How to make a ArrayPlot/MatrixPlot in polar coordinates?

Question

in fact, I want to plot a image like:

Fig1 The picture above is from a paper Zhou K.-J. et al. 2014 about ion up-flow of ionosphere.

In MATLAB, there is a specialized function pcolor which could produce a similar effect：

n = 20;r = (0:n)'/n;
theta = pi*(-n:n)/n;
X = r*cos(theta);Y = r*sin(theta);
C = r*cos(2*theta);
pcolor(X,Y,C)

output:

Fig 2

It's very close to what I want except some details like there are quadrilaterals rather than segments of circle and ugly mesh and ticks, nonetheless, the picture above is acceptable, though not perfect.

However, I hope I could use Mathematica to solve this problem. I thought ArrayPlot or MatrixPlot would help, however, I can't found any options like 'polar coordinates' in these two functions. When I try something like:

n = 20; r = Range[40.]/20; theta = Pi Range[40.]/20;
m = Table[r1 Cos[2. theta1], {theta1, theta}, {r1, r}]

and plot:

ArrayPlot[m, ColorFunction -> "Rainbow", PlotRangePadding -> 0, 
 FrameLabel -> {"theta", "r"}, LabelStyle -> 22]

I only get this rectangle picture:

Fig 3

how can I turn it into a 'pie-chart-style' picture?

Answer 1

Let me use this as example data instead (your m is too big):

m = RandomReal[1, {4, 24}];

Crude Attempt

polararrayplot[array_, colourfunc_] := SectorChart[
  Map[Style[{1, 1}, colourfunc[#]] &, array, {2}],
  SectorSpacing -> None
];
polararrayplot[m, ColorData["Rainbow", #] &]

Finer Attempt

The code is fairly self-explanatory. I'm sure you know where to modify things to suit your needs.

grid[polarticks_, radialticks_, radialaxispos_] := SectorChart[
  {{1, 1}},
  ChartStyle -> Directive[EdgeForm[], Opacity[0]],
  PolarAxes -> True,
  PolarAxesOrigin -> {radialaxispos, 1},
  PolarGridLines -> {False, Range[0, 1, 1/Length[radialticks]]},
  PolarTicks -> {
    Transpose[{
      Most@Range[0, 2 Pi, 2 Pi/Length[polarticks]],
      polarticks
    }],
    Transpose[{
      Rest@Range[0, 1, 1/Length[radialticks]],
      radialticks
    }]
  }
];
polararrayplot[array_, colourfunc_] := SectorChart[
  Map[
    Style[{1, 1/Length[array]}, {EdgeForm[colourfunc[#]], colourfunc[#]}] &,
    array,
    {2}
  ],
  SectorSpacing -> None
];
Show[
  polararrayplot[m, ColorData["Rainbow", #] &],
  grid[{18, 12, 6, 0}, {80, 70, 60, 50}, 14 Pi/8],
  PlotRange -> All
]

Handling Blank Cells

Suppose that your data runs from 200 to 900, and not available is represented by 0:

min = 200;
max = 900;
m = ConstantArray[val, {4, 40}] /. val :> RandomChoice[{RandomReal[{min, max}], 0}];

Blank cells can be handled through a custom colour function, e.g.

colourize[val_] := If[
  val == 0,
  White,
  ColorData["Rainbow", (val - min)/(max - min)]
];

Now,

Show[
  polararrayplot[m, colourize],
  grid[{18, 12, 6, 0}, {80, 70, 60, 50}, 14 Pi/8],
  PlotRange -> All
]

produces

Better Grid

Sadly, SectorChart does not support AxesStyle nor provide PolarAxesStyle as an option, so the look of the polar axes cannot be modified straightforwardly. Only the ticks (i.e. the ticks of the radial axis and the inner circles) can be styled with TicksStyle.

We'd better create our own grid:

grid[polarticks_, radialticks_, radialaxispos_] := Module[
  {
    ticksize, gapsize, polarlabelspace, font, circumference, innercircles,
    tocartesian, gap, ptpos, rtpos
  },
  ticksize = 1/20;
  gapsize = 1/5;
  polarlabelspace = 1/5;
  font = Directive[FontFamily -> "Helvetica", FontSize -> 20];
  circumference = Directive[Black, AbsoluteThickness[1.5]];
  innercircles = Directive[Black, AbsoluteThickness[1]];
  gap[r_] := {
    radialaxispos - 2 Pi + (gapsize/2)/r,
    radialaxispos - (gapsize/2)/r
  };
  tocartesian = CoordinateTransformData["Polar" -> "Cartesian", "Mapping"];
  ptpos = Most@Range[0, 2 Pi, 2 Pi/Length[polarticks]];
  rtpos = Rest@Range[0, 1, 1/Length[radialticks]];
  Graphics[{
    {
      circumference,
      Circle[{0, 0}, 1, gap[1]],
      Line[{tocartesian@{1, #}, tocartesian@{1 + ticksize, #}}] & /@ ptpos
    },
    {
      innercircles,
      Circle[{0, 0}, #, gap[#]] & /@ Most[rtpos]
    },
    {
      font,
      MapThread[
        Text[#1, tocartesian@{#2, radialaxispos}] &,
        {radialticks, rtpos}
      ],
      MapThread[
        Text[
          #1,
          tocartesian@{1 + ticksize, #2},
          tocartesian@{1 + polarlabelspace, Pi + #2}
        ] &,
        {polarticks, ptpos}
      ]
    }
  }]
];

Now,

Show[
  polararrayplot[m, colourize],
  grid[{18, 12, 6, 0}, {80, 70, 60, 50}, 14 Pi/8],
  PlotRange -> All
]

produces

Let me use example data that looks more like that in the paper.

m = ConstantArray[0, {40, 8}];
For[j = 1, j <= 40, j++,
  For[i = 1, i <= 8, i++,
  m[[j, i]] = If[2 < j < 30,
    If[2 < j < 30, If[2 < i < 7,
      RandomReal[{min, max}],
      Which[
        i == 1 || i == 7, foo = RandomChoice[{0, RandomReal[{min, max}]}],
        i == 2, If[foo == 0, bar, RandomReal[{min, max}]],
        i == 8, If[foo == 0, 0, bar]]], 0], 0]]];
m = Transpose@(m /. bar :> RandomChoice[{0, RandomReal[{min, max}]}]);
Show[
  polararrayplot[m, colourize],
  grid[{18, 12, 6, 0}, {80, 70, 60, 50}, 10 Pi/8],
  PlotRange -> All
]

Answer 2

One can get a sort of array plot with ParametricPlot, MeshShading and an appropriate Mesh that seems equivalent to the Matlab plot:

n = 20; r = Range[40.]/20; theta = Pi Range[40.]/20;
m = Table[r1 Cos[2. theta1], {theta1, theta}, {r1, r}];

colorFn = ColorData["Rainbow"];
ParametricPlot[r {Cos[t], Sin[t]}, {r, 0, 1}, {t, 0, 2 Pi},
 Mesh -> Reverse@Dimensions[m] - 1,
 MeshShading -> Map[colorFn, Rescale[m], {2}]]

Answer 3

Here is an approach using Annulus. In this approach element {1,1} starts from horizontal axis and I have not adapted but to start of vertical axis downward but this could be adapted. The ticks have been made to match example and I use m from @Taiki answer. Coloring could be modified and generalized as required.

elem[r_, t_, m_, col_] := If[r > 1,
  {col, Annulus[{0, 0}, {r - 1, r}, {2 Pi (t - 1)/m, 
     2 Pi t/m}]}, {col, 
   Disk[{0, 0}, {1, 1}, {2 Pi (t - 1)/m, 2 Pi t/m}]}]
f[u_, rtc_List, a_, pt_List] := 
 Module[{dim = Dimensions[u], circ, max = Max[Flatten@u], el, 
   tcks},
  circ = Graphics[Table[Circle[{0, 0}, j], {j, dim[[1]]}]];
  el = Graphics[
      elem[##, dim[[2]], 
       If[u[[##]] == 0, White, 
        ColorData["SolarColors"][u[[##]]/max]]]] & @@@ 
    Tuples[Range /@ dim];
  tcks = Graphics[{Table[
      Text[rtc[[j]], j {Cos[a], Sin[a]}, Background -> White], {j, 
       dim[[1]]}], 
     Table[Line[{{0, 0}, 1.1 dim[[1]] {Cos[j], Sin[j]}}], {j, 0, 
       3 Pi/2, Pi/2}], 
     Table[Text[pt[[j + 2]], 
       1.2 dim[[1]] {Cos[j Pi/2], Sin[j Pi/2]}], {j, -1, 2}]}];
  Row[{Show[##, circ, tcks, ImageSize -> 400] &@el, 
    BarLegend[{"SolarColors", {0, max}}]}]

  ]

So for example:

f[m, Range[80, 0, -10], -Pi/4, Range[0, 18, 6]]

yielded:

Answer 4

Using the sector[] function from here (replaceable with Annulus in version 10.2), we can generate a plot that looks like, but is smoother, than the result of MATLAB's pcolor():

n = 20; r = N[Range[0, n]/n]; θ = N[π Range[-n, n]/n];
m = Table[r1 Cos[2 θ1], {r1, r}, {θ1, θ}];

jet[u_?NumericQ] := Blend[{{0, RGBColor[0, 0, 9/16]}, {1/9, Blue}, {23/63, Cyan},
                           {13/21, Yellow}, {47/63, Orange}, {55/63, Red},
                           {1, RGBColor[1/2, 0, 0]}}, u] /; 0 <= u <= 1

Graphics[{EdgeForm[], 
          Transpose[{Map[jet, Rescale[Drop[m, -1, -1]], {2}], 
                     Map[sector[#[[All, 1, 1]], #[[1, All, 2]]] &, 
                         Partition[Outer[List, r, θ], {2, 2}, {1, 1}], {2}]},
                    {3, 1, 2}]},
         BaseStyle -> {"FilledCurveBoxOptions" -> {Method -> {"SplinePoints" -> 30}}}]

(This incorporates the undocumented setting described here by Mr. Wizard for smooth-looking sectors.)

This approach can be combined with Taiki's ticks and labels to give plots similar to the figure in the OP.