How to use MatrixPlot as a DensityPlot?

Question

I have a 1000x1000 matrix the MatrixPlot of which gives this graph:

The code is

matrixplot1 =  MatrixPlot[data, DataReversed -> {True, False},
PlotLegends -> Automatic]

Since this matrix is obtained by sampling a set of data with X and Y coordinates in 1000x1000 grid the number in the i-th and j-th row and column indicates the number of occurrences between i-1 and j-1.

I would like to have the data on the Log10 scale and on the 0-1 interval for both X and Y axes. For these reasons I did the following:

1. I transformed the matrix in a {x,y,z} list as follows

densityData =   Flatten[ParallelTable[{(j - 0.5)/1000, (i - 0.5)/1000,
data[[i, j]]}, {i, 1, 1000}, {j, 1, 1000}], 1];

2. Then I used the DensityListPlot function:

graphDensityPlotLog = 
  ListDensityPlot[densityData,
   PlotLegends -> BarLegend[Automatic, LegendMarkerSize -> 0.4*imagesize],
   ScalingFunctions -> "Log10",
   FrameStyle -> Directive[Thick, Black],
   ColorFunction -> ColorData[{"SunsetColors", "Reverse"}],
   FrameLabel -> {"Quantity X", "Quantity Y"},
   FrameTicksStyle -> Directive[Black, 1.1*fontsize],
   LabelStyle -> Directive[Black, 1.3*fontsize],
   PlotRangePadding -> 0,
   ImageSize -> imagesize];

The result is

Because of the interpolation, there are colored bands in zones that are without any data. So I tried to obtain the same graph directly by the MatrixPlot with the following code:

log10Mod[x_] := If[x == 0, 0, Log10[x]];
log10Mod /@ {0, 1, 2, 3}

graphDensityPlotLog2 = 
 MatrixPlot[data, DataReversed -> {True, False},
  FrameLabel -> {"Quantity X", "Quantity Y"},
  PlotRange -> {{1, 550}, {1, 1000}},
  PlotRangePadding -> 20,
  FrameStyle -> Directive[Thick, Black],
  FrameTicksStyle -> Directive[Black, 1.1*fontsize],
  LabelStyle -> Directive[Black, 1.3*fontsize],
  ImageSize -> imagesize,
  ColorFunction -> (ColorData[{"SunsetColors", "Reverse"}][
      Rescale[log10Mod[#], log10Mod /@ {0, 200}]] &),
  ColorFunctionScaling -> False,
  PlotLegends -> 
   Placed[BarLegend[{(ColorData[{"SunsetColors", "Reverse"}][
         Rescale[#, log10Mod /@ {0, 200}]] &), log10Mod /@ {0, 200}}, 
     LegendMarkerSize -> 400, 
     LabelStyle -> Directive[Black, 25, FontFamily -> "Times"]], {1.0,
      0.5}]]

The result is the following:

I am not able to find a way to obtain the graph I obtain with the ListDensityPlot but from the MatrixPlot and without those wrong bands.

Thank you for your help!

Edit: based on the comment I made the following graph

fticks[min_, max_, bigstep_, smallstep_] := 
 Table[If[Mod[i, bigstep] == 0, {i + min, 
    ToString[If[IntegerQ[i + min], i + min, (i + min)/1.]], {.01, 
     0}}, {i + min, " ", {.005, 0}}], {i, 0, max - min, smallstep}]
fticksNoText[min_, max_, bigstep_, smallstep_] := 
 Table[If[Mod[i, bigstep] == 0, {i + min, " ", {.01, 0}}, {i + min, 
    " ", {.005, 0}}], {i, 0, max - min, smallstep}]
fticksNoBounds[min_, max_, bigstep_, smallstep_] := 
 Table[If[Mod[i, bigstep] == 0 && i != 0 && 
    i != (max - min), {i + min, 
    ToString[If[IntegerQ[i + min], i + min, (i + min)/1.]], {.01, 
     0}}, {i + min, " ", {.005, 0}}], {i, 0, max - min, smallstep}]
fticksReversed[min_, max_, bigstep_, smallstep_] := 
 Table[If[Mod[i, bigstep] == 0, {i + min, 
    ToString[If[IntegerQ[max - i], max - i, (max - i)/1.]], {.01, 
     0}}, {i + min, " ", {.005, 0}}], {i, 0, max - min, smallstep}]

graphDensityPlotLog3 = 
 MatrixPlot[data, DataReversed -> {True, False},
  FrameLabel -> {"Quantity X", "Quantity Y"},
  PlotRangePadding -> 0,
  DataRange -> {{0, 1}, {0, 1}},
  PlotRange -> {{0, 1}, {0, 0.55}},
  FrameStyle -> Directive[Thick, Black],
  FrameTicksStyle -> Directive[Black, 1.1*fontsize],
  FrameTicks -> {{fticksReversed[0, 1, 1/10, 1/100], 
     fticksNoText[0, 1, 1/10, 1/100]}, {fticksNoBounds[0, 1, 2/10, 
      4/100], fticksNoText[0, 1, 2/10, 4/100]}},
  LabelStyle -> Directive[Black, 1.3*fontsize],
  ImageSize -> imagesize,
  AspectRatio -> 1,
  ColorFunction -> (ColorData[{"SunsetColors", "Reverse"}][
      Rescale[log10Mod[#], log10Mod /@ {0, 200}]] &),
  ColorFunctionScaling -> False,
  PlotLegends -> 
   Placed[BarLegend[{(ColorData[{"SunsetColors", "Reverse"}][
         Rescale[#, log10Mod /@ {0, 200}]] &), log10Mod /@ {0, 200}}, 
     LegendMarkerSize -> 400, 
     LabelStyle -> Directive[Black, 25, FontFamily -> "Times"]], {1.0,
      0.5}]]

The result is

I had to define the ticks in order to put them in a MatrixPlot going from zero to one.

The problem of the legend bar remains to be solved.