# Methods of Calculating Relative Background Blur

Started Nov 28, 2012 | Discussions thread
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Methods of Calculating Relative Background Blur
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The subject of this thread relates to mathematical formulas for the purpose of estimating the amount of relative background blur in images (recorded using any camera format). If the use of mathematics happens not to be your personal interest, or if you may feel that the use of mathematics is somehow unrelated to photography, then there is absolutely no relevant need for you to post to this thread.

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Derived a mathematical identity to describe the (diagonal) physical dimension in object-space that an image-frame represents (at the location of background subject-matter of interest):

S = ( (Df) (H) / (L) ) x ( 1 + (M) / (P) )

where:

S is the (diagonal) size of the physical dimension in object-space that image-frame represents;

Df is the Camera (front nodal-plane) to Background Subject Matter Distance;

H is the diagonal dimension of the image-sensor's active-area;

L is the Focal Length (when focused at infinity);

P is the Pupillary Magnification;

M is the Image Magnification.

Note: The Image Magnification (M) is itself equal to the ratio of the diagonal dimension of the image-sensor (H) divided by the unknown variable being solved for (S). However, in cases (such as the specific intended application of this formula) where that ratio is very small, I am ignoring that additional complexity. As M is very small, the effect value of P is also very small, and it seems reasonable (in this application) to consider ( 1 + (M) / (P) ) as simply being equal to unity, yielding:

S ~ (Df) (H) / (L)

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Solved for a mathematical identity describing the diameter of the "Disk of Confusion" in Merklinger's "Object Field" approach, representing the smallest resolvable diameter (in object-space). That smallest diameter can (it seems to me) be thought of as a "blur-disk" diameter:

B = ( (L) / (F) ) x ( (Df) / (D) - 1 )

where:

B is the ("blur-disk") diameter of Merkilnger's "Disk of Confusion";

L is the Focal Length (when focused at infinity);

F is the F-Number;

Df is the Camera (front nodal-plane) to Background Subject Matter Distance;

D is the Camera (front nodal-plane) to Subject (plane-of focus) Distance.

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In order to evaluate the size of the "blur-disk" diameter (B) as a percentage proportion of the (diagonal) physical dimension in object-space that an image-frame represents at the location of background subject-matter of interest (S), the ratio of the two identities ( B / S ) is taken:

BP ~ ( (100) / ( (F) (H) ) ) x ( ( (L) / (D) )^(2) ) x ( Df - D )

where:

BP is the percentage of the image-frame diagonal that the blur-disk represents;

F is F-Number;

H is the diagonal dimension of the image-sensor's active-area;

L is the Focal Length (when focused at infinity);

D is the Camera (front nodal-plane) to Subject (plane-of focus) Distance;

Df is the Camera (front nodal-plane) to Background Subject Matter Distance.

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What I like about the above described approach is that it is derived entirely from object-space quantities. As a result (and unlike approaches that evaluate Depth of Field and Hyperfocal Distance), it is not dependent on the accurate determination of a Circle of Confusion diameter (which is a function of image-sensor size, print/display size, and the viewer's distance from the print/display).

It is certainly possible that I may have made conceptual or mathematical error(s) in my derivation of the above method of quantifying the effects of background-blur in an image. Would welcome any coherent and meaningful thoughts or suggestions from others relating directly to the specific subject of this thread as it is described in the text of this post above.

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