What Properties Make Certain Wavelets "Better" Than Others In Image Compression?
Overview

The short answer is that they have the maximum number of vanishing moments for a given support (i.e number of filter coefficients). That's the "extremal" property which distinguishes Daubechies wavelets in general. Loosely speaking, more vanishing moments implies better compression, and smaller support implies less computation. In fact, the tradeoff between vanishing moments and filter size is so important that it dominates the way that wavelets are named.  For example, you'll often see the D4 wavelet referred to either as D4 or db2.  The 4 refers to the number of coefficients, and the 2 refers to the number of vanishing moments. Both refer to the same mathematical object. Below, I'll explain more about what moments are (and why we want to make them disappear), but for now, just understand that it relates to how well we can "fold up" most of the information in the signal into a smaller number of values. Lossy compression is achieved by keeping those values, and throwing away the others.

Now, you may have noticed that CDF 9/7, which is used in JPEG 2000, has two numbers in the name, rather than one. In fact, it's also referred to as bior 4.4. That's because it's not a "standard" discrete wavelet at all. In fact, it doesn't even technically preserve the energy in the signal, and that property is the entire reason people got so excited about the DWT in the first place! The numbers, 9/7 and 4.4, still refer to the supports and vanishing moments respectively, but now there are two sets of coefficients that define the wavelet. The technical term is that rather than being orthogonal, they are biorthogonal. Rather than getting too deep into what that means mathematically, I'll just review the factors which led to using non-energy-preserving biorthogonal wavelets in the first place.

JPEG 2000

A much more detailed discussion of the design decisions surrounding the CDF 9/7 wavelet can be found in the following paper:


  Usevitch, Bryan E. A Tutorial on Modern Lossy Wavelet Image
  Compression: Foundations of JPEG 2000.


I'll just review the main points here.


Quite often, the orthogonal Daubechies wavelets can actually result in increasing the number of values required to represent the signal. The effect is called coefficient expansion. If we're doing lossy compression that may or may not matter (since we're throwing away values at the end anyway), but it definitely seems counterproductive in the context of compression. One way to solve the problem is to treat the input signal as periodic.
Just treating the input as periodic results in discontinuities at the edges, which are harder to compress, and are just artifacts of the transform. For example, consider the jumps from 3 to 0 in the following periodic extension: $[0,1,2,3] \rightarrow  [...0,1,2,3,0,1,2,3,...]$.  To solve that problem, we can use a symmetric periodic extension of the signal, as follows: $[0,1,2,3] \rightarrow [...,0,1,2,3,3,2,1,0,0,1...]$. Eliminating jumps at the edges is one of the reasons the Discrete Cosine Transform (DCT) is used instead of the DFT in JPEG. Representing a signal with cosines implicitly assumes "front to back looping" of the input signal, so we want wavelets which have the same symmetry property.
Unfortunately, the only orthogonal wavelet which has the required characteristics is the Haar (or D2, db1) wavelet, which only as one vanishing moment. Ugh. That leads us to biorthogonal wavelets, which are actually redundant representations, and therefore don't preserve energy. The reason CDF 9/7 wavelets are used in practice is because they were designed to come very close to being energy preserving. They have also tested well in practice.


There are other ways to solve the various problems (mentioned briefly in the paper), but these are the broad strokes of the factors involved.

Vanishing Moments 

So what are moments, and why do we care about them? Smooth signals can be well approximated by polynomials, i.e. functions of the form:

$$a + bx + cx^2 + dx^3 + ...$$

The moments of a function (i.e. signal) are a measure of how similar it is to a given power of x.  Mathematically, this is expressed as an inner product between the function and the power of x.  A vanishing moment means the inner product is zero, and therefore the function doesn't "resemble" that power of x, as follows (for the continuous case):

$$\int{x^n f(x) dx = 0 }$$

Now each discrete, orthogonal wavelet has two FIR filters associated with it, which are used in the DWT. One is a lowpass (or scaling) filter $\phi$, and the other is a highpass (or wavelet) filter $\psi$. That terminology seems to vary somewhat, but it's what I'll use here. At each stage of the DWT, the highpass filter is used to "peel off" a layer of detail, and the lowpass filter yields a smoothed version of the signal without that detail. If the highpass filter has vanishing moments, those moments (i.e. low order polynomial features) will get stuffed into the complementary smoothed signal, rather than the detail signal. In the case of lossy compression, hopefully the detail signal won't have much information in it, and therefore we can throw most of it away.

Here's a simple example using the Haar (D2) wavelet. There's typically a scaling factor of $1/\sqrt{2}$ involved, but I'm omitting it here to illustrate the concept. The two filters are as follows:
$$ \phi = [1,1] \\  \psi = [1,-1]$$

The highpass filter vanishes for the zero'th moment, i.e. $x^0 = 1$, therefore it has one vanishing moment. To see this, consider this constant signal: $[2,2,2,2]$.  Now intuitively, it should be obvious that there's not much information there (or in any constant signal).  We could describe the same thing by saying "four twos". The DWT gives us a way to describe that intuition explicitly. Here's what happens during a single pass of the DWT using the Haar wavelet:

$$
[2,2,2,2] \rightarrow_{\psi}^{\phi} \left\{ \begin{array}{rr}
\left[2 + 2, 2 + 2\right] = \left[4,4\right] \\ 
\left[2-2,2-2\right] = \left[0,0\right]
\end{array}\right.
$$

And what happens on the second pass, which operates on just the smoothed signal:

$$
[4,4] \rightarrow_{\psi}^{\phi} \left\{ \begin{array}{rr}
\left[4 + 4\right] = \left[8\right] \\ 
\left[4-4\right] = \left[0\right] 
\end{array}\right.
$$

Notice how the constant signal is completely invisible to the detail passes (which all come out to be 0).  Also notice how four values of $2$ have been reduced to a single value of $8$.  Now if we wanted to transmit the original signal, we could just send the $8$, and the Inverse DWT could reconstruct the original signal by assuming that all the detail coefficients are zero. Wavelets with higher-order vanishing moments allow similar results with signals that are well approximated by lines, parabolas, cubics, etc.

Further Reading

I'm glossing over a LOT of detail to keep the above treatment accessible. The following paper has a much deeper analysis:


  M. Unser, and T. Blu, Mathematical properties of the JPEG2000 wavelet
  filters, IEEE Trans. Image Proc., vol. 12, no. 9, Sept. 2003,
  pg.1080-1090.


Footnote

The above paper seems to suggest that the JPEG2000 wavelet is called Daubechies 9/7, and is different from the CDF 9/7 wavelet. 


  We have derived the exact form of the JPEG2000 Daubechies 9/7 scaling
  filters... These filters result from the factorization of the same
  polynomial as $Daubechies_{8}$ [10]. The main difference is that the
  9/7 filters are symmetric. Moreover, unlike the biorthogonal splines
  of Cohen-Daubechies-Feauveau [11], the nonregular part of the
  polynomial has been divided among both sides, and as evenly as
  possible.
  
  [11] A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases
  of compactly supported wavelets,” Comm. Pure Appl. Math., vol. 45, no.
  5, pp. 485–560, 1992.


The draft of the JPEG2000 standard (pdf link) that I've browsed also calls the official filter Daubechies 9/7. It references this paper:


  M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding
  using the wavelet transform,” IEEE Trans. Image Proc. 1, pp. 205-220,
  April 1992.


I haven't read either of those sources, so I can't say for sure why Wikipedia calls the JPEG2000 wavelet CDF 9/7. It seems like there may be a difference between the two, but people call the official JPEG2000 wavelet CDF 9/7 anyway (because it's based on the same foundation?).  Regardless of the name, the paper by Usevitch describes the one that's used in the standard.