How to speed up algorithm (Binarization, integral image) - delphi

I wrote this procedure based on integral image algorithm described at this url
http://people.scs.carleton.ca/~roth/iit-publications-iti/docs/gerh-50002.pdf
Is there any way to do this code faster?
Pointers are much faster as dynamic arrays?
procedure TForm1.bBinarizationClick(Sender: TObject);
var
iX1, iY1,
iX2, iY2,
ii, jj,
s, s2,
iSum, iCount, index,
iHeight, iWidth : Integer;
iSize: Integer;
row : ^TRGBTriple;
black : TRGBTriple;
aIntegralIm: array of Integer;
aGrays : array of Byte;
startTime : Cardinal;
begin
iWidth := bBitmap.Width;
iHeight := bBitmap.Height;
iSize := iWidth * iHeight;
SetLength(aGrays, iSize);
SetLength(aIntegralIm, iSize);
black.rgbtRed := (clBlack and $0000FF);
black.rgbtGreen := (clBlack and $00FF00) shr 8;
black.rgbtBlue := (clBlack and $FF0000) shr 16;
bBitmap2.Canvas.Brush.Color := clWhite;
bBitmap2.Canvas.FillRect(Rect(0, 0, bBitmap2.Width, bBitmap2.Height));
s := Round(iWidth / TrackBar2.Position);
s2 := Round(s / 2);
startTime := GetTickCount();
index := 0;
for ii := 0 to iHeight - 1 do begin
row := bBitmap.ScanLine[ii];
for jj := 0 to iWidth - 1 do begin
aGrays[index] := ((row.rgbtRed * 77 + row.rgbtGreen * 150 + row.rgbtBlue * 29) shr 8);
inc(index);
inc(row);
end;
end;
for ii := 0 to iWidth - 1 do begin
iSum := 0;
for jj := 0 to iHeight - 1 do begin
index := jj*iWidth+ii;
iSum := iSum + aGrays[index];
if ii = 0 then aIntegralIm[index] := iSum
else aIntegralIm[index] := aIntegralIm[index - 1] + iSum;
end;
end;
for jj := 0 to iHeight - 1 do begin
row := bBitmap2.ScanLine[jj];
for ii := 0 to iWidth - 1 do begin
index := jj*iWidth+ii;
iX1 := ii-s2;
iX2 := ii+s2;
iY1 := jj-s2;
iY2 := jj+s2;
if (iX1 < 0) then iX1 := 0;
if (iX2 >= iWidth) then iX2 := iWidth-1;
if (iY1 < 0) then iY1 := 0;
if (iY2 >= iHeight) then iY2 := iHeight-1;
iCount := (iX2 - iX1) * (iY2 - iY1);
iSum := aIntegralIm[iY2*iWidth+iX2]
- aIntegralIm[iY1*iWidth+iX2]
- aIntegralIm[iY2*iWidth+iX1]
+ aIntegralIm[iY1*iWidth+iX1];
if (aGrays[index] * iCount) < (iSum * (100 - TrackBar1.Position) / 100) then row^ := black;
inc(row);
end;
end;
ePath.Text := 'Time: ' + inttostr(GetTickCount() - startTime) + ' ms';
imgOryginal.Picture.Bitmap.Assign(bBitmap2);
end;

You can at least do a few simple things:
precalculate (100 - TrackBar1.Position) into a variable
Instead of division: / 100 use * 100 on the other side. You might not need any floating point values.
Use lookup tables for the following (care to explain the identation btw?):
Code:
if (iX1 < 0) then iX1 := 0;
if (iX2 >= iWidth) then iX2 := iWidth-1;
if (iY1 < 0) then iY1 := 0;
if (iY2 >= iHeight) then iY2 := iHeight-1;
Try to keep the index and icremnet, decrement istead of multiplication: index := jj*iWidth+ii;

My guess is that the second loop is the slow bit.
The trick would be to avoid to recalculate everything in the second loop all the time
If S is constant (relative to the loop I mean, not absolute)
iy1,iy2 only change with the main(jj) loop and so do iy1*width (and iy2*width).
Precalculate them, or optimize them away in the same way you do with row. (precalculate once per line, increment inbetween)
change the ii loop into three loops:
the first bit where ix1=0
the second where ix1=ii-s ix2=ii+s;
the third where ix1=ii-s and ix2=iwidth-1
this removes a lot of checks out of the loops, to be done only once.
make a dedicated loop for the condition if (aGrays[index] * iCount) < (iSum * (100 - TrackBar1.Position) / 100) then row^ := black; so that it isn't evaluated for each pixel, since you can precalculate the area's where this happens ?
introduce pointers into the gray calculating loop so that you don't have to recalculate the index each pixel (but e.g. only for the row loop, incrementing a ptr per pixel)
If you are hardy, you can also precalculate the jump between lines. Keep in mind that abs(scanline[j]-scanline[i])-width is a metric for the number of alignment bytes per row.
Even more advanced is optimizing for cache effects on the level of your algorithm. See
rotating bitmaps. In code
to get an idea how this works. Some pointer tricks are demonstrated there too (but only for 8-bit elements)

I would first use a profiler to find out the CPU usage repartition, to figure out the smallest part(s) of code that would benefit the most from optimisation.
Then I would adapt the effort according to the results. If some code represents 90% of the CPU load and is executed zillions of times, even extreme measures (recoding a few sequences using inline assembly language) might make sense.

Use the excellent and free SamplingProfiler to find out the bottleneck in your code. Then optimize and run the profiler again to find the next bottleneck. This approach is much better than guessing what's need to be optimized because even experts are often wrong about that.

Related

Scan Line Out of Range Error for Bitmap. TJanDrawImage Component for a Paint-like Program

I am using the free JansDraw Components and when the executable runs, it throws AV error. I could locate the error to specifically the loop block of the code (not the initial block of assignment statements).
procedure TjanDrawImage.colorcircle(var bm:TBitmap;center:tpoint;radius,mode:integer);
var p,p0,p1:pbytearray;
dx,x,y,w,h,i,j,sum,c:integer;
cm,tm:tbitmap;
Rs,Rd:trect;
begin
x:=center.x;
y:=center.y;
w:=bm.width;
h:=bm.height;
cm:=tbitmap.create;
cm.width:=2*radius;
cm.height:=2*radius;
cm.PixelFormat :=FPixelFormat;
tm:=tbitmap.create;
tm.width:=2*radius;
tm.height:=2*radius;
tm.PixelFormat :=FPixelFormat;
tm.canvas.brush.color:=clblack;
tm.canvas.Ellipse (0,0,tm.width-1,tm.height-1);
tm.transparent:=true;
tm.TransparentColor :=clblack;
Rd:=rect(0,0,cm.width,cm.height);
Rs:=rect(x-radius,y-radius,x+radius,y+radius);
cm.canvas.CopyRect (Rd,bm.canvas,RS);
for j:=0 to cm.height-1 do begin
p:=cm.scanline[j];
if j>0 then p0:=cm.scanline[j-1];
if j<(h-1) then p1:=cm.scanline[j+1];
for i:=0 to cm.width-1 do begin
case mode of
0: //blue
begin
p[i*3+1]:=0;
p[i*3+2]:=0;
end;
1: //green
begin
p[i*3]:=0;
p[i*3+2]:=0;
end;
2: //red
begin
p[i*3]:=0;
p[i*3+1]:=0;
end;
3: //not blue
begin
p[i*3]:=0;
end;
4: //not green
begin
p[i*3+1]:=0;
end;
5: //not red
begin
p[i*3+2]:=0;
end;
6: //half blue
begin
p[i*3]:=p[i*3]*9 div 10;
end;
7: //half green
begin
p[i*3+1]:=p[i*3+1]*9 div 10;
end;
8: //half red
begin
p[i*3+2]:=p[i*3+2]*9 div 10;
end;
9:// darker
begin
p[i*3]:=round(p[i*3]*10 /11);
p[i*3+1]:=round(p[i*3+1]*10 / 11);
p[i*3+2]:=round(p[i*3+2]*10 /11);
end;
10:// lighter
begin
p[i*3]:=round(p[i*3]*11 / 10);
p[i*3+1]:=round(p[i*3+1]*11 / 10);
p[i*3+2]:=round(p[i*3+2]*11 / 10);
end;
11:// gray
begin
sum:=round((p[i*3]+p[i*3+1]+p[i*3+2])/ 3);
p[i*3]:=sum;
p[i*3+1]:=sum;
p[i*3+2]:=sum;
end;
12:// mix
begin
c:=p[i*3];
p[i*3]:=p[i*3+1];
p[i*3+1]:=p[i*3+2];
p[i*3+2]:=c;
end;
13://smooth
begin
if ((j>0) and (j<(h-1))and (i>0)and (i<(w-1))) then begin
p[i*3]:=round((p[(i-1)*3]+p[(i+1)*3]+p0[i*3]+p1[i*3]) /4);
p[i*3+1]:=round((p[(i-1)*3+1]+p[(i+1)*3+1]+p0[i*3+1]+p1[i*3+1]) /4);
p[i*3+2]:=round((p[(i-1)*3+2]+p[(i+1)*3+2]+p0[i*3+2]+p1[i*3+2]) / 4);
end;
end;
end;
end;
end;
cm.canvas.Draw (0,0,tm);
cm.transparent:=true;
cm.transparentcolor:=clwhite;
bm.Canvas.draw(x-radius,y-radius,cm);
cm.free;
tm.free;
end;
A linked question which is helpful is this - implementing scan line of bitmap corectly. It suggests to cast the pointers to NativeInt. The OP changed his code after answers, making it difficult to correlate old code with new code. I understand that my problem is due to some hard coded sequential access of pointers but I am really beginner to make sense of scan line or pointers. If you help me port this, these components will continue to be useful to everyone.
update after comment from #Renate Schaaf:
all the brush modes of the janDrawImage are working now, except for the below one. I was expecting a bigger problem but that didn't turn out to be the case. So modified the title of the question. #Renate Schaaf Can you please help fix the below one too. I tried but failed.
procedure TjanDrawImage.rimple(src,dst:tbitmap;amount:extended);
var ca,sa,a,dx,dy,r,rx,ry,sr,fr:extended;
w,h,x,y,cx,cy,i,j,c,ci:NativeInt;
p1,p2:pbytearray;
begin
w:=src.width;
h:=src.height;
cx:=w div 2;
cy:=h div 2;
if amount<1 then amount:=1;
fr:=cx/amount;
for y:=0 to h-1 do begin
p1:=src.ScanLine[y];
for x:=0 to w-1 do begin
dx:=x-cx;dy:=-(y-cx);
r:=sqrt(sqr(dx)+sqr(dy));
sr:=fr*sin(r/cx*amount*2*pi);
if (r+sr<cx) and (r+sr>0) then begin
a:=arctan2(dy,dx);
sincos(a,sa,ca);
i:=cx+round((r+sr)*ca);
j:=cy+round((r+sr)*sa);
p2:=dst.scanline[j];
c:=x*3;ci:=i*3;
p2[ci]:=p1[c];
p2[ci+1]:=p1[c+1];
p2[ci+2]:=p1[c+2];
end;
end;
end;
end;
When you do scanline operations, you always need to make sure that your pixel location is within the boundaries of your bitmap, particularly if you make geometric transformations.
So, in the last example you must clamp j to [0,h-1] and i to [0,w-1] using max(min(..)).
Also, you should set the size and pixelformat of src and dst to the same at the beginning.
I didn't really bother to find out what this ripple is supposed to do, but when I run an example it doesn't look like it's doing whatever it does right. For geometric transformations you need to work backwards, running through the pixels of the destination and figure out which pixel of the source needs to go there. Otherwise you end up with a destination that has holes, like here.
Edit:
Since I'm stuck with my project: I think this is the routine you really want to use. Note that I just switched the roles of src and dst and corrected some errors. It now adds a water ripple effect to the bitmap.
procedure rimple(src, dst: TBitmap; amount: extended);
var
ca, sa, a, dx, dy, r, sr, fr: extended;
w, h, x, y, cx, cy, i, j, c, ci: NativeInt;
p1, p2: pbytearray;
bits: integer;
begin
Assert(src.PixelFormat in [pf24bit, pf32bit],
'Device independent bitmap needed');
dst.PixelFormat := src.PixelFormat;
bits := 3;
if src.PixelFormat = pf32bit then
bits := 4;
w := src.width;
h := src.height;
dst.SetSize(w, h);
cx := w div 2;
cy := h div 2;
// in case somebody enters a negative amount
if abs(amount) < 1 then
amount := 1;
fr := cx / amount;
for y := 0 to h - 1 do
begin
// switched src and dst
p1 := dst.scanline[y]; // src.scanline[y];
for x := 0 to w - 1 do
begin
dx := x - cx;
// Corrected from dy:=-(y-cx)
dy := (y - cy);
r := sqrt(sqr(dx) + sqr(dy));
sr := fr * sin(r / cx * amount * 2 * pi);
// Omitted the following check
// if (r + sr < cx) and (r + sr > 0) then
begin
a := arctan2(dy, dx);
sincos(a, sa, ca);
i := max(min(cx + round((r + sr) * ca), w - 1), 0);
j := max(min(cy + round((r + sr) * sa), h - 1), 0);
// switched src and dst
p2 := src.scanline[j];
c := x * bits;
ci := i * bits;
p1[c] := p2[ci];
p1[c + 1] := p2[ci + 1];
p1[c + 2] := p2[ci + 2];
end;
end;
end;
end;

Are points on max. two lines?

I have a time problem with my program. Given a set of points, it has to say whether all of those points are lying on two different lines.
I wrote code, which has points in array and removes one by one and try calculate it's vector.
But this solution is slow, because it must control all cases of lines. On input with 10,000 points it takes over 10 seconds.
Can someone please tell me if, is here better solution for this problem?
I made this code in Pascal:
uses
math;
type
TPoint = record
x, y: real;
end;
TList = array of TPoint;
function xround(value: real; places: integer): real;
var
muldiv: real;
begin
muldiv := power(10, places);
xround := round(value * muldiv) / muldiv;
end;
function samevec(A, B, C: TPoint): boolean;
var
bx, by: real; // vec A -> B
cx, cy: real; // vec A -> C
lb, lc: real; // len AB, len AC
begin
bx := B.x - A.x;
by := B.y - A.y;
cx := C.x - A.x;
cy := C.y - A.y;
lb := sqrt(bx * bx + by * by);
lc := sqrt(cx * cx + cy * cy);
// normalize
bx := xround(bx / lb, 3);
by := xround(by / lb, 3);
cx := xround(cx / lc, 3);
cy := xround(cy / lc, 3);
samevec := ((bx = cx) and (by = cy)) or ((bx = -cx) and (by = -cy));
end;
function remove(var list: TList; idx: integer): TPoint;
var
i: integer;
begin
remove.x := 0;
remove.y := 0;
if idx < length(list) then
begin
remove := list[idx];
for i := idx to length(list) - 2 do
list[i] := list[i + 1];
setlength(list, length(list) - 1);
end;
end;
var
i, j, lines: integer;
list, work: TList;
A, B: TPoint;
begin
while not eof(input) do
begin
setlength(list, length(list) + 1);
with list[length(list) - 1] do
readln(x, y);
end;
if length(list) < 3 then
begin
writeln('ne');
exit;
end;
lines := 0;
for i := 1 to length(list) - 1 do
begin
work := copy(list, 0, length(list));
lines := 1;
B := remove(work, i);
A := remove(work, 0);
for j := length(work) - 1 downto 0 do
if samevec(A, B, work[j]) then
remove(work, j);
if length(work) = 0 then
break;
lines := 2;
A := remove(work, 0);
B := remove(work, 0);
for j := length(work) - 1 downto 0 do
if samevec(A, B, work[j]) then
remove(work, j);
if length(work) = 0 then
break;
lines := 3; // or more
end;
if lines = 2 then
writeln('YES')
else
writeln('NO');
end.
Thanks, Ferko
APPENDED:
program line;
{$APPTYPE CONSOLE}
uses
math,
sysutils;
type point=record
x,y:longint;
end;
label x;
var
Points,otherPoints:array[0..200001] of point;
n,n2,i,j,k,i1,i2:longint;
function sameLine(A,B,C:point):boolean;
var
ABx,ACx,ABy,ACy,k:longint;
begin
ABx:=B.X-A.X;
ACx:=C.X-A.X;
ABy:=B.Y-A.Y;
ACy:=C.Y-A.Y;
k:=ABx*ACy-ABy*ACx;
if (k=0) then sameLine:=true
else sameLine:=false;
end;
begin
readln(n);
if (n<=4) then begin
writeln('YES');
halt;
end;
for i:=1 to n do readln(Points[i].x,Points[i].y);
for i:=1 to 5 do for j:=i+1 to 5 do for k:=j+1 to 5 do if not (sameLine(Points[i],Points[j],Points[k])) then begin
i1:=i;
i2:=j;
goto x;
end;
writeln('NO');
halt;
x:
n2:=0;
for i:=1 to n do begin
if ((i=i1) or (i=i2)) then continue;
if not sameLine(Points[i1],Points[i2],Points[i]) then begin
inc(n2,1);
otherPoints[n2]:=Points[i];
end;
end;
if (n2<=2) then begin
writeln('YES');
halt;
end;
for i:=3 to n2 do begin
if not sameLine(otherPoints[1],otherPoints[2],otherPoints[i]) then begin
writeln('NO');
halt;
end;
end;
writeln('YES');
end.
Three points A, B and C lie on the same straight line, if vectors AB and AC are collinear or anti-collinear. We can check for collinearity using cross product of vectors - it should be zero.
#LU RD already described this approach is comment, but author probably missed it.
Note that method doesn't suffer from division by zero - there is no division at all.
ABx := B.X - A.X;
ACx := C.X - A.X;
ABy := B.Y - A.Y;
ACy := C.Y - A.Y;
Cross := ABx * ACy - ABy * ACx;
// for integer coordinates
if Cross = 0 then
A,B,C are collinear
If coordinates are float, one must consider some tolerance level. Variants:
//better if available:
if Math.IsZero(Cross)
if Math.SameValue(Cross, 0)
//otherwise
if Abs(Cross) <= SomeEpsilonValue
If coordinate range is very large, numerical error might be significant, so it is worth to normalize tolerance by squared magnitude of coordinate differences:
if Math.IsZero(Cross / Max(ABx * ABx + ABy * ABy, ACx * ACx + ACy * ACy))
I guess the answer to the Q should be devided into two parts.
I. How to know that the given three points belong to the same line?
The answer to this part of the Q was given by #Lurd and then expanded by Mbo.
Let us name their solution function BelongToOneLine(Pnts: array [1..3] of TPoint): boolean; We can consider this part solved.
II. How to decrease time consumption of the algorithm or in other words: how to avoid calling BelongToOneLilne with every possible combination of points as parameters?
Here is the algorithm.
We select 5 distinct points from the task set. 5 is enough (check combination possibilities).
We find the answer to the question if there are at least three points from given five that belong to a single line.
if No - then we do not need to iterate the remaining poins - the answer is that we require more then two lines.
if Yes - (say poins Pt1, Pt2 and Pt3 belong to the same line and Pt4 and Pt5 - don't).
Then we store the points that do not belong to the line Pt1-Pt2-Pt3 from the group-of-five in a distinct array of "outsider" points (or store their indexes in the main array). It may have Length = 0 by the end of this step. This will not affect the rest of the algo.
We get the boolean result of the function BelongToOneLine([Pt1, Pt2, Pt[i]]).
if Yes - we skip the point - it belongs to the line Pt1-Pt2-Pt3.
if No - we store this point in the "outsiders" array.
We watch the length of the OutsidersArray.
if it is <= 2 then the answer to the whole Q is Yes, they do belong to 2 or less lines.
if >2 then we iterate the function BelongToOneLine([OutsiderPt1, OutsiderPt2, OutsiderPt[i]]) until High(OutsiderArray) or until when OutsiderPt[i] does not belong to OutsiderPt1-OutsiderPt2 line. All points of OutsiderArray must belong to the same line otherwise the answer to the whole Q will be negative.
Math note
Without optimization the inerations count will be n! / ((n - k)! * k!).
With the optimization it will be:
5! / ((5-3)! * 3!) + (n - 3) + P(q)outsiders * n that is about 15000 for n = 10000. Most negative count - about 20000.
And another optimization note
Replace declaration of TPoint with integer variables.
Search Results
Featured snippet from the web
For n=1: you need two lines to intersect, so the maximum number of intersections is 0. n=2: Two distinct lines will always intersect in at most one point irrespective of dimensions. ... Explanation: Each set of 2 lines can intersect at one point. Or one point is common intersection for 2 lines.

Lift UInt64 limits with strings in Delphi

I'm reaching my limit with UInt64 and I was wondering if there are functions which do simple operating options such as +/- , etc. with just strings because they can store just as much RAM as you have... (theoretically)
For example I would like to calculate
24758800785707605497982484480 + 363463464326426 and get the result as a string.
I kinda know how to solve this problems with strings using the number system 0123456789 and kinda do digit by digit and overflow the next position - which would cost a lot more power, but I wouldn't mind this issue...
I would like to have this ability to do such calculations until my RAM just blows up (which would be the real limit...)
Are there such functions which already do that?
Arbitrarily large integers are not supported at the language level in Delphi, but a bit of Googling turns up http://www.delphiforfun.org/programs/Library/big_integers.htm, which can support them as alibrary.
On super computers, its called BCD math (Binary Coded Decimals) and each half-byte of RAM represents a decimal digit [0..9] - not an efficient use of RAM, but huge computations take minimal time (i.e. about 3 mSecs to multiply 2 million digit numbers. A BCD Emulator on a fast PC takes 5 or 6 minutes.
I never need to add big numbers, but I do multiply. Actually I call this routine iteratively to compute for example, 1000000 factorial (a 5,565,709 million digit answer. Str6Product refers to how it chops up a pair of string numbers. s1 and s2 have a practical length limit of about 2^31. The function is limited by what a "string can hold". Whatever that limit is, I've never gotten there.
//==============================================================================
function Str6Product(s1: string; s2: string): string; // 6-13 5:15 PM
var
so,snxt6 : string;
z1,z3, i, j, k : Cardinal; // Cardinal is 32-bit unsigned
x1,x3,xm : Cardinal;
countr : Cardinal;
a1, a2, a3 : array of Int64;
inum, icarry : uInt64; // uInt64 is 64-bit signed
begin
s1 := '00000'+s1;
s2 := '00000'+s2;
z1 := length(s1); // set size of Cardinal arrays
z3 := z1 div 6;
x1 := length(s2); // set size of Cardinal arrays
x3 := x1 div 6;
xm := max(x3,z3);
SetLength(a1,xm+1);
SetLength(a2,xm+1);
// try to keep s1 and s2 about the
// same length for best performance
for i := 1 to xm do begin // from rt 2 lft - fill arrays
// with 4-byte integers
if i <= z3 then a1[i] := StrToInt(copy (s1, z1-i*6+1, 6));
if i <= x3 then a2[i] := StrToInt(copy (s2, x1-i*6+1, 6));
if i > z3 then a1[i] := 0;
if i > x3 then a2[i] := 0;
end;
k := max(xm-x3, xm-z3); // k prevents leading zeroes
SetLength(a3,xm+xm+1);
icarry := 0; countr := 0;
icMax := 0; inMax := 0;
for i := 1 to xm do begin // begin 33 lines of "string mult" engine
inum := 0;
for j := 1 to i do
inum := inum + (a1[i-j+1] * a2[j]);
icarry := icarry + inum;
if icMax < icarry then icMax := icarry;
if inMax < inum then inMax := inum;
inum := icarry mod 1000000;
icarry := icarry div 1000000;
countr := countr + 1;
a3[countr] := inum;
end;
if xm > 1 then begin
for i := xm downto k+1 do begin // k or 2
inum := 0;
for j := 2 to i do
inum := inum + (a1[xm+j-i] * a2[xm-j+2]);
icarry := icarry + inum;
if icMax < icarry then icMax := icarry;
if inMax < inum then inMax := inum;
inum := icarry mod 1000000;
icarry := icarry div 1000000;
countr := countr + 1;
a3[countr] := inum;
end;
end;
if icarry >= 1 then begin
countr := countr + 1;
a3[countr] := icarry;
end;
so := IntToStr(a3[countr]);
for i := countr-1 downto 1 do begin
snxt6 := IntToStr(a3[i]+1000000);
so := so+ snxt6[2]+ snxt6[3]+ snxt6[4]+ snxt6[5]+ snxt6[6]+ snxt6[7];
end;
while so[1] = '0' do // leading zeroes may exist
so := copy(so,2,length(so));
result := so;
end;
//==============================================================================
Test call:
StrText := Str6Product ('742136061320987817587158718975871','623450632948509826743508972875');
I should have added that you should be able to add large numbers using the same methodology - From right to left, fragment the strings into 16 byte chunks then convert those chunks to uInt64 variables. Add the least significant digits first and if it produces a 17th byte, carry that over to the 2nd least significant chunk, add those two PLUS any carry over etc. When otherwise done, convert each 16-byte chunk back to string and concatenate accordingly.
The conversions to and from integer to string and vice-versa is a pain, but necessary for big number arithmetic.

Integration of values in a buffer with Delphi

Following on from my question on differentiation:
Differentiation of a buffer with Delphi
I'm now looking at doing the integration. I can't quite get my head around this one. The situation is that I receive a buffer of data periodically that contains a number of values that are a fixed distance in time apart. I need to differentiate them. It is soo long since I did calculus at school ....
What I have come up with is this:
procedure IntegrateBuffer(ABuffer: TDoubleDynArray;
var AOutBuffer: TDoubleDynArray;
AVPS: integer);
const
SumSum: double = 0.0;
LastValue: double = NaN;
var
i: integer;
dt, aa, hl, hr: double;
begin
// protect from divide by zero
if (AVPS < 1) then exit;
dt := 1 / AVPS;
for i := 0 to high(ABuffer) do begin
if (i = 0) then begin
if (IsNaN(LastValue)) then begin
hl := ABuffer[0];
hr := ABuffer[0];
end else begin
hl := LastValue;
hr := ABuffer[i];
end;
end else begin
hl := ABuffer[i -1];
hr := ABuffer[i];
end;
aa := 0.5 * dt * (hl + hr);
SumSum := SumSum + aa;
AOutBuffer[i] := SumSum;
end;
// remember the last value for next time
LastValue := ABuffer[high(ABuffer)];
end;
I'm using the trapezium rule, hl and hr ar the left and right heights of the trapezium. dt is the base.
AVPS is values per second. A typical value for this would be between 10 and 100. The length of the buffers would typically be 500 to 1000 values.
I call the buffer time after time with new data which is continuous with the previous block of data, hence keeping the last value of the block for next time.
Is what I have done correct? ie, will it integrate the values properly?
Thank you.
Looks like you need some help with testing the code. Here, as discussed in comments, is a very simple test.
{$APPTYPE CONSOLE}
uses
SysUtils, Math;
type
TDoubleDynArray = array of Double;
var
SumSum: double;
LastValue: double;
procedure Clear;
begin
SumSum := 0.0;
LastValue := NaN;
end;
procedure IntegrateBuffer(
ABuffer: TDoubleDynArray;
var AOutBuffer: TDoubleDynArray;
AVPS: integer
);
var
i: integer;
dt, aa, hl, hr: double;
begin
// protect from divide by zero
if (AVPS < 1) then exit;
dt := 1 / AVPS;
for i := 0 to high(ABuffer) do begin
if (i = 0) then begin
if (IsNaN(LastValue)) then begin
hl := ABuffer[0];
hr := ABuffer[0];
end else begin
hl := LastValue;
hr := ABuffer[i];
end;
end else begin
hl := ABuffer[i -1];
hr := ABuffer[i];
end;
aa := 0.5 * dt * (hl + hr);
SumSum := SumSum + aa;
AOutBuffer[i] := SumSum;
end;
// remember the last value for next time
LastValue := ABuffer[high(ABuffer)];
end;
var
Buffer: TDoubleDynArray;
OutBuffer: TDoubleDynArray;
begin
// test y = 1 for a single call, expected output = 1, actual output = 2
Clear;
Buffer := TDoubleDynArray.Create(1.0, 1.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Writeln(OutBuffer[high(OutBuffer)]);
Readln;
end.
I'm integrating the function y(x) = 1 over the range [0..1]. So, the expected output is 1. But the actual output is 2.
So, what's wrong? You can work it out in the debugger, but it's easy enough to see by inspecting the code. You are summing a triangle on the very first sample. When IsNaN(LastValue) is true then you should not make a contribution to the integral. At that point you've not covered any distance on the x axis.
So to fix the code, let's try this:
....
if (IsNaN(LastValue)) then begin
hl := 0.0;//no contribution to sum
hr := 0.0;
end else begin
hl := LastValue;
hr := ABuffer[i];
end;
....
That fixes the problem.
Now let's extend the test a little and test y(x) = x:
// test y = x, expected output = 12.5
Clear;
Buffer := TDoubleDynArray.Create(0.0, 1.0, 2.0, 3.0, 4.0, 5.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Writeln(OutBuffer[high(OutBuffer)]);
So, that looks good.
OK, what about multiple calls:
// test y = x for multiple calls, expected output = 18
Clear;
Buffer := TDoubleDynArray.Create(0.0, 1.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Buffer := TDoubleDynArray.Create(2.0, 3.0, 4.0, 5.0, 6.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Writeln(OutBuffer[high(OutBuffer)]);
And how about one value at a time?
// test y = x for multiple calls, one value at a time, expected 0.5
Clear;
Buffer := TDoubleDynArray.Create(0.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Buffer := TDoubleDynArray.Create(1.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Writeln(OutBuffer[high(OutBuffer)]);
What about passing an empty array?
// test y = x for multiple calls, some empty arrays, expected 0.5
Clear;
Buffer := TDoubleDynArray.Create(0.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Buffer := nil;
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Buffer := TDoubleDynArray.Create(1.0);
SetLength(OutBuffer, Length(Buffer));
IntegrateBuffer(Buffer, OutBuffer, 1);
Writeln(OutBuffer[high(OutBuffer)]);
Uh, oh, access violation. Better protect that by simply skipping the function at the start if the buffer is empty:
if (AVPS < 1) then exit;
if (Length(ABuffer) = 0) then exit;
OK, now that last test passes
Hopefully you get the idea now. I've just used noddy Writeln based testing but that does not scale. Get yourself a unit test framework (I recommend DUnitX) and build proper test cases. This will also force you to factor your code so that it is well designed. One of the often unexpected benefits of making code testable is that it usually results in the design of the interface being improved.
For your next question, I request that you supply an SSCCE with the test code! ;-)
Some comments on the code:
Pass dynamic arrays by const or by var. In your case you want to pass the input buffer by const.
Don't use writeable typed constants. Use either parameters, or some other more sane state management.
Again, as I said in the previous question, write tests to prove code, as well as checking it by eye. The key to writing tests is to start with the very simplest thing you can possibly think of. Something so simple that you know for 100% sure the answer. Then, once you get that to work, expand the testing to more complex cases.

Convert Int64 to Base30 and Back

For a registration code I want to convert an Int64 to base30 (30 so that only uppercase characters and excluding 0,O,I,1,etc.) and back.
This is not too difficult using functions like:
const
Base = 30;
Base30CharSet = '23456789ABCDEFGHJKLMNPRSTVWXYZ';
function ConvertIntToBase30(ANumber: Int64): string;
begin
if(ANumber = 0) then
Result := Copy(Base30CharSet, 1, 1)
else begin
Result := '';
while(ANumber <> 0) do begin
Result := Copy(Base30CharSet, (ANumber mod Base)+1, 1) + Result;
ANumber := ANumber div Base;
end;
end;
end;
function ConvertBase30ToInt(ANumber: string): Int64;
var
i: integer;
begin
Result := 0;
for i := 1 to Length(ANumber) do begin
Result := Result + (Pos(ANumber[i], Base30CharSet)-1);
if(i < Length(ANumber)) then
Result := Result * Base;
end;
end;
The snag is that I am interested in the Int64's bits, so I could be dealing with a number like $FFFFFFFFFFFFFFFF = -1.
To work around this I thought I would store and remove the sign (abs()) and include the sign as an extra character appended to the base30 result. The problem the occurs at the lower limit of Int64 as calling abs(-9223372036854775808) results in an overflow.
Does anyone have a solution or better algorithm to solve this problem?
The way to deal with it is having a character to indicate it is a negative number so that you can decode back. For negative number, just flip the bit from 1 to 0 and remove the sign bit before encoding and when decode, do a flip back and add the sign bit. Below is working codes
function InvertIntOff(const ANumberL, ANumberH: Integer): Int64;
asm
XOR EAX,$FFFFFFFF
XOR EDX,$FFFFFFFF
end;
function InvertIntOn(const ANumberL, ANumberH: Integer): Int64;
asm
XOR EAX,$FFFFFFFF
XOR EDX,$FFFFFFFF
OR EDX,$80000000
end;
function ConvertIntToBase(ANumber: Int64): string;
const
CBaseMap: array[0..31] of Char = (
'2','3','4','5','6','7','8','9', //0-7
'A','B','C','D','E','F','G','H', //8-15
'J','K','L','M','N', //16-20
'P','Q','R','S','T','U','V','X','W','Y','Z'); //21-31
var
I: Integer;
begin
SetLength(Result, 15);
I := 0;
if ANumber < 0 then
begin
Inc(I);
Result[I] := '1';
ANumber := InvertIntOff(ANumber and $FFFFFFFF, (ANumber and $FFFFFFFF00000000) shr 32);
end;
while ANumber <> 0 do
begin
Inc(I);
Result[I] := CBaseMap[ANumber and $1F];
ANumber := ANumber shr 5;
end;
SetLength(Result, I);
end;
function ConvertBaseToInt(const ABase: string): Int64;
var
I, Index: Integer;
N: Int64;
begin
Result := 0;
if Length(ABase) > 0 then
begin
if ABase[1] = '1' then
Index := 2
else
Index := 1;
for I := Index to Length(ABase) do
begin
case ABase[I] of
'2'..'9':
N := Ord(ABase[I]) - Ord('2');
'A'..'H':
N := Ord(ABase[I]) - Ord('A') + 8;
'J'..'N':
N := Ord(ABase[I]) - Ord('J') + 16;
'P'..'Z':
N := Ord(ABase[I]) - Ord('P') + 21;
else
raise Exception.Create('error');
end;
if I > Index then
Result := Result or (N shl ((I - Index) * 5))
else
Result := N;
end;
if ABase[1] = '1' then
Result := InvertIntOn(Result and $FFFFFFFF, (Result and $FFFFFFFF00000000) shr 32);
end;
end;
procedure TestBase32;
var
S: string;
begin
S := ConvertIntToBase(-1);
ShowMessage(S + ' / ' + IntToStr(ConvertBaseToInt(S)) + ' ? -1');
S := ConvertIntToBase(-31);
ShowMessage(S + ' / ' + IntToStr(ConvertBaseToInt(S)) + ' ? -31');
S := ConvertIntToBase(1);
ShowMessage(S + ' / ' + IntToStr(ConvertBaseToInt(S)) + ' ? 1');
S := ConvertIntToBase(123456789);
ShowMessage(S + ' / ' + IntToStr(ConvertBaseToInt(S)) + ' ? 123456789');
S := ConvertIntToBase(-123456789);
ShowMessage(S + ' / ' + IntToStr(ConvertBaseToInt(S)) + ' ? -123456789');
end;
I think you are almost there by considering abs()...
But rather than using abs() why not simply ignore the sign for processing the value of the Int64 itself ? As far as I can tell, you are in fact already doing this so only one minor addition is needed to the encoding routine:
if aNumber < 0 then
// negative
else
// positive;
The only problem then is the LOSS of sign information in the resulting Base30 string. So treat that as a separate problem to be solved using the new information gained from the aNumber < 0 test...
I see you have excluded all chars that could be confused for 0 or 1 but have also excluded 0 and 1 themselves. You could therefore use 0 and 1 to indicate positive or negative (or vice versa).
Depending on the purpose of these routines, the placement of the 0/1 in the result could be entirely arbitrary (if you wished to obfuscate things and make the placement of the 0/1 random rather than a consistent lead/trail character).
When encoding simply drop a sign indicator into the result string at random, and when decoding handle the 0/1 character whenever as the sign marker it is encountered, but skipped for the purposes of decoding the value.
Of course, if obfuscation is not an issue then simply consistently pre or post fix the sign indicator.
You could even simply choose to use '1' to indicate negative and the LACK of a '1' to indicate/assume positive (this would simplify the zero value case a little I think)
The easy answer is to turn range checking off, even just for the method that you're calling abs in.
If you don't care about an extra char or two you could split the int64 into words or dwords and string those together. I would be more tempted to go to base32 and use bit shifts for speed and ease of use. Then your encoding becomes
Base32CharSet[(ANumber shr 5) % 32]
and a similar pos() based approach for the decode.

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