows how shutter speed would need to be increased from the safe infinity focus value.

180mm f3.5 Macro

Assumed dp=120mm, P=0.3

Presenting just the summary data.

300mm f4

Assumed dp=194mm,  P=0.3 Extra length of extension tubes is neglected.

Conclusions

From this model IS seems to provide an advantage at higher magnifications, although this is reduced from the infinity case.

The effect is more pronounced at shorter focal lengths, longer focal length lenses are more dominated by the angular vibration component even at high magnifications so the IS has more effect in combating vibration.

For macro lenses in the region of 100-200mm a 2 stop IS system provides about 1-1.4 stop of advantage at life size. Although the most recent IS systems are advertised as 3 of 4 stop, measurements of the 3 stop 24-105 f4L IS suggest that the real advantage for good sharpness criteria is more like 2 stops. 

Finally, it must be remembered that this is only a model with a number of simplifying assumptions and so may not accurately represent the situation.

Even is a 4 stop IS is used it is a situation of diminishing returns with magnification, the situation gets worse if the size of the degrees of freedom #1 and #2 is larger than estimated. This is illustrated in the below plot.

Last Updated 05/06/2008

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