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1 . Similarity and Invariance; Searching for Similar
Images
36-350: Data Mining
2 September 2009
Reading: Section 14.5 in the textbook.
So far, we have seen how to search and categorize texts by representing them
as feature vectors — bag-of-word vectors, appropriately scaled and weighted.
Except for page-rank, all of the procedures we have used have depended only on
the feature vectors, not on the underlying text. (Page-rank doesn’t even depend
on the text, just on the links.) We can use the same techniques on any other kind
of data, if we can just define numerical features for them — and as long as we are
content to just use numerical features. The nature of the underlying objects, and
even the method of representation, play no role in things like similarity searching
or multi-dimensional scaling. They don’t care where vectors come from, or what
they actually represent, just that they’re vectors. Using feature vectors rather
than raw data is a kind of abstraction, in the sense in which we use that term
in math or computer programming, and one of the advantages of abstraction is,
exactly, facilitating the re-use of methods and procedures. Whether recycling
our techniques from text to some other domain will work well depends largely
on how well we chose our features, i.e., the mode of representation we use to
hide data and produce abstract vectors. As an example, let’s look at images.
1 Image Representation
At the lowest level, computers represent images as matrices of pixels; the matrix
entries give the color of each pixel.1 So let’s talk about colors.
1.1 Colors
Normal human eyes contain three different sorts of color-sensitive cells, each
tuned to a different part of the visible spectrum. These give us the three primary
1 Actually, most graphics formats, like jpeg, use other representations, which are more
compact for typical images. All formats get translated into pixel matrices when they’re
displayed, however, so we’ll ignore this complication.
1
2 .colors, red, green and blue, and others colors are combinations of them.2 The
RGB encoding of a color is a three-dimensional vector indicating how much of
red, green and blue needs to be blended to make the color, from 0 to 1 on each
axis. Black is (0, 0, 0) and white is (1, 1, 1); all the other colors fall somewhere
in the cube. There are other representations of color space, but RGB is by far
the most common, and we’ll stick with it.3
Many physically distinct colors, from different points in the RGB cube, are
psychologically indistinguishable4 , or differ only trivially. Rather than use the
full color space available to the computer, it is often desirable to quantize color
down to a smaller number of values, effectively merging similar colors (so that
we just have one “red”, and not millions of shades of red). Geometrically, this
means partitioning the RGB cube, cutting it into cells, and treating all colors
in a cell as equivalent, calling them with a single name. (This corresponds
to stemming for words, or, if the cells are very big and coarse, going beyond
stemming to lumping together words with similar meanings, like “steel” and
“iron”.)
1.2 Back to Images
Just as with documents, we would now like to do similarity searching for images.
The user should be able to give the system an initial image and tell it “find me
more which look like this”.
The most straightforward approach would use the computer’s representation
of the image directly. If it’s a color image of M × N pixels, we have a list of
3M N numbers; call this our feature vector, and calculate Euclidean distances as
before. This can actually work OK for some purposes, like undoing the effects of
some kinds of camera distortions, but it’s not very good at finding meaningfully
similar images. For instance, in Figure 1, flower2 is a close-up of the middle
flower from flower1, but their Euclidean distance will be very large. (Why?)
Since the bag-of-words representation served us well with documents, we
might try the same thing here; it’s called the bag of colors. We first chose
a quantization of the color space, and then, for each quantized color, count
the number of pixels of that color in the image (possibly zero). That is, our
procedure is as follows:
1. The user gives us an image Q. Convert it into a vector of color counts.
2. For each image in the collection, measure the distance to Q.
3. Return the k images closest to Q.
2 Red-green color-blind people effectively have only two primary colors, because of muta-
tions changing the tuning of their color-sensitive cells. Some other species of animals have
more than three primary colors.
3 Color-encoding schemes for printing are different, because the physics of blending inks
and dyes is different from that of blending light.
4 Indistinguishable by whom? On average, women tend to be better at color discrimination
than men, and there is some evidence that this is due to hormonal differences, but the overlap
is large, and most people can get better with practice.
2
3 .Figure 1: The images flower1 (left) and flower2 (right) from our data set.
flower2 is a close-up photo of the middle flower from flower1.
Today’s data are photographs of flowers, photographs of tigers, and pho-
tographs of the ocean. Table 1 is part of the color-count representation of the
data.
Just as with documents, we found it useful to emphasize rare words, it can
be helpful to emphasize rare colors, by using the inverse picture frequency,
IPF:
N
IP F (c) = log
nc
Colors which occur frequently, like a black border, will have a small IP F (c).
(Notice that both flower1 and flower2 have such a border.)
Taking a set of 30 pictures, equally divided into scenes of the ocean, of
flowers and of tigers, and ask how often a picture’s nearest neighbor is of the
same type, we get the results in Table 2 depending on how we normalize the
bags-of-colors (cf. Figure 2).
As a sheer classification task, this example is a bit trivial. Our ancestors
spent the last forty-odd million years getting very good at sorting flowers from
big cats at a glance.5 On the other hand, we do not have the same evolutionary
priming for other kinds of image classification, like looking at a microscope slide
and telling whether the cells on it are normal or cancerous...
2 Invariance
When trying to decide whether or not a certain representation is going to be
useful, it’s often helpful to think about the invariances of the representation.
That is, what sorts of changes could we make to the object, without changing
its representation? (The representation is invariant under those changes.)
Invariances tell us what the representation ignores, what concrete aspects of
the object it abstracts away from. Abstraction as such is neither good nor bad;
particular abstractions are more or less useful for particular tasks.
5 More precisely, those of our relatives who could not do this are tragically under-represented
among our ancestors.
3
4 . violetred4 goldenrod lightskyblue4 gray58.2
flower1 59 0 0 0
flower2 128 0 0 0
flower3 166 75 0 0
tiger1 0 4 457 85
tiger2 0 0 0 2
tiger3 0 0 0 115
ocean1 0 0 4326 433
ocean2 0 0 2761 142
ocean3 0 0 1179 8596
Table 1: Part of the bag-of-colors vector for the data.
Multidimensional scaling
flower ocean tiger
ocean5 ocean6
1.0
ocean1
flower4 flower9 ocean4 ocean3
0.5
ocean7
flower7
flower3 tiger6 ocean2
V2
0.0
flower2
flower6 tiger4
flower8
tiger2
tiger1
−0.5
flower1
tiger7
tiger8
tiger9
flower5
tiger5
−1.0
tiger3
−1.0 −0.5 0.0 0.5 1.0
V1
Figure 2: Multi-dimensional scaling of the flower/tiger/ocean images, using
bag-of-color vectors, Euclidean distance with length normalization, and inverse-
picture-frequency weighting.
4
5 . Normalization Equal weight IPF weight
None 8 4
Picture size 8 0
Euclidean length 7 0
Table 2: How often an image’s nearest neighbor belongs to a different class,
measuring distance between bag-of-color vectors with different normalizations
and weightings of colors. There are 30 images total.
The bag-of-words representation is invariant to changes in punctuation, word
order, and general grammar. If it is normalized, it is also invariant to changes
in document length. If inverse-document-frequency weighting is used, it is in-
variant under changes in the exact counts of the most common words (“the”,
“of”, “and”, etc.) which appear in any text written in English. Stemming, and
grouping together synonyms, would add extra invariances. These invariances
make the representations good at finding documents with similar topics, even
though they might be written by different people. (By the same token, these
invariances make it hard to, say, find all the stories by a given author.)
What invariances do we get for images in the bag-of-colors representation?
• Position of objects, pose, small changes in camera angle
• Small amounts of zooming
• Small amounts of unusual colors
• Differences in texture: we can scramble the pixels in any region, even the
whole image, without effect. (Plaids and stripes seem the same.)
Some other invariances we might like, but don’t get with color counts:
• Lighting and shadows; time of day
• Occlusion (one object covers another), placement in 3D
• Blurring, changes in focus
If there is an invariance we don’t want, we can “break” that invariance by
adding additional features. There are various schemes for representing textures,
so that plaids, stripes, spots, “snow”, solid colors, etc. can be distinguished,
and many systems for content-based image searching combine color-counting
with textures.
It’s harder to add an invariance than to break one; generally you need to go
back to the beginning and re-design your representation. The sorts of invariances
we I just said we’d like to have, but don’t, with this representation are all very
hard to achieve; roughly 40% of the human cortex is devoted to visual processing
for a very good reason.
5
6 . You should think about what invariances the following representations have,
and how they compare to the invariances of the bag-of-colors representation.
The first one is much smaller than the bag of colors, and the second one is
larger.
1. Represent an image by its average pixel color.
2. Represent an image by dividing it into four quarters, and doing color-
counts in each quarter. (Is there any reason why four equal parts should
be special?)
3 Practice
Content-based image retrieval systems are used commercially. All the ones I
know of include the bag-of-colors representation, but generally also a number
of other features. For example, you should go to http://labs.systemone.at/
retrievr/ and think about what features they are using, and how well they
work. (Cf. Figure 3.)
Google’s image search is very popular, but it’s a total cheat, and doesn’t
analyze the images at all. Instead, it searches on the phrases you give it, takes
the pages which get a high rank, and assumes that images which appear on
those pages close you your query phrase are probably pictures of what you are
looking for. This works well for a lot of things, but can give curious results
when there aren’t, actually, many pictures of what you are looking for.
6
7 .Figure 3: Example of similarity-search for images in action, using http://labs.
systemone.at/. The query image, on the left, is an old snapshot of my cat.
The returned images were obtained from flickr.com. What features are being
used here? Are they working?
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