Applying shading radially outwards on a circle

Hello, wondering if you all can help me with an issue I'm having - at my wits end try to get it to work!

I'm doing a study of the limb darkening profile of the Sun, and part of this involves creating a 'fake sun' with a perfect profile. This essientially means I have to use an equation to add the shading to the fake sun I have created.

| needed a raduis value for each pixel and calculated cos(thetha) for each pixel to use the equation.

I have started with a binary image, with a plain circle for the Sun, of the correct size and position. So in my code this is called 'NewbinaryImage'. So its all zeros but with 1's to create the Sun.

I have then tried to create a point matrix with all the details I would need to add the shading correctly. I have the x and y values of all the pixels in the image, the radius, costhetha value, the binary image pixel value, and the calculated shading. I intended to mutliply the shading value across the 1's in the binary image to create the perfect radial profile.

I've included the relevent section of my code here.

[xdata,ydata]=meshgrid(1:size(NewbinaryImage,1), 1:size(NewbinaryImage,2)); %assign an x and y value for each pixel 
result=[ydata(:) xdata(:) NewbinaryImage(:)];  %create long vevtor for each result, with x,y and the pixel values from the image 

r = sqrt(((xdata(:)-2604).^2+(ydata(:)-1738).^2)); %find r for each x and y (the offset is due to centring the circle) 

normr = r./(max(r)); %normalise so 0 at centre and 1 at max (at edge of circle)

costheta = sqrt(1-(normr.^2)); %assign a cosine value for all r in image 

%Limb darkening (LD) is applied with a+bcostheta. Choose values for a and b

a= 0.4; %select values for a and b 
b= 0.6;

LD = a+(b*costheta); %limb darkening equation 

shading = (NewbinaryImage(:)).*LD; %apply shading - each 1 in the binary image will change

pointmatrix = [xdata(:) ydata(:) normr costheta LD NewbinaryImage(:) shading]; %create point matrix with all info needed 

shadedimage = reshape(shading, [3476,5208]); %reshape the long vector to the same size as the original image

imshow(shadedimage) 

But the final image has shading that is like stripes across the sun? Rather than being brighter in the centre and darker at the edges.

Can anyone see where I'm going wrong?

[Jan]

Please use the buttons on top of the field for editing messages to format your code. This will increase the readability massively.

Can you provide some input data and show, what you currently get?

Why do you create the matrix result, when it is not used anywhere?

[Claire Park]

Hi, thanks for replying!

I can get that for you. I use the university computers for this, so I can get it tomorrow. It just starts as a basic white circle on a black background, but ends up with grey stripes all over the circle.

um, I think i just did that so I could get the 3 results and use them easily? I actually han't realised it was unnecessary sorry!

[Claire Park]

Okay, so the binary image I start with is this:

[Claire Park]

And the final fake sun image comes out like this:

[Claire Park]

So the final image should have a bright centre and then fade out to a darker edges, radially around the centre. I can't seem to get it to work, and I've kinda exhausted all the help I can get at uni

Would be very grateful for any help you can give! :)

Answer 1

[Star Strider]

I’m notr exactly certain what you want, however this should get you started:

rmax = 100;
r = linspace(0, 1, rmax);                               % Define Radius & Radius Gradient Vector
a = linspace(0, 2*pi, 150);                             % Angles (Radians)
[R,A] = ndgrid(r, a);                                   % Create Grid
Z = -R;                                                 % Create Gradient Matrix
[X,Y,Z] = pol2cart(A,R,Z);                              % Convert To Cartesian

figure
surf(X, Y, Z)
view(0, 90)
axis('equal')
colormap(bone)
shading('interp')

producing:

If you want to eliminate the background grid and axes and just show the ‘sun’, add this set call after the plot:

set(gca, 'Visible','off')

to get this result:

Experiment to get the result you want.

[Claire Park]

Oh wow, that looks promising!

I need to apply a specific shading for each pixel though, so it wont be quite as uniform as that - is there a way to change this to be able to use the values I've calculated?

Thanks so much!

[Star Strider]

My pleasure.

I need for you to go into some detail as to what you want to do.

What values have you calculated? What parts of the image my code created would you want to change? I’m not an astronomer, so you’ll need to be specific.

[Claire Park]

Okay, so I need the sun to be the same size as my image which should be fine, but I also need the shading to be made via the equation.

So I used the radius of each point, and then used the equation to work out what value of shading there needs to be at each point. So the final image will look similar, the centre is bright and find towards the edges - but it’s done by the equations, so not something I can set myself.

Does that help at all?

[Star Strider]

I don’t understand ‘the equation’.

Please espress it in terms of my code, or something that could work with it.

What do you want to do?

Do you want the ‘Z’ value to be an inverse-square relation, or something else? Do you have a specific relation to the ‘Z’ value in my code that you want to implement?

[Claire Park]

Okay sorry, I’m not explaining it the best!

So it needs to be a gradient according to this equation: LD= a+(b*costhetha). So a and b are values that I set, and I’ve assigned a costhetha value for each pixel in the sun, going radially outwards.

[Star Strider]

I’m lost.

I would appreciate a somewhat more detailed description of what you want to do, and what your calculations are.

So ‘r’ in your code is the radius, essentially the ‘Z’ in my code? Or is ‘LD’ the radius?

Do you have some sort of image depicting the desired result?

—————

EDIT — (1 Apr 2019 at 15:20)

I still don’t know what you want, or what your equations mean.

Try this:

rmax = 100;
r = linspace(0, 1, rmax);                               % Define Radius & Radius Gradient Vector
a = linspace(0, 2*pi, 150);                             % Angles (Radians)
[R,A] = ndgrid(r, a);                                   % Create Grid
rm = sqrt((R-2604).^2 + (A-1738).^2);
normr = rm./max(rm);
costheta = sqrt(1 - normr.^2);
a = 0.4;
b = 0.6;
Z = -(a + b*costheta);                                  % Create Gradient Matrix
[X,Y,Z] = pol2cart(A,R,Z);                              % Convert To Cartesian

figure
surf(X, Y, Z)
view(0, 90)
axis('equal')
colormap(gray)
shading('interp')
set(gca, 'Visible','off')

producing:

Experiment to get the result you want.

[Claire Park]

Thanks so much for all your help with this! I appreciate it :)

And yeah, r is the radius and LD is the gradient for the shading.

I’ll have a play about with this, definitely working better than my code!

[Star Strider]

My pleasure!

If my Answer helps you solve your problem, please Accept it!