Analysis 

Fitting Routine 

Use to convert b from inverse pixels to inverse mm 

units = 19.79331; 

Include the Sqrt[2 Pi]: 

 
f[x_] := Exp[- x^2/2]/Sqrt[2 Pi] ; 

 
g[x_] :=  f[units b x] (c0 HermiteH[0,units b x] + c1  
HermiteH[1, units b x] + c2 HermiteH[2, units b x] + c3  
HermiteH[3, units b x] + c4 HermiteH[4, units b x]) 

 
<<Calculus`FourierTransform` 

Hmmm... not using a := here seems to work better. 

fg[w_] = FourierTransform[g[x], x, w]; 

 
ax=D[(1/2 +(1/Pi) ArcTan[b x]),x]  

 
gf[w_]=FourierTransform[ax,x,w] 

 
<<Algebra`ReIm` 

 
a4[w_] := Re[fg[w]] 

 
b4[w_] := Im[fg[w]] 

 
mtf4[w_] := Sqrt[(a4[w]^2 + b4[w]^2) /( a4[0]^2+b4[0]^2)] 

 
mtf3[w_] := mtf4[w] /. c4 -> 0; 

 
mtf2[w_] := mtf4[w] /. {c4 -> 0, c3 -> 0}; 

 
mtf1[w_] := mtf4[w] /. {c4 -> 0, c3 -> 0, c2 -> 0}; 
 

mtf0[w_] := mtf4[w] /. {c4 -> 0, c3 -> 0, c2 -> 0, c1 -> 0}; 

****************** Now do the fits ******************** 

 
<<Statistics`NonlinearFit` 

On line cut out of image file, halfway down 

dat=ReadList["2080x.pgm",Byte]; 

Parms for this data: 

jump = 397 

start = 118 

stop = 150 

big = 225 

bot=15 

read out data averaging over 8 squares 

dta[m_]:=Flatten[Table[{i + .56790123 j + jump m,dat[[i+ jump m+100 j]] 
},{i,start, stop},{j,0,5}],1]; 

 
points = ListPlot[dta[4],Prolog->AbsolutePointSize[2.3],PlotLabel->  
FontForm[" Typical data averaged over 8 squares",{"Times-Bold",15}]]  
 

This file reads in with the name data 

<< dalsa.data 

 
ddat = ListPlot[data, PlotRange->{{0,20.1},{0,1.1}}, AxesOrigin->{0,0},  
Frame->True,Prolog->AbsolutePointSize[3]] 
 

Zeroth Order Fit 

 
********************** Apply PL Smiths approach *************************** 

Fit will give b in pixels, so dont use ``units'' here 

 
z[x_, l_] := b (x - (((stop - start)/2) + c + jump l + start)); 
 

******* Zeroth order fit to data  ************* 

 
 
fitfun0[x_,l_] := c0 Erf[z[x, l]]  + d; 

 
terms0 = NonlinearFit[dta[4],fitfun0[x,4],{b, c, d, c0}, x,\ 
	 InitialGuess->{.46, 3.55 , (big + bot)/2, (big - bot)/2} 
,MaxIterations->40]; 

 
plot0 = Plot[fitfun0[x, 4] /. terms0, {x, 1710, 1740},PlotLabel-> 
FontForm["0th order fit",{"Times-Bold",15}],PlotStyle->{Thickness[.0001]}] 
 

 
fit0=Show[points, plot0,PlotLabel->FontForm["0th order fit", 
{"Times-Bold",15}]] 
 

 
m0 = Plot[mtf0[x] /. terms0, {x, 0, 20}, \ 
	PlotRange->{{0,20.1}, {0,1.1}}, Frame->True, AxesOrigin->{0,0}, 
PlotLabel->FontForm["MTF for the 0th order fit",{"Times-Bold",15}]]   

 
mfit0=Show[ddat, m0, PlotRange->{{0,20.1}, {0,1.1}}, Frame->True,  
AxesOrigin->{0,0},PlotLabel->FontForm["MTF for the 0th order", 
{"Times-Bold",15}]]  
 

 
Show[GraphicsArray[{{mfit0},{fit0}}]] 
 

First Order Fit 

********** First order ******************** 

 
fitfun1[x_,l_] := c0 Erf[z[x, l]] - \ 
	(Exp[-(z[x,l]^2)/2]/Sqrt[2 Pi]) (c1 HermiteH[0, z[x,l]]) + d; 
 

 
terms1 = NonlinearFit[dta[4],fitfun1[x,4],{b, c, d, c0, c1}, x,\ 
	 InitialGuess->{.464911, 3.42083 , (big + bot)/2, (big - bot)/2, 10}]; 
 

 
plot1 = Plot[fitfun1[x, 4] /. terms1, {x, 1710, 1740},PlotLabel-> 
FontForm["Fit function for the 1st order",{"Times-Bold",15}],PlotStyle-> 
{Thickness[.0001]}] 
 

 
fit1=Show[points, plot1,PlotLabel->FontForm[ "1st order fit", 
{"Times-Bold",15}]] 
 

 
m1 = Plot[Evaluate[mtf1[x]] /. terms1, {x, 0, 20}, PlotRange-> 
{{0,20.1},{0,1.1}},PlotLabel->FontForm["MTF for the 1st order fit", 
{"Times-Bold",15}]]  
 

 
mfit1=Show[ddat, m1, PlotRange->{{0,20.1}, {0,1.1}},  
	Frame->True,AxesOrigin->{0,0},PlotLabel->FontForm 
["MTF for 1st order",{"Times-Bold",15}]] 
 

 
Show[GraphicsArray[{{mfit1},{fit1}}]] 
 

Second Order Fit 

************* Second order ****************** 

 
fitfun2[x_,l_] := c0 Erf[z[x, l]] - \ 
	 (Exp[-(z[x,l]^2)/2]/Sqrt[2 Pi]) (c1 + c2 HermiteH[1,z[x,l]])  
          + d; 
 

 
terms2=NonlinearFit[dta[4],fitfun2[x,4],{b,c,d,c0,c1,c2},x,\ 
InitialGuess->{.464229,3.26988,(big+bot)/2,(big-bot)/2,19.1599,10}] 
 

 
plot2 = Plot[fitfun2[x, 4] /. terms2, {x, 1710, 1740},PlotStyle->{Thickness[.0001]}]  
 

 
fit2=Show[points, plot2,PlotLabel->FontForm["2nd order fit",{"Times-Bold",15}]] 
 

 
m2 = Plot[mtf2[x] /. terms2, {x, 0, 20}, PlotRange->{{0,20.1},{0,1.1}}]  
 

 
mfit2=Show[m2, ddat, PlotRange->{{0,20.1}, {0,1.1}},  
	Frame->True, AxesOrigin->{0,0},PlotLabel->FontForm 
["MTF for the 2nd order",{"Times-Bold",15}]] 
 

 
Show[GraphicsArray[{{mfit2},{fit2}}]]//PSPrint 
 

Third Order Fit 

************ Third order ************* 

fitfun3[x_,l_] := c0 Erf[z[x, l]] -  
		(Exp[-(z[x,l]^2)/2]/Sqrt[2 Pi]) *( 
		c1 HermiteH[0, z[x, l]]  
	 	+ c2 HermiteH[1, z[x, l]]  
		+ c3 HermiteH[2, z[x, l]] ) + d; 
 

terms3 = NonlinearFit[dta[4],fitfun3[x,4],{b, c, d, c0, c1, c2, c3}, 
 x, InitialGuess->{.583477, 3.18453 , (big + bot)/2, (big - bot)/2,  
24.2164,21.8726, 5}]>>>TERMS; 

plot3 = Plot[fitfun3[x, 4] /. terms3, {x, 1710, 1740},PlotStyle-> 
{Thickness[.0001]}] 

 
fit3=Show[points, plot3,PlotLabel->FontForm["3rd order fit", 
{"Times-Bold",15}]] 

 
m3 = Plot[mtf3[x] /. terms3, {x, 0, 20}, PlotRange->{{0,20.1}, 
{0,1.1}}]  

 
mfit3=Show[m3, ddat, PlotRange->{{0,20.1}, {0,1.1}},  
	Frame->True, AxesOrigin->{0,0},PlotLabel->FontForm 
["MTF for the 3rd order",{"Times-Bold",15}]] 

 
Show[GraphicsArray[{{mfit3},{fit3}}]] 

Fourth Order Fit 
 

************ Fourth order ********************** 

 
 
fitfun4[x_,l_] := c0 Erf[z[x, l]] -  
		(Exp[-(z[x,l]^2)/2]/Sqrt[2 Pi]) *( 
		c1 HermiteH[0, z[x, l]]  
		+ c2 HermiteH[1, z[x, l]]  
		+ c3 HermiteH[2, z[x, l]]  
		+ c4 HermiteH[3, z[x, l]]  ) + d; 
 

terms4 = NonlinearFit[dta[4],fitfun4[x,4],{b, c, d, c0, c1, c2, c3, c4}, 
 x,InitialGuess->{.596284, 3.4371 , (big + bot)/2, (big - bot)/2, 
 -7.97937, 23.6496, 2.09462, 5},MaxIterations->100]>>>TERMS; 
 

plot4 = Plot[fitfun4[x, 4] /. terms4, {x, 1710, 1740}, 
PlotStyle->{Thickness[.0001]}] 

 
fit4=Show[points, plot4,PlotLabel->FontForm[" 4th order fit", 
{"Times-Bold",15}]]  

 
m4=Plot[mtf4[x] /. terms4, {x, 0, 20}, PlotRange->{{0,20.1},{0,1.1}}]  

 
mfit4=Show[m4,ddat,PlotRange->{{0,20.1},{0,1.1}},Frame->True, 
PlotLabel->FontForm["MTF for the 4th order",{"Times-Bold",15}]] 
 

Show[GraphicsArray[{{mfit4},{fit4}}]] 

 
Show[GraphicsArray[{{mfit0,fit0},{mfit1,fit1},{mfit2,fit2}, 
{mfit3,fit3},{mfit4,fit4}}]]