This is actually a deceptively good question because, as we all know, estimates of variance are extremely sensitive to sampling frequency, sampling intervals, and lags. This is because not all stock prices perfectly adhere to Brownian Motion (i.e., the variance doesn't adhere strongly to the root time rule). It is also not entirely clear from the paper how exactly intraday volatilty is being measured. Therefore, real-world interpretations and implementations could vary significantly.
Taking your excerpt and comparing to the following expert, from page 12, suggests that Almgren intends intraday volatilty to mean one which is sampled from intraday data across 15-minute intervals:
Ten-day average intraday volume profile (upper) and volatilty profile
(lower), on 15-minute intervals.
I infer this to mean that "intraday volatility" for the puprose of this paper means something which resembles the following:
${P_t} = $ average logarithmic price sampled over each $t$ minute interval
$\mu_{(P,t)} = $ average logarithmic price of each $t$ minute interval sampled from $t$ to $T$.
$$\mathbb{E}[\sigma^2_{intraday}] \approx \frac{ \sum_{t=0}^{15*24*60} \big( ({P_{15t}}-{P_{15t-15}}) -({\mu_{P,15t }-\mu_{P,15t -15}}) \big)^2}{T}$$
This verbose implementation is supported by Almgren's footnotes on high frequency volatility estimation: http://cims.nyu.edu/~almgren/timeseries/notes7.pdf.
Almgren also references simplified implementations using the Garman-Klass (GK) and Yhang-Zhang (YZ) estimators.
In practice, I implement the Almgren impact model using the YZ OHLC estimator. For one, it has been shown to be more efficient than GK. In addition, it's a great example of the Pareto 80/20 rule: YZ gets about 80% of the benefit of intraday data but it requires roughly 20% of the effort (i.e., 5x less work) to implement -- plus OHLC data is low-low-cost.
A precis on the YZ estimator is found here: Understanding Yang-Zhang Volatility Estimator.
