[Blindmath] Basic mathematical concept question

Sat Mar 24 21:17:42 UTC 2012

I personally don't think it's the mechanics that gives blind math
students trouble. It's rather the intuition concerning a topic or
concept as it's usually conveyed in a highly visual manner.

Your example, for instance, is a linear transformation of a matrix of
pixels, in which you take differences between successive columns. The
intuition behind such operations is expressed as vectors in R^n. The
actual "picture" of the R^3 vector space, as an example isn't ever
really mentioned in the kind of explicit detail that I feel new blind
students need. Having the very tangible notion that you're "smoothing"
out a particular axis/dimension of the vectors is more
powerful/long-lasting than say taking differences.

On 3/24/12, Richard Baldwin <baldwin at dickbaldwin.com> wrote:
> Now for the practical.
>
> Many real-life "curves" cannot be analytically defined in precise
> mathematical terms without using an infinite series. Even for those that
> can be analytically defined, you might find this information useful.
>
> Consider, for example, the set of values that define the red color values
> in a single row of pixels in a digital image. It is highly unlikely that
> you can, with any reasonable number of terms, fit a mathematical expression
> to to that shape. However, using a computer, you can approximate the first
> derivative of that shape by successively subtracting the red value of one
> pixel from the red value of the next pixel from the left end of the row to
> the right end of the row and saving those difference values along the way.
>
> If you plot those values, you will be plotting an estimate of the first
> derivative of the curve.
>
> Viewed from a completely different viewpoint, you will have applied a
> high-pass frequency filter to the curve. This filter will suppress the low
> frequency values and accentuates the high frequency values. In general, you
> can expect the new function to appear to be more "ragged" than the first.
>
> If the set of values to which the process is applied happen to be samples
> of audio data, and you listen to the "before" and "after" versions, you
> might be able to audibly detect the change in the frequency spectrum.
>
> Perform that process for all three color values for every row of pixels in
> the image, normalize the results to guarantee that all of the new values
> fall in the range from 0 to 255 inclusive, display those new values as an
> image, and you will have produced a new image in which the features tend to
> be sharper than in the original image on a horizontal basis. Since you
> didn't do the same thing vertically, you won't experience the increased
> sharpness along the vertical axis of the image.
>
> Math is a wonderful thing. I continue to be amazed as to how mathematical
> concepts in one area tie into mathematical concepts in an apparently
> different area if you keep an open mind for such connections.
>
> Now back to the practical. When I took differential calculus about 100
> years ago, there was no such thing as a personal computer and barely such a
> thing as a mainframe computer. However, with the availability of personal
> computers, you now have the opportunity to estimate and graph the
> derivative of a given function and to compare that graph with a graph of
> the derivative function that you produce analytically. If they don't match
> -- it's back to the analytic drawing board.
>
> Here is the procedure:
>
> 1. Evaluate the original function at a set of equally and closely spaced
> points and save those values as a sequence of sampled data values.
>
> 2. Perform the difference operation that I described above on those sampled
> data values, save, and graph those results. (Be aware that you cannot
> perform the difference operation on the last data value.)
>
> 3. Graph your analytically derived differential function and compare the
> two.
>
> Hope this helps.
> Dick Baldwin
>
> On Sat, Mar 24, 2012 at 2:30 PM, David Tseng <davidct1209 at gmail.com> wrote:
>
>> With regard to accessible materials I would start with seeking a
>> non-lossy medium for your textbook. When I went through my
>> undergraduate work in mathematics, I employed a variety of techniques
>> to accomplish this. If you can find it, audio recordings of Calculus
>> books (Stewart comes to mind) are available in the U.S. Despite the
>> very sequential (tape) nature of this method, humans reading the book
>> render the text precisely and insert appropriate descriptions for
>> figures/graphs when appropriate. This becomes invaluable as you move
>> through Calculus in its differential, integral, and multi variable
>> variants.
>>
>> There are also now Daisy equivalents of the above. So, you can
>> download an audio book that has chapter/section indecies.
>>
>> Alternatively, you can try to have relevant materials hand-annotated
>> by whatever resources your college may offer.
>>
>> Finally, wikipedia, though it reads more like a reference than say a
>> tutorial, is quite accessible. Images have alt tags that expose the
>> raw LaTeX of the expression under discussion.
>>
>> Hth,
>> David
>>
>> On 3/24/12, Andrew Stacey <andrew.stacey at math.ntnu.no> wrote:
>> > That was a nearly perfect explanation.
>> >
>> > Just one small technicality.  The properties that Richard refers to as
>> > "continuous" and "discontinuous" are actually "differentiable" and
>> > "non-differentiable".  The rough-and-ready description of a function
>> being
>> > continuous is that its graph can be drawn without taking the pencil off
>> the
>> > paper.  The actual definition is a little more complicated, but that's
>> the
>> > basic idea.  A continuous function has no jumps, no gaps.  It is
>> > a differentiable function that has a well-defined slope everywhere.
>> >
>> > A differentiable function is automatically continuous, but not vice
>> versa.
>> >
>> > Some examples:
>> >
>> > 1. A discontinuous function: define f(t) = 0 if t < 0 and f(t) = 1
>> > otherwise.
>> > This has a jump at 0 so is discontinuous.
>> >
>> > 2. A continuous function that isn't differentiable: f(t) = 0 if t < 0
>> > and
>> > f(t)
>> > = t otherwise.  This is continuous, but it has no well-defined slope at
>> > 0 since as we approach 0 from the left the slope appears to be 0 (flat)
>> > whereas if we approach it from the right then the slope appears to be 1.
>> >
>> > Most functions that you encounter are differentiable, in fact probably
>> even
>> > better than that.  But properly defined, most functions that exist are
>> not
>> > differentiable anywhere.  It's just that since differentiable functions
>> are
>> > so
>> > easy to study (compared to arbitrary functions), we use them if at all
>> > possible.
>> >
>> > Andrew Stacey
>> >
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>
>
>
> --
> Richard G. Baldwin (Dick Baldwin)
> Home of Baldwin's on-line Java Tutorials
> http://www.DickBaldwin.com
>
> Professor of Computer Information Technology
> Austin Community College
> (512) 223-4758
> mailto:Baldwin at DickBaldwin.com
> http://www.austincc.edu/baldwin/
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