The difficulty in finding planets is that they are intrinsically very faint objects extremely close to their much brighter parent star. Going to large telescopes increases the resolution (one's ability to distinguish objects close together on the sky), but Earth's atmosphere blurs out the profile of the parent star, smearing its light over any fainter companions. This can be corrected for by adaptive optics, where a deformable mirror is bent to correct for these distortions, restoring the star to (almost) a point of light. At this point, the "size" of the star is simply a function of the size of your telescope (doubling the diameter of your primary mirror cuts the star's size on your detector in half). Unfortunately, this is not the end of it.
When the halo of the star is removed, one discovers the field very close to the center of the star is filled with features called "speckles," some of which have bizarre shapes and are easily identified, others appear almost identical to a planet. The source of these speckles is expected to be internal reflections within one's optics, so that some of the light from the bright primary star ends up going slightly off-axis, giving a second, much fainter image of the star on your detector, somewhat offset from the main image. This happens many times, and the field becomes full of speckles. Since the optics change shape somewhat over the course of the night, these speckles appear and disappear, and wander throughout the image during one's observations. Since it's an instrumental problem, even space telescopes like Hubble must deal with the speckle problem.
This image (courtesy of Beth Biller) shows the results of trying to find planets without first addressing the speckle problem. These are forty minutes of data from the VLT (we use one of four 8.2 meter telescopes in Chile), with a state-of-the-art adaptive optics system. We simultaneously observe in four wavelength bands (more on that below), and this image is the straight combination of light from these filters, somewhat analagous to observing in the "H" infrared band (1.6 microns, or 0.0000016 meters). The broad halo of the star has been subtracted off, and now the image is dominated by speckles, many of which might be mistaken for a planet. The white bar at the top-right represents a distance of one arcsecond on the sky, which would be the separation between the sun and Jupiter, when our system is viewed from a star 5.2 parsecs (17 light years) away. Most good target star for planet-finding are further away than this, meaning we need to look at the worst part of this image to get our results.
Overcoming speckles requires understanding their nature, as they are functions of time (speckles appear and disappear as optics change shape, so two images taken just a few minutes apart will look quite different), wavelength (the speckle pattern at 1.6 microns will be significantly different than the one at 2.2 microns), and optical path (speckles are produced by imperfections in one's optics, so if you split the light into two beams, each of which go through different filters and mirrors and such, then you again get different speckle patterns). SDI as a technique aims to address all three of these concerns so that we can get past the speckles and detect faint companions at tight separations.
Our main weapon in fighting speckles is their source: they're scattered light from the star. All the speckles will have the same color (that is, the ratio of the amount of blue light to the amount of red light, say) as the host star, but the planet's (or brown dwarf's) color will be very different. Unfortunately, most techniques to try to use this fact to our advantage end up failing because of the three aspects of speckles I listed above. Your typical telescope comes equipped with a filter wheel, so one can roll in different filters between the starlight and the detector. For example, an H-band filter only lets in light between about 1.5 and 1.8 microns (red light, for reference, is at about 0.8 microns). So, one might consider observing with an H-filter (1.6 microns), moving the filter wheel to J (1.2), repeating the observation, and so on, then looking for "speckles" with different colors than the rest, as these will be real objects. Unfortunately, you run into all three difficulties with this method. You take your observations at different times, so the speckle pattern has had time to shift. You're observing at very different colors, so your speckle pattern changes. And while your optical path is similar, each filter is a different optical element, giving rise to different speckle patterns.
There are typical tricks astronomers use to get past some of these issues, but it's very difficult to bypass all three. For example, one might use closely-spaced narrow-band filters. Instead of a broad-band filter like J (which lets in light between 1.1 microns and 1.4 microns), one might use a filter tuned to the Pauchen gamma line of hydrogen, which only transmits light between 1.087 and 1.102 microns, then find other narrowband filters at nearby wavelengths. Thus, the colors are close enough to get the same speckle pattern, while still far enough apart to be able to tell the difference between planets and speckles (starlight). One tries to adjust for temporal variation by beamsplitters, which are filters at an angle. These pass one range of wavelengths on to a detector, while reflecting the rest elsewhere in one's instrument. So one can observe at different colors at the same time, and so take care of the temporal problem of speckles. But again, we're now putting the light through different sets of optics, and so we end up with different speckle patterns at the end.
Above right (image from Markus Hartung and Laird Close, see A new VLT surface map of Titan at 1.575 microns) is Saturn's moon Titan, as observed through our VLT device in February 2004. The image is a single frame, which contains four images of Titan (the bottom two are barely visible, since they are in a methane absorption feature, and the methane in Titan's atmosphere absorbs most of this light), at three different colors. These have been combined to make the central, three-color image of Titan's surface.
This image (from Laird Close and Beth Biller) shows close-ups of a star in all three beams. The field is crowded with speckles, but if you look closely, you can see that these speckles patterns are nearly identical in all four images. This is exactly what we want: three-color images of every speckle, so that we can tell the difference between the speckles and real planets.
So, the remaining question is where do we pick our closely-spaced, narrow-band filters in order to most easily spot the difference between planets and speckles? The answer is to exploit physics: carbon and hydrogen are very prevelent in nearby stars (and so companions to these stars), and so at low temperatures (as in planets and brown dwarfs) we expect to see methane in the atmosphere. The methane molecule (four hydrogens and a carbon) absorbs most light redward of about 1.62 microns, so cool objects like planets and low-mass brown dwarfs would be much fainter at 1.63 microns than they are at 1.60 microns. Stars, whose atmospheres are too hot for the methane molecule to form, will be about the same brightness across this range.
The above image shows the theoretical spectrum of a young planet (from David Sudarsky). This is the light the planet is emitting on its own, with light squeezing out through absorption bands across the infrared due to many different types of molecules. The red, green, and blue curves show the locations of our SDI filters, and demonstrate how we can subtract an on-methane band (like F3) from an off-methane band (like F2), removing all the speckles (the star, and so also the speckles, should have a flat spectrum across all three filters), while leaving behind the planet (crudely, much_flux_in_F2 - less_flux_in_f3 = leftover_flux).
Now, if we make use of this clever subtraction (or, more specificly, we use Beth's very intricate data-reduction routines), we can greatly improve upon the image at the top of the page (again, this is just adding the four images together). The result is shown above (image from Beth Biller and Laird Close). Traces of the speckle pattern remain, but the speckles are much weaker than they were before. In fact, we can now see fake planets that were added into the data at an early stage. These objects are almost 50,000 times fainter than the primary star, just fractions of an arcsecond away, yet are clearly visible in these subtracted images. We use one final trick, where we rotate the telescope on the sky. A real planet would then rotate on the image, whereas a speckle (which is produced by the instruments) is unaffected by telescope rotation, and so remains fixed on the image.