More geometry with inverse images

Yesterday’s post was on inverse images of functions as sets, and ways to visualize them.  Today, I realized that despite my early series of posts on the Jacobian derailing, I probably have enough background to describe the area and coarea formulae.  The two give a relationship between the “size” of the fibers and the derivative of the map.  The first thing I’ll need to do is define the Jacobian for maps f: \mathbb{R}^m \to \mathbb{R}^n.  The definition will be slightly different depending on whether m or n is larger, but if n \geq m, then

|Jf(x)| := \sqrt{|Df(x)^T \cdot Df(x)|},

and if m \geq n, then

|Jf(x)| := \sqrt{|Df(x) \cdot Df(x)^T|}

where Df is the n x m matrix of partial derivatives of f, and we use the absolute value bars to indicate a determinant.  Notice that if m = n, then the definitions agree, and it is just the absolute value of the determinant of the matrix of partial derivatives.  If n = 1 so that f is a real-valued function, then the Jacobian is the length of the gradient of f.

Now then, the area formula says that for a Lipschitz f:\mathbb{R}^m \to \mathbb{R}^n with m \leq n, and any Lebesgue measurable U \subset \mathbb{R}^m,

\int_U |Jf(x)| d\mathcal{L}^m(x) = \int_{\mathbb{R}^n} \#(f^{-1}(y) \cap U)~d\mathcal{H}^m(y),

A hyperboloid projecting onto a circle.

where, for a set S, \#(S) denotes the cardinality of S, i.e., how many points are in S.  We expect this number to be finite (for most functions f I think of, each inverse image has cardinality either 1 or 0).  Indeed, notice that if f is a smooth embedding, then f is one-to-one, and the right hand side of the above is always 1 if f maps there.  Hence the right hand side will be \mathcal{H}^m(f(U)), the area of the image of U under f.  This explains why it is called the area formula- it agrees with the classical area of parametrized surfaces.

The coarea formula (the subject of my research) keeps all the conditions above, but now f maps from high dimensions into a lower one, so m \geq n.  We have

\int_U |Jf(x)| d\mathcal{L}^m(x) = \int_{\mathbb{R}^n} \mathcal{H}^{m-n}(f^{-1}(y) \cap U)~d\mathcal{H}^n(y).

In plain English, the integral of the Jacobian of f is equal to the integral of the length of the fibers of f.  [Technical sentences coming up!] One surprising fact is that this coarea formula was first proven in 1959 in Herbert Federer’s paper “Curvature Measures“, while the area formula was (is) a basic calculus fact, at least for smooth functions.  The formula has since played a role in image processing, as when f is a real valued function, the quantity is usually referred to as the total variation.  De Giorgi showed that the fibers of functions which minimized the left hand side are actually minimal surfaces.

A string hyperboloid.

I’ve included a few illustrations of how the coarea formula might relate to “projections” of hyperboloids onto the circle.  The first shows such a hyperboloid, along with the fibers of the “projection”.  The coarea of this map will be the surface area of the hyperboloid.  Such a hyperboloid can be made with string from a cylinder, and twisting the top.  See the second figure.  The final .gif illustrates continuing to twist the top, and the resulting surfaces.  In each case, integrating the function whose level sets are these straight lines will return the surface area of the hyperboloid

Twisting hyperboloids


One comment on “More geometry with inverse images

  1. […] posted earlier on a way of visualizing the fibers of certain maps from high dimensions to low dimensions. […]

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