Curve fitting

[Photo Engineer]

Interesting, but as a project engineer for that film, I would still be looking at the raw data. Those bumps may indicate any one of several problems. And, it will distort the print quality if real!

PE

 

[alanrockwood]

Here is a list of the residual between the fit and the data for the four parameter fit I showed earlier.

RESIDUALS
-0.00814735474025491
0.00443299720689611
-0.00495683903640237
0.00927281290335008
0.0186755876170858
-0.0117224326914512
-0.0187317246975552
-0.00515886458521414
-0.00172260138239366
0.0141963929648516
0.0225380760765277
-0.00312415700747659
0.00257906197888336
0.0359646349921837
-0.0544397383232357
-0.0420528610374982
-0.00949162093982636
0.0151039044630277
-0.0271638726211845
0.00310327956330925
-0.00466891187437768
0.00313234397646411
0.0109405486984975
0.057643304979889
0.0535096713490784

The maximum deviation is about 1/5 stop, which is not a lot.

For the six parameter model the residuals are as follows.

OTHER_RESIDUAL
-0.00865538809797933
0.00429587801755249
-0.00471582502816537
0.00982849469006986
0.0192905480795434
-0.0113471586589236
-0.0188551747281989
-0.00592387433963665
-0.00290941838266201
0.0128966866227697
0.0215282306281418
-0.00353167589170789
0.00302865642946382
0.0375730821164881
-0.0516656584672155
-0.0380676412445947
-0.00432637463678009
0.021395202826574
-0.0202538528298055
0.00981449003702384
-0.000141232715030082
0.00223204673597954
-0.00124480933729565
0.0172293681725331
-0.0147493797907858

The maximum residual between the fit and the data points approximately 1/8 stop.

A fifth order polynomial gave a maximum residual of about 1/7 stop, which is close but not quite as close as the six parameter equation. The visual comparison between the fifth order polynomial fit and the six parameter fit is pretty comparable (data not shown).

As a point of comparison, a fifth order polynomial has six degrees of freedom (six adjustable parameters), which is the same number of degrees of freedom as my six parameter equation.

Based on this, it looks like there is a not a lot to choose between the polynomial fit and the six parameter equation. Nevertheless, I tend to prefer my equation because it is constrained to have the right qualitative shape, whereas the polynomial is not.

Spline functions are, of course, another story altogether.

 

[alanrockwood]

Interesting, but as a project engineer for that film, I would still be looking at the raw data. Those bumps may indicate any one of several problems. And, it will distort the print quality if real!

PE

No argument from me that it would be a good idea to investigate the bumps and so forth.

By the way, the film is arista edu ultra (re-branded fomapan 200), and the developer is LMax, which is supposed to be a clone of T-Max developer. The dilution was 1+5.5. The time was 3 min. The temperature: 74 F. Agitation: continuous rotary.

The x-axis is "relative exposure", which is calculated from minus log of the meter setting, so it is not an absolute exposure level. The object was a blank wall illuminated with an even illuminating source. These details of the exposure are not actually important for evaluating the fit, but I give it for completeness.

Anyway, I would love for other people to list their other density results so I (and others) can practice fitting equations to it.

By the way, why bother? Because if you are evaluating parameters from the curve a fit is smoother than the raw data and therefore more likely to give an accurate answer than taking the measurements directly from the raw points.

 

[Bradmhughes]

Melbourne Event Photography

Curve fitting[1][2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,[3] possibly subject to constraints.[4][5] Curve fitting can involve either interpolation,[6][7] where an exact fit to the data is required, or smoothing,[8][9] in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis,[10][11] which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization,[12][13] to infer values of a function where no data are available,[14] and to summarize the relationships among two or more variables.[15] Extrapolation refers to the use of a fitted curve beyond the range of the observed data,[16] and is subject to a degree of uncertainty[17] since it may reflect the method used to construct the curve as much as it reflects the observed data.




.........................................
Melbourne Event Photography

 

[RalphLambrecht]

I use a software,called Deltagraph for film-curve fitting.It does an excellent job and even spits out the best fitting equation.

 

[RalphLambrecht]

I use DeltaGraph 7 from http://www.redrocksw.com/ which comes in both Mac and PC versions.

I do too.It seemed to be the most flexible to me and allows for user-defined equations but I never found the one-fits-all equation with it either.

 

[ic-racer]

Again, to emphasize utility (or perhaps lack there of). Whatever method one uses for a curve fit, the equation is only of value if solving it for an unknown film density at any input luminance is easier than just looking at the H&D graph!

I have my own solution, I use Excel to automatically crawl through the curve and give me speed points from the curve via interpolation. This automatic 'point-and-click' solution works very well.

(there was a url link here which no longer exists)

 

[alanrockwood]

I use PsiPlot, which has some powerful curve fitting capabilities, including the ability to fit data to equations defined by the user.

I have even used PsiPlot as the primary computational tool in publishing several scientific papers.