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%% Exploring a Conformal Mapping
% This example shows how to explore a conformal mapping. Geometric image
% transformations are useful in understanding a conformal mapping that is
% important in fluid-flow problems, and the mapping itself can be used to
% transform imagery for an interesting special effect.
% Copyright 2000-2014 The MathWorks, Inc.
%% Step 1: Select a Conformal Transformation
%
% Conformal transformations, or mappings, have many important properties
% and uses. One property relevant to image transformation is the
% preservation of local shape (except sometimes at isolated points).
%
% This example uses a 2-D conformal transformation to warp an image. The
% mapping from output to input, |g: R^2 -> R^2|, is defined in terms of a
% complex analytic function |G: C -> C|, where
%
% G(z) = (z + 1/z) / 2.
%
% We define |g| via a direct correspondence between each point |(x,y)| in
% |R^2| (the Euclidean plane) and the point |z = x + i*y| in |C| (the
% complex plane),
%
% g(x,y) = (Re(w),Im(w)) = (u,v)
%
% where
%
% w = u + i*v = G(x + i*y).
%
% This conformal mapping is important in fluid mechanics because it
% transforms lines of flow around a circular disk (or cylinder, if we add a
% third dimension) to straight lines. (See pp. 340-341 in Strang, Gilbert,
% Introduction to Applied Mathematics, Wellesley-Cambridge Press,
% Wellesley, MA, 1986.)
%
% A note on the value of complex variables: although we could express the
% definition of |g| directly in terms of |x| and |y|, that would obscure
% the underlying simplicity of the transformation. This disadvantage would
% come back to haunt us in Step 3 below. There, if we worked purely in
% real variables, we would need to solve a pair of simultaneous nonlinear
% equations instead of merely applying the quadratic formula!
%% Step 2: Warp an Image Using the Conformal Transformation
%
% We start by loading the peppers image, extracting a 300-by-500 subimage,
% and displaying it.
A = imread('peppers.png');
A = A(31:330,1:500,:);
figure
imshow(A)
title('Original Image','FontSize',14)
%%
% Then use |maketform| to make a custom |tform| struct with a handle to
% function |conformalInverse| as its |INVERSE_FCN| argument:
conformal = maketform('custom', 2, 2, [], @conformalInverse, []);
%%
% To view |conformalInverse| use:
type conformalInverse.m
%%
% Horizontal and vertical bounds are needed for mapping the original and
% transformed images to the input and output complex planes. Note that the
% proportions in |uData| and |vData| match the height-to-width ratio of the
% original image (3/5).
uData = [ -1.25 1.25]; % Bounds for REAL(w)
vData = [ 0.75 -0.75]; % Bounds for IMAG(w)
xData = [ -2.4 2.4 ]; % Bounds for REAL(z)
yData = [ 2.0 -2.0 ]; % Bounds for IMAG(z)
%%
% We apply |imtransform| using the |SIZE| parameter to ensure an aspect
% ratio that matches the proportions in |xData| and |yData| (6/5), and view
% the result.
B = imtransform( A, conformal, 'cubic', ...
'UData', uData,'VData', vData,...
'XData', xData,'YData', yData,...
'Size', [300 360], 'FillValues', 255 );
figure
imshow(B)
title('Transformed Image','FontSize',14)
%%
% Compare the original and transformed images. Except that the edges are
% now curved, the outer boundary of the image is preserved by the
% transformation. Note that each feature from the original image appears
% twice in the transformed image (look at the various peppers). And there
% is a hole in the middle of the transformed image with four regular cusps
% around its edges.
%
% In fact, every point in the input w-plane is mapped to two points in the
% output |z|-plane, one inside the unit circle and one outside. The copies
% inside the unit circle are much smaller than those outside. It's clear
% that the cusps around the central hole are just the copies of the four
% image corners that mapped inside the unit circle.
%% Step 3: Construct Forward Transformations
%
% If the transformation created with |maketform| has a forward function,
% then we can apply |tformfwd| to regular geometric objects (in particular,
% to rectangular grids and uniform arrays of circles) to obtain further
% insight into the transformation. In this example, because |G| maps two
% output points to each input point, there is no unique forward
% transformation. But we can proceed if we are careful and work with two
% different forward functions.
%
% Letting |w = (z + 1/z)/2| and solving the quadratic equation that
% results,
%
% z^2 + 2*w*z + 1 = 0,
%
% we find that
%
% z = w +/- sqrt{(w^2 - 1).
%
% The positive and the negative square roots lead to two separate forward
% transformations. We construct the first using |maketform| and a handle to
% the function, |conformalForward1|.
t1 = maketform('custom', 2, 2, @conformalForward1, [], []);
%%
% To view |conformalForward1| use:
type conformalForward1.m
%%
% We construct the second transformation with another function that is
% identical to |conformalForward1| except for a sign change.
t2 = maketform('custom', 2, 2, @conformalForward2, [], []);
type conformalForward2.m
%% Step 4: Explore the Mapping Using Grid Lines
%
% With the two forward transformations, we can illustrate the mapping of a
% grid of lines, using additional helper functions.
f3 = figure('Name','Conformal Transformation: Grid Lines');
axIn = conformalSetupInputAxes( subplot(1,2,1));
axOut = conformalSetupOutputAxes(subplot(1,2,2));
conformalShowLines(axIn, axOut, t1, t2)
% Reduce wasted vertical space in figure
f3.Position = [1 1 1 0.7] .* f3.Position;
%%
% You can see that the grid lines are color-coded according to their
% quadrants in the input plane before and after the transformations. The
% colors also follow the transformed grids to the output planes. Note that
% each quadrant transforms to a region outside the unit circle and to a
% region inside the unit circle. The right-angle intersections between grid
% lines are preserved under the transformation -- evidence of the
% shape-preserving property of conformal mappings -- except for the points
% at +1 and -1 on the real axis.
%% Step 5: Explore the Mapping Using Packed Circles
%
% Under a conformal transformation, small circles should remain nearly
% circular, changing only in position and size. Again applying the two
% forward transformations, this time we map a regular array of
% uniformly-sized circles.
f4 = figure('Name','Conformal Transformation: Circles');
axIn = conformalSetupInputAxes( subplot(1,2,1));
axOut = conformalSetupOutputAxes(subplot(1,2,2));
conformalShowCircles(axIn, axOut, t1, t2)
% Reduce wasted vertical space in figure
f4.Position = [1 1 1 0.7] .* f4.Position;
%%
% You can see that the transform to a circle packing where tangencies have
% been preserved. In this example, the color coding indicates use of the
% positive (green) or negative (blue) square root of |w^2 - 1|. Note that
% the circles change dramatically but that they remain circles
% (shape-preservation, once again).
%% Step 6: Explore the Mapping Using Images
%
% To further explore the conformal mapping, we can place the input and
% transformed images on the pair of axes used in the preceding examples and
% superpose a set of curves as well.
%
% First we display the input image, rendered semi-transparently, over the
% input axes of the conformal map, along with a black ellipse and a
% red line along the real axis.
figure
axIn = conformalSetupInputAxes(axes);
conformalShowInput(axIn, A, uData, vData)
title('Original Image Superposed on Input Plane','FontSize',14)
%%
% Next we display the output image over the output axes of the conformal
% map, along with two black circles and one red circle. Again, the
% image is semi-transparent.
figure
axOut = conformalSetupOutputAxes(axes);
conformalShowOutput(axOut, B, xData, yData)
title('Transformed Image Superposed on Output Plane','FontSize',14)
%%
% MATLAB(R) graphics made it easy to shift and scale the original and
% transformed images to superpose them on the input (|w-|) and output
% (|z-|) planes, respectively. The use of semi-transparency makes it easier
% to see the ellipse, line, and circles. The ellipse in the w-plane has
% intercepts at 5/4 and -5/4 on the horizontal axis and 3/4 and -3/4 on the
% vertical axis. |G| maps two circles centered on the origin to this
% ellipse: the one with radius 2 and the one with radius 1/2. And, as shown
% in red, |G| maps the unit circle to the interval [-1 1] on the real axis.
%% Step 7: Obtain a Special Effect by Masking Parts of the Output Image
%
% If the inverse transform function within a custom |tform| struct returns
% a vector filled with |NaN| for a given output image location, then
% |imtransform| (and also |tformarray|) assign the specified fill value at
% that location. In this step we repeat Step 1, but modify our inverse
% transformation function slightly to take advantage of this feature.
type conformalInverseClip.m
%%
% This is the same as the function defined in Step 2, except for the two
% additional lines:
%
% q = 0.5 <= abs(Z) & abs(Z) <= 2;
% W(~q) = complex(NaN,NaN);
%
% which cause the inverse transformation to return |NaN| at any point not
% between the two circles with radii of 1/2 and 2, centered on the origin.
% The result is to mask that portion of the output image with the specified
% fill value.
ring = maketform('custom', 2, 2, [], @conformalInverseClip, []);
Bring = imtransform( A, ring, 'cubic',...
'UData', uData, 'VData', vData,...
'XData', [-2 2], 'YData', yData,...
'Size', [400 400], 'FillValues', 255 );
figure
imshow(Bring)
title('Transformed Image With Masking','FontSize',14);
%%
% The result is identical to our initial transformation except that the
% outer corners and inner cusps have been masked away to produce a ring
% effect.
%% Step 8: Repeat the Effect on a Different Image
%
% Applying the "ring" transformation to an image of winter greens (hemlock
% and alder berries) leads to an aesthetic special effect.
%
% Load the image |greens.jpg|, which already has a 3/5 height-to-width
% ratio, and display it.
C = imread('greens.jpg');
figure
imshow(C)
title('Winter Greens Image','FontSize',14);
%%
% Transform the image and display the result, this time creating a square
% output image.
D = imtransform( C, ring, 'cubic',...
'UData', uData, 'VData', vData,...
'XData', [-2 2], 'YData', [-2 2],...
'Size', [400 400], 'FillValues', 255 );
figure
imshow(D)
title('Transformed and Masked Winter Greens Image','FontSize',14);
%%
% Notice that the local shapes of objects in the output image are
% preserved. The alder berries stayed round!
%%